Spin $3/2$ pentaquarks in anisotropic lattice QCD

High-precision mass measurements of a pentaquark (5Q) ${\Theta }^{+}$ in the $J^P=3/2^{\pm }$ channels are performed in anisotropic quenched lattice QCD. A large number of gauge configurations ($N_{\mathrm{conf}}=1000$) are prepared for the standard Wilson gauge action at $\beta =5.75$ and the $O(a)$ improved Wilson (clover) quark action is employed for $\kappa =0.1210(0.0010)0.1240$ on a ${12}^3\times 96$ lattice with the renormalized anisotropy as $a_s/a_t=4$. The Rarita-Schwinger formalism is adopted for interpolating fields. We examine several interpolating fields with isospin $I=0$, such as (a) the $N{K}^{*}$-type, (b) the color-twisted $N{K}^{*}$-type, and (c) the diquark-type operators. After chiral extrapolation, we obtain massive states, $m_{5\mathrm{Q}}\simeq 2.1–2.2\mathrm{GeV}$ in $J^P=3/2^{-}$, and $m_{5\mathrm{Q}}=2.4–2.6\mathrm{GeV}$ in $J^P=3/2^{+}$. Analyses using the hybrid boundary condition method are performed to determine whether these states are compact 5Q resonances or two-hadron scattering states. No compact 5Q resonance state is found below 2.1 GeV.

Figure 1

The 5Q effective mass plots in the $J^P=3/2^{-}$ channel in the standard PBC for three types of interpolating fields, i.e., (a) the $N{K}^{*}$ type, (b) the twisted $N{K}^{*}$ type, and (c) the diquark type. Equation (31) is adopted as a typical set of hopping parameters. The statistical error is estimated with the jackknife prescription. The dotted lines indicate the $s$-wave $N{K}^{*}$ and the $d$-wave $NK$ threshold in the spatial lattice size $L\simeq 2.15\mathrm{fm}$. Note that they accidentally coincide with each other. The solid lines denote the results of the single-exponential fit performed in each plateau region.

Figure 2

$m_{5\mathrm{Q}}$ in the $J^P=3/2^{-}$ channel against $m_{\pi }^2$ for the two interpolating fields, i.e., (circle) the $N{K}^{*}$ type, (square) the twisted $N{K}^{*}$ type. Open symbols denote the direct lattice QCD data, while the closed symbols and the solid lines represent the results of the chiral extrapolations to the physical quark mass region.

Figure 3

The 5Q effective mass plots in the $J^P=3/2^{+}$ channel with PBC for three types of interpolating fields, i.e., (a) the $N{K}^{*}$ type, (b) the twisted $N{K}^{*}$ type, and (c) the diquark type. Equation (31) is adopted as a typical set of hopping parameters. The dotted lines indicate the $s$-wave $N^{*}{K}^{*}$, the $p$-wave $N{K}^{*}$, and the $p$-wave $NK$ thresholds in the spatial lattice size $L\simeq 2.15\mathrm{fm}$. The solid lines denote the results of the single-exponential fit performed in each plateau region.

Figure 4

$m_{5\mathrm{Q}}$ in the $J^P=3/2^{+}$ channel against $m_{\pi }^2$ for the three interpolating fields, i.e., (circle) the $N{K}^{*}$ type, (square) the twisted $N{K}^{*}$ type, and (triangle) a diquark type [Eq. (3)]. Open symbols denote the direct lattice QCD data, while the closed symbols and the solid lines represent the results of the chiral extrapolations to the physical quark mass region.

Figure 5

The 5Q effective mass plots in the $J^P=3/2^{-}$ channel with HBC for three types of interpolating fields, i.e., (a) the $N{K}^{*}$ type, (b) the twisted $N{K}^{*}$ type, and (c) the diquark type. The meanings of the dotted lines and the solid lines are the same as in Fig. 1.

Figure 6

5Q effective mass plots in the $J^P=3/2^{+}$ channel with HBC for three types of interpolating fields: (a) the $N{K}^{*}$ type, (b) the twisted $N{K}^{*}$ type, and (c) the diquark type. The meanings of the dotted lines and the solid lines are the same as in Fig. 3.