Pentaquark baryon in anisotropic lattice QCD

The pentaquark (5Q) baryon is studied in anisotropic quenched lattice QCD with renormalized anisotropy $a_s/a_t=4$ for a high-precision mass measurement. The standard Wilson action at $\beta =5.75$ and the $O(a)$ improved Wilson quark action with $\kappa =0.1210(0.0010)0.1240$ are employed on a ${12}^3\times 96$ lattice. Contribution of excited states is suppressed by using a smeared source. We investigate both the positive- and negative-parity 5Q baryons with $I=0$ and spin $J=1/2$ using a non-NK-type interpolating field. After chiral extrapolation, the lowest positive-parity state is found to have a mass, $m_{5\mathrm{Q}}=2.25\mathrm{G}\mathrm{e}\mathrm{V}$, which is much heavier than the experimentally observed ${\Theta }^{+}(1540)$. The lowest negative-parity 5Q state appears at $m_{5\mathrm{Q}}=1.75\mathrm{G}\mathrm{e}\mathrm{V}$, which is near the s-wave NK threshold. To distinguish spatially-localized 5Q resonances from NK-scattering states, we propose a new general method imposing a “Hybrid Boundary Condition (HBC),” where the NK threshold is artificially raised without affecting compact five-quark states. The study using the HBC method shows that the negative-parity state observed on the lattice is not a compact 5Q state but an s-wave NK-scattering state.

Figure 1

The effective mass plot of positive-parity pentaquark (5Q). Equation (10) is adopted as a typical set of hopping parameters. The statistical error is obtained with jackknife error estimate. The solid line denotes the result of the single-exponential fit performed over the interval $25\le t\le 35$. The dotted line denotes the p-wave NK threshold energy corresponding to the spatial lattice size $L\simeq 2.15\mathrm{f}\mathrm{m}$.

Figure 2

The effective mass plot for negative-parity pentaquark (5Q) for the typical set of hopping parameters (Eq. (10)). The solid line denotes the result of the single-exponential fit performed over the interval $25\le t\le 35$. The dotted line denotes the s-wave NK threshold.

Figure 3

The masses of the pentaquark states $m_{5\mathrm{Q}}$ for both parities plotted against $m_{\pi }^2$. The triangles denote positive parity, while the circles denote negative parity. The open symbols denote the direct lattice QCD data, whereas the closed symbols denote the results of the chiral extrapolation. The dotted lines indicate the NK thresholds for p-wave (upper) and s-wave (lower) cases.

Figure 4

The effective mass plots of $\Sigma$($uds$) baryon under the standard BC, and under the HBC. Equation (10) is adopted as a typical set of the hopping parameters. The solid lines denote the results of the single-exponential fit performed in the region $20\le t\le 30$. No significant difference is observed between the two results.

Figure 5

The effective mass plot for the negative-parity pentaquark (5Q) states under the HBC. Equation (10) is adopted as a typical set of hopping parameters. The solid line denotes the result of the single-exponential fit performed within the interval $25\le t\le 35$. The dotted line denotes the s-wave NK threshold energy modified due to the HBC. This figure should be compared with Fig. 2.

Figure 6

Comparison of results of standard BC (left-hand side) and HBC (right-hand side) for each hopping parameter $\kappa$. Closed circles denote results obtained from the best-fit analysis. The solid lines denote the corresponding NK threshold.