QCD with two light dynamical chirally improved quarks: Baryons

We present a study of baryon ground states and low lying excitations of nonstrange and strange baryons. The results are based on seven gauge field ensembles with two dynamical light chirally improved quarks corresponding to pion masses between 255 and 596 MeV and a strange valence quark with mass fixed by the $\Omega$ baryon. The lattice spacing varies between 0.1324 and 0.1398 fm. Given in lattice units, the bulk of our results are for size ${16}^3\times 32$; for two ensembles with light pion masses (255 and 330 MeV) we also use ${24}^3\times 48$ lattices and perform an infinite volume extrapolation. We derive energy levels for the spin $1/2$ and $3/2$ channels for both parities. In general, our results in the infinite volume limit compare well with experiment. We analyze the flavor symmetry content by identifying the singlet/octet/decuplet contributions of the resulting eigenstates. The ground states’ compositions agree with quark model expectations. In some cases the excited states, however, disagree and we discuss possible reasons.

Figure 1

Eigenvalues for the four lowest states in the $\Sigma$ $1/2^{+}$ channel for ensembles A50, C72 and A66 (top to bottom) which cover the whole range of pion masses considered.

Figure 2

Energy levels for nucleon spin $1/2$, positive (upper) and negative parity (lower). Black, red and blue denote values of $\beta$ equal to 4.70, 4.65 and 4.58, respectively. The solid lines give the mean values of the fits in $m_{\pi }^2$; the dashed ones indicate the region of one $\sigma$.

Figure 3

Eigenvectors for nucleon spin $1/2$ negative parity ground state and first excitation, ensemble A50 (upper) and B70 (lower). Note the different composition of the states at the different pion masses.

Figure 4

Energy levels for nucleon spin $3/2$, positive (upper) and negative parity (lower).

Figure 5

Energy levels for $\Delta$ spin $1/2$, positive (upper) and negative parity (lower).

Figure 6

Energy levels for $\Delta$ spin $3/2$, positive (upper) and negative parity (lower).

Figure 7

Energy levels for $\Sigma$ spin $1/2$, positive (upper) and negative parity (lower).

Figure 8

Eigenvectors for $\Sigma$ spin $1/2$ negative parity ground state and first excitation (upper) and second and third excitation (lower) for ensemble B70. Note the dominance of decuplet interpolators for the second excitation, which is a low lying state (see Fig. 7). Details are discussed in the text.

Figure 9

Energy levels for $\Sigma$ spin $3/2$, positive (upper) and negative parity (lower).

Figure 10

Energy levels for $\Xi$ spin $1/2$, positive (upper) and negative parity (lower).

Figure 11

Energy levels for $\Xi$ spin $3/2$, positive (upper) and negative parity (lower).

Figure 12

Energy levels for $\Omega$ spin $1/2$, positive (upper) and negative parity (lower).

Figure 13

Energy levels for $\Omega$ spin $3/2$, positive (upper) and negative parity (lower).

Figure 14

Systematic error of the nucleon ground state energy level. The levels are shown for different choices of interpolators and fit ranges, labeled on the horizontal axis; e.g., “A,4” denotes the set of interpolators “A” and a fit range for the eigenvalues from $t=4a$ to the onset of noise. “A” denotes a set of interpolators (1, 2, 9, 10, 19, 20); “B” denotes (3, 4, 11, 12, 19, 20). For each set of interpolators and fit range, results for small to large lattices (spatial size 16, 24 for ensemble A66, and 12, 16, 24 for C77) are shown from left to right, the corresponding infinite volume limit rightmost.

Figure 15

Volume dependence of the nucleon mass for the set of interpolators (1, 2, 9, 10, 19, 20) and $t_{\min }=5a$ [(A,5) of Fig. 14].

Figure 16

Energy levels for the nucleon spin $\frac{1}{2}\genfrac{}{}{0ex}{}{{\mathrm{}}^{+}}{}$ (upper) and $\Delta$ spin $\frac{3}{2}\genfrac{}{}{0ex}{}{{\mathrm{}}^{+}}{}$ (lower) in the infinite volume limit. After infinite volume extrapolation [(A,5) of Fig. 14 respective Fig. 19], we extrapolate to physical pion masses. We obtain $m_N=954(16)\mathrm{MeV}$ and $m_{\Delta }=1268(32)\mathrm{MeV}$, which both match the experimental values within roughly $1\sigma$.

Figure 17

Systematic error of the nucleon spin $1/2^{-}$ ground state mass, analogous to Fig. 14. “A” denotes the set of interpolators (5, 11, 17); “B” denotes (1, 2, 9, 10, 17, 18). For each set of interpolators and fit range, results for small to large lattices are shown from left to right, the corresponding infinite volume limit rightmost.

Figure 18

Systematic error of the nucleon spin $1/2^{-}$ first excited energy level, analogous to Fig. 14. “A” denotes the set of interpolators (5, 11, 17); “B” denotes (1, 2, 9, 10, 17, 18). For each set of interpolators and fit range, results for small to large lattices are shown from left to right, the corresponding infinite volume limit rightmost.

Figure 19

Systematic error of the $\Delta$ spin $3/2^{+}$ mass, analogous to Fig. 14. “A” denotes the set of interpolators (1, 4, 5); “B” denotes (1, 5, 8). For each set of interpolators and fit range, results for small to large lattices are shown from left to right, the corresponding infinite volume limit rightmost.

Figure 20

Systematic error of the $\Omega$ spin $3/2^{+}$ mass, analogous to Fig. 14. “A” denotes the set of interpolators (1, 5, 8); “B” denotes (1, 3, 4). For each set of interpolators and fit range, results for small to large lattices are shown from left to right, the corresponding infinite volume limit rightmost. For definiteness we choose (B,4).

Figure 21

Energy levels for $\Omega$ spin $3/2^{+}$ in the infinite volume limit. After infinite volume extrapolation we extrapolate to physical pion masses, obtaining $m_{\Omega }=1650(20)\mathrm{MeV}$. For discussion please refer to the text.

Figure 22

Energy levels for the $\Sigma$ spin $\frac{1}{2}\genfrac{}{}{0ex}{}{{\mathrm{}}^{+}}{}$ (upper pane) and the $\Xi$ spin $\frac{1}{2}\genfrac{}{}{0ex}{}{{\mathrm{}}^{+}}{}$ (lower pane) ground states in the infinite volume limit. After infinite volume extrapolation we extrapolate to physical pion masses. We obtain $m_{\Sigma }=1176(19)(+07)\mathrm{MeV}$ and $m_{\Xi }=1299(16)(+15)\mathrm{MeV}$.

Figure 23

Systematic error of the $\Sigma$ spin $\frac{1}{2}\genfrac{}{}{0ex}{}{{\mathrm{}}^{+}}{}$ mass, analogous to Fig. 14. “A” denotes the set of interpolators (1, 2, 9, 10, 25, 26); “B” denotes (2, 3, 10, 11, 19, 20, 26, 27). For each set of interpolators and fit range, results for small to large lattices are shown from left to right, the corresponding infinite volume limit rightmost.

Figure 24

Systematic error of the $\Xi$ spin $1/2^{+}$mass, analogous to Fig. 14. “A” denotes the set of interpolators (1, 2, 9, 10, 25, 26); “B” denotes (2, 3, 10, 11, 19, 20, 26, 27). For each set of interpolators and fit range, results for small to large lattices are shown from left to right, the corresponding infinite volume limit rightmost.

Figure 25

Energy levels for $\Sigma$ spin $3/2^{+}$ (upper pane) and $\Xi$ spin $3/2^{+}$ (lower pane) ground states in the infinite volume limit. After infinite volume extrapolation we extrapolate to physical pion masses. We obtain $m_{\Sigma }=1431(25)(+07)\mathrm{MeV}$ and $m_{\Xi }=1540(22)(+15)\mathrm{MeV}$.

Figure 26

Energy levels for positive parity (top) and negative parity baryons (bottom). All values are obtained by chiral extrapolation linear in the pion mass squared. Horizontal lines or boxes represent experimentally known states; dashed lines indicate poor evidence, according to Ref. [48]. The statistical uncertainty of our results is indicated by bands of $1\sigma$, that of the experimental values by boxes of $1\sigma$. The strange quarks are implemented in valence approximation. Grey symbols denote a poor ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ of the chiral fits (see Tables 5 and 6).

Figure 27

Energy levels of baryons in the infinite volume limit at physical pion mass. Horizontal lines and boxes represent experimentally known states [48]. The statistical uncertainty of our results is indicated by bands of $1\sigma$.