Figure 1

Principal correlator fits according to Eq. (17). For ten states of the Nucleon $H_u$ irrep for $m_{\pi }=524\mathrm{MeV}$, the plots show ${\lambda }^{\mathrm{n}}(t)\times e^{m_{\mathrm{n}}(t-t_0)}$ data and the fit for $t_0=9$. Data used in the fit are shown in black, while points excluded from the fit are in grey.Reuse & Permissions

Figure 2

The magnitude of matrix elements in a matrix of correlation functions, $C_{ij}/\sqrt{C_{ii}{C}_{jj}}$, at time-slice 5 is shown according to the scale at the lower right. The matrix is for the Nucleon $H_u$ irrep, with 28 [$J=\frac{3}{2}$] operators, 16 [$J=\frac{5}{2}$] operators and 4 [$J=\frac{7}{2}$] operators.Reuse & Permissions

Figure 3

Histograms of spectral overlaps, $Z$, are shown for a selection of eight operators (shown at the top using the naming convention of Eq. (13)). The 4 operators labeled with $J=\frac{3}{2}$ (color red) have subductions only to the $H_u$ irrep. The 3 operators labeled with $J=\frac{5}{2}$ (color green) and the 1 labeled with $J=\frac{7}{2}$ (color blue) have subductions to both $H_u$ and $G_{2u}$ irreps. Each histogram is labeled by the value of mass $m$ of the state (in units of $m_{\Omega }$) and has 8 vertical bars showing the relative $Z$ values for each of the operators. The data are from the $m_{\pi }=524\mathrm{MeV}$ ensemble and $Z$’s are normalized so that the largest value across all states, for a given operator, is equal to 1. The lighter area at the head of each bar represents the one sigma statistical uncertainly. Note that for each state only one or two operators have a large relative $Z$ value, and it is the same operators appearing across each of the irreps. Note also that nearly the same energies are obtained in $H_u$ and $G_{2u}$ irreps for state subduced from one $J$ value.Reuse & Permissions

Figure 4

Selected $Z$ values across irreps ${\Lambda }_u$ are shown for states suspected of being $J=\frac{5}{2}$ (left panel) and $\frac{7}{2}$ (right panel), based on the $m_{\pi }=524\mathrm{MeV}$ ensemble. The boxes at the top show the mass for the various states. There are two states of $J={\frac{5}{2}}^{-}$. The operators in the left panel, all projected onto $J={\frac{5}{2}}^{-}$, are left to right, $N_{\mathsf{M}}\otimes ({\frac{1}{2}}^{-}{)}_{\mathsf{M}}^1\otimes D_{\mathsf{L}=2,\mathsf{S}}^{[2]}$, $N_{\mathsf{M}}\otimes ({\frac{1}{2}}^{-}{)}_{\mathsf{S}}^1\otimes D_{\mathsf{L}=2,\mathsf{M}}^{[2]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{+}{)}_{\mathsf{M}}^1\otimes D_{\mathsf{L}=1,\mathsf{M}}^{[1]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{+}{)}_{\mathsf{S}}^1\otimes D_{\mathsf{L}=1,\mathsf{M}}^{[1]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{-}{)}_{\mathsf{M}}^1\otimes D_{\mathsf{L}=2,\mathsf{S}}^{[1]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{-}{)}_{\mathsf{S}}^1\otimes D_{\mathsf{L}=2,\mathsf{M}}^{[1]}$. Overlaps of these operators after subduction into $H_u$ and $G_{2u}$, agree well for each of the two states shown. The operators in the right panel, all projected onto $J={\frac{7}{2}}^{-}$, are left to right, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{-}{)}_{\mathsf{M}}^1\otimes D_{\mathsf{L}=2,\mathsf{M}}^{[2]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{-}{)}_{\mathsf{M}}^1\otimes D_{\mathsf{L}=2,\mathsf{S}}^{[2]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{-}{)}_{\mathsf{S}}^1\otimes D_{\mathsf{L}=2,\mathsf{M}}^{[2]}$, $N_{\mathsf{M}}\otimes ({\frac{3}{2}}^{-}{)}_{\mathsf{S}}^2\otimes D_{\mathsf{L}=2,\mathsf{M}}^{[2]}$. Similarly, the operator overlaps for this state agree well across $G_{1u}$, $H_u$, and $G_{2u}$.Reuse & Permissions

Figure 5

Fit to the three subduced principal correlators of lowest-lying ${\frac{7}{2}}^{-}$ Nucleon using a common mass. Results are from the $m_{\pi }=524\mathrm{MeV}$ ensemble. These levels correspond to the sixth excited $G_{1u}$, the seventh $H_u$ and the fourth $G_{2u}$. Plotted is $\lambda (t)·e^{m(t-t_0)}$. Grey points not included in the fit.Reuse & Permissions

Figure 6

Extracted Nucleon $H_u$ mass spectrum for various operator bases. (a) Full basis ($\dim =48$), (b) Only $J=\frac{3}{2}$ operators ($\dim =28$), (c) Only $J=\frac{5}{2}$ operators ($\dim =16$), (d) Only $J=\frac{7}{2}$ operators ($\dim =4$). Results are from the $m_{\pi }=524\mathrm{MeV}$ ensemble. The dimensionality of the operator basis in each $J$ is shown in Table .Reuse & Permissions

Figure 7

Extracted Nucleon $H_u$ mass spectrum as a function of $t_0$. Horizontal bands serve to guide the eye. For clarity of presentation, only one of the highly excited $J={\frac{3}{2}}^{-}$ and $J={\frac{5}{2}}^{-}$ levels are shown.Reuse & Permissions

Figure 8

Extracted Delta $H_g$ mass spectrum as a function of number of distillation vectors in the $m_{\pi }=524\mathrm{MeV}$ dataset. For $N\lesssim 32$, spin identification is lost for the highest $J={\frac{5}{2}}^{+}$ and $J={\frac{7}{2}}^{+}$ levels. Colors are red ($\frac{3}{2}$), green ($\frac{5}{2}$), blue ($\frac{7}{2}$), and orange is undetermined ($J$).Reuse & Permissions

Figure 9

Extracted Nucleon spectra by irrep for $m_{\pi }=524\mathrm{MeV}$. Colors are black ($J=\frac{1}{2}$), red ($\frac{3}{2}$), green ($\frac{5}{2}$), blue ($\frac{7}{2}$). Masses are shown in ratios of the $\Omega$ baryon mass.Reuse & Permissions

Figure 10

Extracted Delta spectra by irrep for $m_{\pi }=524\mathrm{MeV}$. Colors are black ($J=\frac{1}{2}$), red ($\frac{3}{2}$), green ($\frac{5}{2}$), blue ($\frac{7}{2}$). Masses are shown in ratios of the $\Omega$ baryon mass.Reuse & Permissions

Figure 11

Spin-identified spectrum of Nucleons and Deltas from the lattices at $m_{\pi }=524\mathrm{MeV}$, in units of the calculated $\Omega$ mass.Reuse & Permissions

Figure 12

Spin-identified spectrum of Nucleons and Deltas from the lattices at $m_{\pi }=396\mathrm{MeV}$, in units of the calculated $\Omega$ mass.Reuse & Permissions

Figure 13

Spectral overlaps for the negative-parity Nucleon operators in Table  within the lowest-lying states for the $m_{\pi }=524\mathrm{MeV}$ ensemble. The operators are subduced into the $G_{1u}$, $H_u$ and $G_{2u}$ irreps, and their spectral overlaps are shown for the lowest-lying states with the masses listed in units of the $\Omega$ baryon mass. In the spectroscopic notation, these are the $N^2{P}_{\mathsf{M}}{\frac{1}{2}}^{-}$ and $N^4{P}_{\mathsf{M}}{\frac{1}{2}}^{-}$ subduced into $G_{1u}$, the $N^2{P}_{\mathsf{M}}{\frac{3}{2}}^{-}$ and $N^4{P}_{\mathsf{M}}{\frac{5}{2}}^{-}$ subduced into $H_u$, and $N^4{P}_{\mathsf{M}}{\frac{5}{2}}^{-}$ subduced into both $H_u$ and $G_{2u}$. As can be seen, for each level one operator is dominant.Reuse & Permissions

Figure 14

Lightest few Nucleon $J={\frac{1}{2}}^{+}$, ${\frac{3}{2}}^{+}$, ${\frac{5}{2}}^{+}$, and ${\frac{7}{2}}^{+}$ states. Also shown in $N{\frac{1}{2}}^{+}$ is the threshold for $N\pi$ and $N\pi \pi$. The influence in the spectrum from these thresholds is complicated by the use of a finite spatial cubic box. Further discussion is in Sec. 9.Reuse & Permissions

Figure 15

Lightest few Nucleon $J={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{5}{2}}^{-}$, and ${\frac{7}{2}}^{-}$ states. Also shown in $N{\frac{1}{2}}^{-}$ is the threshold for $N\pi$ and $N\pi \pi$. The influence in the spectrum from these thresholds is complicated by the use of a finite spatial cubic box. Further discussion is in Sec. 9.Reuse & Permissions

Figure 16

Lightest few Delta $J={\frac{1}{2}}^{+}$, ${\frac{3}{2}}^{+}$, ${\frac{5}{2}}^{+}$, and ${\frac{7}{2}}^{+}$ states. Also shown in $\Delta {\frac{3}{2}}^{+}$ is the threshold for $N\pi$ and $N\pi \pi$. The influence in the spectrum from these thresholds is complicated by the use of a finite spatial cubic box. Further discussion is in Sec. 9.Reuse & Permissions

Figure 17

Lightest few Delta $J={\frac{1}{2}}^{-}$, ${\frac{3}{2}}^{-}$, ${\frac{5}{2}}^{-}$, and ${\frac{7}{2}}^{-}$ states. Also shown in $\Delta {\frac{1}{2}}^{-}$ is the threshold for $N\pi$ and $N\pi \pi$. The influence in the spectrum from these thresholds is complicated by the use of a finite spatial cubic box. Further discussion is in Sec. 9.Reuse & Permissions

Figure 18

Comparison of results for the Nucleon $J={\frac{1}{2}}^{+}$ channel. The results shown in grey are from Ref. 32, while those in orange are from Ref. 4. Note that data are plotted using the scale-setting scheme in the respective papers. Results from this paper are shown in red (the ground state), green and blue. At the lightest pion mass, there is a clustering of four states as indicated near 2 GeV, while there are three nearly degenerate states close to 2.7 GeV. Operators featuring the derivative constructions discussed in this paper feature prominently in these excited states.Reuse & Permissions