Search for the possible $S=+1$ pentaquark states in quenched lattice QCD

We study spin $\frac{1}{2}$ hadronic states in quenched lattice QCD to search for a possible $S=+1$ pentaquark resonance. Simulations are carried out on $8^3\times 24$, ${10}^3\times 24$, ${12}^3\times 24$, and ${16}^3\times 24$ lattices at $\beta =5.7$ at the quenched level with the standard plaquette gauge action and the Wilson quark action. We adopt a Dirichlet boundary condition in the time direction for the quark to circumvent the possible contaminations due to the (anti)periodic boundary condition for the quark field, which are peculiar to the pentaquark. By diagonalizing the $2\times 2$ correlation matrices constructed from two independent operators with the quantum numbers $(I,J)=(0,\frac{1}{2})$, we successfully obtain the energies of the lowest state and the 2nd-lowest state in this channel. The analysis of the volume dependence of the energies and spectral weight factors indicates that a resonance state is likely to exist slightly above the nucleon-Kaon ($NK$) threshold in $(I,J^P)=(0,{\frac{1}{2}}^{-})$ channel.

Figure 1

Schematic figure for the explanation on the possible contaminations of the particles propagating over the temporal boundary. $H(t)$ and $\Theta (t)$ are the interpolating operators and their arguments are the distances from the source point. $H(t)$ is a generic hadronic operator, which creates and annihilates the particles which cannot decay in quenched QCD. The wave lines represent the propagations of states. Resonance states like ${\Theta }^{+}$ are represented by “5Q” in the figure, as well as $NK$ scattering states. The five-quark state can dissociate into forward-propagating nucleon (Kaon) and backward-propagating Kaon (nucleon).

Figure 2

The “effective mass” plot $E_i(T)$ as the function of $T$, the separation between source operators and sink operators, in $(I,J^P)=(0,1/2^{-})$ channel with the hopping parameters $({\kappa }_{u,d},{\kappa }_s)=(0.1600,0.1600)$ on $8^3\times 24$, ${10}^3\times 24$, ${12}^3\times 24$, ${16}^3\times 24$ lattice at $\beta =5.7$. The stability of each $E_i(T)$ against $T$ means the smallness of the unwanted higher excited-state contaminations. The solid line and the dashed lines represent the central value and the error of the fitted masses $E_0^{-}$ and $E_1^{-}$.

Figure 3

The “effective mass” plot $E_i(T)$ as the function of $T$, the separation between source operators and sink operators, in $(I,J^P)=(0,1/2^{-})$ channel with the hopping parameters $({\kappa }_{u,d},{\kappa }_s)=(0.1600.0.1650)$ on $8^3\times 24$, ${10}^3\times 24$, ${12}^3\times 24$, ${16}^3\times 24$ lattice at $\beta =5.7$.

Figure 4

The “effective mass” plot $E_i(T)$ as the function of $T$, the separation between source operators and sink operators, in $(I,J^P)=(0,1/2^{-})$ channel with the hopping parameters $({\kappa }_{u,d},{\kappa }_s)=(0.1650,0.1600)$ on $8^3\times 24$, ${10}^3\times 24$, ${12}^3\times 24$, ${16}^3\times 24$ lattice at $\beta =5.7$.

Figure 5

The “effective mass” plot $E_i(T)$ as the function of $T$, the separation between source operators and sink operators, in $(I,J^P)=(0,1/2^{-})$ channel with the hopping parameters $({\kappa }_{u,d},{\kappa }_s)=(0.1625,0.1650)$ on $8^3\times 24$, ${10}^3\times 24$, ${12}^3\times 24$, ${16}^3\times 24$ lattice at $\beta =5.7$.

Figure 6

The “effective mass” plot $E_i(T)$ as the function of $T$, the separation between source operators and sink operators, in $(I,J^P)=(0,1/2^{-})$ channel with the hopping parameters $({\kappa }_{u,d},{\kappa }_s)=(0.1650,0.1650)$ on $8^3\times 24$, ${10}^3\times 24$, ${12}^3\times 24$, ${16}^3\times 24$ lattice at $\beta =5.7$.

Figure 7

The black (gray) filled squares denote the lattice QCD data of the 2nd-lowest state in $(I,J^P)=(0,{\frac{1}{2}}^{-})$ channel extracted with $N_{\mathrm{fit}}-1$ ($N_{\mathrm{fit}}$) data plotted against the lattice extent $L$. The filled circles represent the lattice QCD data $E_0^{-}$ of the lowest state in $(I,J^P)=(0,{\frac{1}{2}}^{-})$ channel. The open symbols are the sum $E_N^{|\overrightarrow{\mathbf{n}}|=1}+E_K^{|\overrightarrow{\mathbf{n}}|=1}$ of energies of nucleon $E_N^{|\overrightarrow{\mathbf{n}}|=1}$ and Kaon $E_K^{|\overrightarrow{\mathbf{n}}|=1}$ with the smallest lattice momentum $|\overrightarrow{\mathbf{p}}|=\frac{2\pi }{L}|\overrightarrow{\mathbf{n}}|=2\pi /L$. The upper line represents $\sqrt{M_N^2+|\mathbf{p}{|}^2}+\sqrt{M_K^2+|\mathbf{p}{|}^2}$ with $|\mathbf{p}|=2\pi /L$ the smallest relative momentum on the lattice. The lower line represents the simple sum $M_N+M_K$ of the masses of nucleon $M_N$ and Kaon $M_K$. We adopt the central values of $M_N$ and $M_K$ obtained on the largest lattice to draw the two lines.

Figure 8

The spectral weight factors defined in Sec. 7 are plotted against the lattice volume $V$. The left figure shows $W_0$ for the lowest state in $(I,J^P)=(0,{\frac{1}{2}}^{-})$ channel and the right figure shows $W_1$ for the 2nd-lowest state. In the case when the weight factor $W_i$ for the $i$th state $|i⟩$ in a point-point correlator shows no volume dependence, $|i⟩$ is likely to be a resonance state. On the contrary, when the $i$th state $|i⟩$ is a two-particle state, $W_i$ scales according to $1/V$.

Figure 9

The lattice QCD data in the $(I,J^P)=(0,{\frac{1}{2}}^{+})$ channel are plotted against the lattice extent $L$. The solid line denotes the simple sum $M_{N^{*}}+M_K$ of the masses of the lowest-state negative-parity nucleon $M_{N^{*}}$ and Kaon $M_K$ obtained with the largest lattice.

Figure 10

The spectral weight factor of the extracted state in $(I,J^P)=(0,{\frac{1}{2}}^{+})$ channel with the hopping parameters $({\kappa }_{u,d},{\kappa }_s)=(0.1600,0.1600)$ is plotted against the lattice volume $V$. We note here that the $1/V$-like volume dependence is not always the characteristic of scattering states when we do not adopt point-point correlators as is the present case. To conclude from this dependence, we need to determine the precise volume dependences of the weight factors in wall-point correlators.

Figure 11

The effective mass plots constructed from $\sum _{\overrightarrow{\mathrm{x}}}⟨{\Theta }^1(\overrightarrow{\mathrm{x}},T+t_{\mathrm{src}}){\overline{{\Theta }^1}}_{\mathrm{wall}}(t_{\mathrm{src}})⟩$, $\sum _{\overrightarrow{\mathrm{x}}}⟨{\Theta }^2(\overrightarrow{\mathrm{x}},T+t_{\mathrm{src}}){\overline{{\Theta }^2}}_{\mathrm{wall}}(t_{\mathrm{src}})⟩$, $\sum _{\overrightarrow{\mathrm{x}}}⟨{\Theta }^1(\overrightarrow{\mathrm{x}},T+t_{\mathrm{src}})\overline{{\Theta }^1}(t_{\mathrm{src}})⟩$, $\sum _{\overrightarrow{\mathrm{x}}}⟨{\Theta }^2(\overrightarrow{\mathrm{x}},T+t_{\mathrm{src}})\overline{{\Theta }^2}(t_{\mathrm{src}})⟩$, and $\sum _{\overrightarrow{\mathrm{x}}}⟨{\Theta }^3(\overrightarrow{\mathrm{x}},T+t_{\mathrm{src}})\overline{{\Theta }^3}(t_{\mathrm{src}})⟩$, in $(I,J^P)=(0,{\frac{1}{2}}^{-})$ channel in ${16}^3\times 24$ lattice at $\beta =5.7$ employing the hopping parameters $({\kappa }_{u,s},{\kappa }_s)=(0.1600,0.1600)$ are plotted, along with the dashed line which denotes the lowest-state energy $E_0^{-}$.

Figure 12

A comparison of the chiral extrapolations in $(I,J^P)=(0,{\frac{1}{2}}^{-})$ channel on ${12}^3\times 24$ (left) and ${16}^3\times 24$ (right) lattices for the $NK$ threshold energy $M_K+M_N$ and the 2nd-lowest state energy $E_1^{-}$. $M_K+M_N$ and $E_1^{-}$ are plotted against $M_{\pi }^2$. The filled circles (triangles) denote the energies of the $NK$ threshold (2nd-lowest state) in the chiral limit. ${\kappa }_s$ is fixed to be ${\kappa }_s\sim 0.1620$ so that the physical Kaon mass is reproduced in the chiral limit.

Figure 13

The left figure shows $\det ([{\mathcal{C}}_{IJ}^{\mathrm{PS}}(T){\mathcal{C}}_{IJ}^{\mathrm{PS}}(T+1{)}^{-1}{]}^T{\mathcal{C}}_{IJ}^{\mathrm{PS}}(T))$ as the function of $T$ on each volume. The middle figure is the plot of $\det ([{\mathcal{C}}_{IJ}^{\mathrm{PW}}(T){\mathcal{C}}_{IJ}^{\mathrm{PW}}(T+1{)}^{-1}{]}^T{\mathcal{C}}_{IJ}^{\mathrm{PW}}(T))$ against $T$. $\mathrm{Det}([{\mathcal{C}}_{IJ}^{\mathrm{PW}}(T){]}^{-1}{\mathcal{C}}_{IJ}^{\mathrm{PS}}(T))$ is plotted in the right figure. $\mathrm{Det}([{\mathcal{C}}_{IJ}^{\mathrm{PS}}(T){\mathcal{C}}_{IJ}^{\mathrm{PS}}(T+1{)}^{-1}{]}^T{\mathcal{C}}_{IJ}^{\mathrm{PS}}(T))$, $\det ([{\mathcal{C}}_{IJ}^{\mathrm{PW}}(T){\mathcal{C}}_{IJ}^{\mathrm{PW}}(T+1{)}^{-1}{]}^T{\mathcal{C}}_{IJ}^{\mathrm{PW}}(T))$, and $\det ([{\mathcal{C}}_{IJ}^{\mathrm{PW}}(T){]}^{-1}{\mathcal{C}}_{IJ}^{\mathrm{PS}}(T))$ show plateaus in the large $T$ region and coincide with $\det (C^{\mathrm{P}†}{C}^{\mathrm{S}})$, $\det (C^{\mathrm{P}†}{C}^{\mathrm{W}})$, and $\det ((C^{\mathrm{W}}{)}^{-1}{C}^{\mathrm{S}})$, respectively.

Figure 14

$\mathrm{Det}(C^{\mathrm{P}†}{C}^{\mathrm{S}})$ is plotted as the function of the lattice volume $V$. The solid line denotes the best-fit curve by $A_1/V$ and the dashed line does the best-fit curve by $A_2/V^2$. The best-fit parameters are $A_1=1.35$ and $A_2=2.17$, and ${\chi }^2/N_{\mathrm{df}}$ is 1.72 and 7.13, respectively. These data behave consistently in accordance with $1/V$.