Lattice study of pentaquark states

We present a study of the pentaquark system in quenched lattice QCD using diquark-diquark and kaon-nucleon local and smeared interpolating fields. We examine the volume dependence of the spectral weights of local correlators on lattices of size ${16}^3\times 32$, ${24}^3\times 32$, and ${32}^3\times 64$ at $\beta =6.0$. We find that a reliable evaluation of the volume dependence of the spectral weights requires accurate determination of the correlators at large time separations. Our main result from the spectral weight analysis in the pentaquark system is that within our variational basis and statistics we cannot exclude a pentaquark resonance. However, our data also do not allow a clear identification of a pentaquark state since only the spectral weights of the lowest state can be determined to sufficient accuracy to test for volume dependence. In the negative parity channel, the mass extracted for this state is very close to the kaon-nucleon threshold, whereas in the positive parity channel it is about 60% above.

Figure 1

The upper graph shows the effective mass for the nucleon (positive parity) and the lower graph for the $N^{*}$ (negative parity) at ${\kappa }_l=0.153$. The crosses show the results using the local-local correlator, $C(t)$, and Dirichlet b.c. and the open triangles using the local-smeared correlator, $\tilde{C}(t)$ and antiperiodic b.c. shifted to the right for clarity. In the case of the nucleon, we also show results obtained using the smeared-smeared correlator $\widehat{C}(t)$ (asterisks) shifted to the left for clarity. The dotted and dashed lines are fits to the effective masses obtained using $C(t)$ and $\tilde{C}(t)$, respectively, assuming two state dominance.

Figure 2

We show the ratio $C_{16}(t)/C_{24}(t)$ (crosses) and the ratio $R_{16:24}^H$ (open triangles with a small shift to the right for clarity) as a function of the time separation in lattice units. The upper graph shows results for the nucleon and the lower for $N^{*}$ at ${\kappa }_l=0.153$.

Figure 3

The effective mass at ${\kappa }_l=0.153$ as a function of $t/a$ for the lattice of size ${16}^3\times 32$ (top), ${24}^3\times 32$ (middle), and ${32}^3\times 64$ (bottom). The set of results with the lower value are obtained using one pion interpolating fields, ${\mathcal{J}}_1^{\pi }$ (crosses), ${\mathcal{J}}_2^{\pi }$ (asterisks), and ${\mathcal{J}}_3^{\pi }$ (open triangles). The set with the higher value corresponds to the two-pion interpolating fields, ${\mathcal{J}}_1^{2\pi }$ (crosses), ${\mathcal{J}}_2^{2\pi }$ (asterisks), ${\mathcal{J}}_4^{2\pi }$ (open triangles), and ${\mathcal{J}}_5^{2\pi }$ (circles). The dotted lines are the two lowest two-pion scattering states $E_{2\pi }^0$ and $E_{2\pi }^1$.

Figure 4

The effective mass at ${\kappa }_l=0.153$ extracted from the correlator using the interpolating field ${\mathcal{J}}_1^{2\pi }$ (crosses) and from the two largest eigenvalues ${\Lambda }_0$ (open triangles) and ${\Lambda }_1$ (asterisks) for the lattice of size ${16}^3\times 32$ (upper graph), ${24}^3\times 32$ (middle graph), and ${32}^3\times 64$ (lower graph). The dotted lines are the two lowest energy two-pion scattering states $E_{2\pi }^0$ and $E_{2\pi }^1$. The dashed line is the lowest energy two-rho scattering state $E_{2\rho }^0$.

Figure 5

The mass at ${\kappa }_l=0.153$ extracted from fitting the plateau of the effective mass of the lowest energy eigenvalue (filled triangles) and the second higher energy eigenvalue (filled circles shifted to right for clarity) as a function of the spatial length of the lattice $L$. Also shown are results for twice the mass extracted from the pion correlator (crosses) and rho correlator (asterisks). The dotted line is the two-pion scattering energy $E_{2\pi }^1$.

Figure 6

The upper graph shows the effective mass for the lowest energy eigenstates and the lower graph the effective mass for the second higher energy eigenstate. In both graphs we show data with projection to zero relative momentum using Eq. (21) (asterisks shifted to the right for clarity) and without explicit projection (open triangles) for the lattice of size ${16}^3\times 32$ at ${\kappa }_l=0.153$. We also show twice the effective mass extracted from a single pion (crosses in upper graph) and rho correlator (circles in lower graph). The dotted line is the two-pion scattering state $E_{2\pi }^1$.

Figure 7

The ratio $R_{16:24}$ at ${\kappa }_l=0.153$ for one and two-pion states using interpolating field ${\mathcal{J}}_1^{\pi }(x)$ (crosses) and ${\mathcal{J}}_1^{2\pi }(x)$ (asterisks) as a function of $t/a$. The ratio $R_{16:24}$ extracted from correlators using the optimal combination $J_{\mathrm{optimal}}(x)$ for the lowest eigenstate of the two-pion system is shown by the open triangles shifted to the right for clarity. The dotted lines show the expected value of the ratio for a single particle state and a two-particle scattering state.

Figure 8

The ratio of spectral weights $w_{16}/w_{24}$ (upper graph) and $w_{16}/w_{32}$ (lower graph) as a function of the lower fit range $t_i/a$ at ${\kappa }_l=0.153$. Asterisks denote results extracted from the correlator with ${\mathcal{J}}_1^{2\pi }(x)$, open triangles and filled circles from the correlator with $J_{\mathrm{optimal}}(x)$ fitted to a single exponential and to a sum of two exponentials, respectively. For comparison we also show these ratios for the single pion state using ${\mathcal{J}}_1^{\pi }(x)$$ (crosses). The upper fit range is fixed to 26 for the ${16}^3\times 32$ lattice and to 56 for the ${32}^3\times 64$ lattice. On the lower graph with the filled triangles we show results for the ratio of spectral weights $w_{16}/w_{32}$ when the upper fit range for the large lattice is fixed to 26. The dotted lines show the expected value of the ratios for a single particle state and a two-particle scattering state.

Figure 9

The ratio of spectral weights $w_{16}/w_{24}$ extracted from the correlators with interpolating field $J_{\mathrm{optimal}}(x)$ (asterisks) for the first excited state as a function of the lower fit range $i_i/a$. The corresponding ratio $w_{16}/w_{32}$ is shown with the open triangles. The dotted lines show the expected value of the ratio for a single particle state and a two-particle scattering state. The crosses show for comparison the ratio of spectral weights for one pion state extracted from the correlator using interpolating field ${\mathcal{J}}_1^{\pi }(x)$.

Figure 10

The effective mass for the pentaquark in the negative parity channel for lattices of size ${16}^3\times 32$ (upper graph), ${24}^3\times 32$ (middle graph), and ${32}^3\times 64$ (lower graph) at ${\kappa }_l=0.153$. The crosses show the results obtained from correlators using ${\mathcal{J}}_{DD}$ and the open triangles using ${\mathcal{J}}_{KN}$. The dotted lines show $m_N+m_K$ and $E_{KN}^1$.

Figure 11

The effective mass for the pentaquark in the positive parity channel at ${\kappa }_l=0.153$. The dotted lines show $E_{KN}^1$ and $E_{KN}^2$ and the dashed line $m_K+m_{N^{*}}$. The rest of the notation is the same as that of Fig. 10.

Figure 12

The upper graph shows the pentaquark effective mass for the negative parity channel and the lower graph for the positive parity channel at ${\kappa }_l=0.153$ on the lattice of size ${32}^3\times 64$. The crosses show results obtained from the local-local correlator $C(t)$, the open triangles results from the local-smeared correlator $\tilde{C}(t)$ shifted to the right, and the asterisks results from the smeared-smeared correlator $\widehat{C}(t)$. The diquark-diquark interpolating field is used in all correlators. The dashed line is a two-exponential fit to the local-smeared results. The dotted line shows $E_{KN}^0$ for the negative and $E_{KN}^1$ for the positive parity channel.

Figure 13

The upper graphs show the effective mass for the negative parity and the lower graphs for the positive parity channels at ${\kappa }_l=0.155$ on the lattice of size ${32}^3\times 64$. The squares on the left two graphs denote the results extracted from the correlators with the local interpolating field ${\mathcal{J}}_{DD}$. The crosses on all four graphs show the effective mass, derived from the eigenvalue ${\Lambda }_0(t)$ of ${\mathcal{C}}_{DD;KN}$. On the graphs on the right, the open triangles denote results derived from ${\Lambda }_1(t)$, whereas the circles and asterisks are results extracted from the eigenvalue ${\tilde{\Lambda }}_0(t)$ and ${\tilde{\Lambda }}_1(t)$ from the diagonalization of ${\tilde{\mathcal{C}}}_{DD;KN}$, respectively. The dotted lines show $E_{KN}^0$ for the negative parity channel and $E_{KN}^1$ for the positive parity channel.

Figure 14

The energy gap between the two lowest energy eigenstates as a function of $t/a$ for the pentaquark in the positive parity channel at ${\kappa }_l=0.153$ (upper graph) and ${\kappa }_l=0.155$ (lower graph) for lattices of size ${16}^3\times 32$ (asterisks), ${24}^3\times 32$ (crosses), and ${32}^3\times 64$ (open triangles). The dashed line shows the plateau value determined on the lattice of size ${16}^3\times 32$.

Figure 15

The upper graph shows the effective mass for the lowest energy eigenstate of the two-pion system as a function of $t/a$ at ${\kappa }_l=0.153$. The middle graph shows the corresponding effective mass for the pentaquark system in the negative parity channel and the lower graph for the pentaquark in the positive parity channel for the lattices of size ${16}^3\times 32$ (asterisks), ${24}^3\times 32$ (crosses), and ${32}^3\times 64$ (open triangles).

Figure 16

The ratio $R_{16:24}$ versus $t/a$ for the negative parity channel for the lowest energy eigenstate at ${\kappa }_l=0.153$. Asterisks show results obtained with correlators using the diquark-diquark interpolating field ${\mathcal{J}}_{DD}$ and open triangles with ${\mathcal{J}}_{\mathrm{optimal}}$. For comparison we also show the ratio for the nucleon (crosses). The dotted lines show the expected value of the ratios for a single particle state and a two-particle scattering state.

Figure 17

We show the spectral weights for the lowest eigenstate at ${\kappa }_l=0.153$ versus the lower fitting range $t_i/a$ keeping the upper range fixed to 26 for the two smaller lattices and to 56 for the large lattice. The upper graph shows results for the two-pion system and the lower graph for the pentaquark in the negative parity channel for the lattices of size ${16}^3\times 32$ (asterisks for a single exponential fit), ${24}^3\times 32$ (crosses for a single exponential fit), and ${32}^3\times 64$ (open triangles for a single exponential fit). In the case of the two-pion system, we also show results for the weights determined from fitting the correlator to a sum of two exponentials for lattices of size ${16}^3\times 32$ (squares), ${24}^3\times 32$ (circles), and ${32}^3\times 64$ (filled triangles). The solid lines in the upper graph show the plateau values.

Figure 18

The mass squared of the kaon (crosses), nucleon (open triangles), pentaquark in the negative parity channel (filled circles), and in the positive parity channel (asterisks) plotted versus the pion mass squared. We use the nucleon mass at the chiral limit to set the scale obtaining $a^{-1}=2.2\mathrm{GeV}$. The dashed lines show the linear extrapolations to the chiral limit. The dotted line shows the KN threshold at the chiral limit.

Figure 19

The mass of the lowest eigenstate of the pentaquark system in the negative parity channel as a function of the pion mass squared. Filled squares show the results of this work, crosses are data from Ref. 12, open triangles from Ref.  13, open circles from Ref. 18, open squares from Ref. 17, and filled circles from Ref. 19. In each case we give the spatial length of the lattice.