Lattice QCD determination of patterns of excited baryon states

Energies for excited isospin $I=\frac{1}{2}$ and $I=\frac{3}{2}$ states that include the nucleon and $\Delta$ families of baryons are computed using quenched, anisotropic lattices. Baryon interpolating field operators that are used include nonlocal operators that provide $G_2$ irreducible representations of the octahedral group. The decomposition of spin $\frac{5}{2}$ or higher spin states is realized for the first time in a lattice QCD calculation. We observe patterns of degenerate energies in the irreducible representations of the octahedral group that correspond to the subduction of the continuum spin $\frac{5}{2}$ or higher. The overall pattern of low-lying excited states corresponds well to the pattern of physical states subduced to the irreducible representations of the octahedral group.

Figure 1

Pion dispersion relation based on tuned parameters $m_0$ and ${\nu }_s$. The renormalized anisotropy is 3.0 and the lattice volume is ${16}^3\times 64$ with $\beta =6.1$. Lines show the $c=1$ dispersion relation. The meaning of symbols is given in Table .

Figure 2

Energies in lattice units versus dimension of the matrices of correlation functions. Data for the ${24}^3\times 64$ lattice are used. Upper panels show energies for positive parity states and lower panels show energies for negative parity states. The $I=\frac{3}{2}$, $G_{2g/u}$ states are excluded because correlation functions are based on a single operator.

Figure 3

Eigenvector components $v_k^{(0)}(t,0)$ of the ground state for (a) the $I=\frac{1}{2}$, $G_{1g}$ channel using a $10\times 10$ matrix and (b) the $I=\frac{1}{2}$, $H_u$ channel using an $8\times 8$ matrix, both on the ${24}^3\times 64$ lattice. The vertical axis is the value of the coefficient of a basis operator and the horizontal axis is time. Each basis operator corresponds to a symbol in the plot; the four most important operators are listed below the plot together with their symbols. ${\overline{N}}_{S,S_z}^{\lambda k}$ is our notation for a smeared three-quark operator with spin $S$, spin projection $S_z$, irrep $\Lambda$, and embedding $k$. The symmetry (MA, MS, or S) of Dirac indices for each three-quark operator is written next to the operator. Explicit forms of the operators ${\overline{N}}_{S,S_z}^{\lambda k}$ are provided in Ref. 17.

Figure 4

The energies obtained for each symmetry channel of $I=\frac{1}{2}$ baryons are shown based on the ${24}^3\times 64$ lattice data. The scale on the left side shows energies in lattice units and the scale on the right side shows energies in GeV. The scale was set using the string tension. Errors are indicated by the vertical size of the box.

Figure 5

The lowest energies obtained for each symmetry channel of $I=\frac{1}{2}$ and $I=\frac{3}{2}$ baryons are shown in columns 2 and 4 and the experimental levels are shown in columns 1 and 3. The ${24}^3\times 64$ lattice data are shown and the left two columns show the energies of positive-parity states and the right two columns show the energies of negative-parity states. The vertical axis shows energy in MeV.