Spectrum of heavy baryons in the quark model

Single- and double-heavy baryons are studied in the constituent quark model. The model Hamiltonian is chosen as a standard one with two exceptions: (1) the color-Coulomb term depends on quark masses and (2) an antisymmetric $LS$ (spin-orbit) force is introduced. Model parameters are fixed by the strange baryon spectra, $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ baryons. The masses of the observed charmed and bottomed baryons are, then, fairly well reproduced. Our focus is on the low-lying negative-parity states, in which the heavy baryons show specific excitation modes reflecting the mass differences of heavy and light quarks. By changing quark masses from the SU(3) limit to the strange quark mass, and, further, to the charm and bottom quark masses, we demonstrate that the spectra change from the SU(3) symmetry patterns to the heavy-quark-symmetry ones.

Figure 1

Heavy-quark mass dependence of excited energies of the $\lambda$ mode (red solid line) and the $\rho$ mode (blue solid line) in Eq. (1).

Figure 2

Jacobi coordinates for the three-body system. We place the heavy quark as the third particle in the case of single-heavy baryons, while the first and second particles are heavy quarks in double-heavy baryons.

Figure 3

Convergence of the energy of the lowest ${\mathrm{\Lambda }}_c(3/2^{-})$ for increasing the number of bases functions.

Figure 4

The $\rho$- and $\lambda$-mode excitations of the single-heavy baryon.

Figure 5

Calculated energy spectra of ${\mathrm{\Lambda }}_c$, ${\mathrm{\Sigma }}_c$, and ${\mathrm{\Omega }}_c$ for $1/2^{+}$, $3/2^{+}$, $5/2^{+}$, $1/2^{-}$, $3/2^{-}$, and $5/2^{-}$(solid line) together with experimental data (dashed line). Several thresholds are also shown by dotted lines.

Figure 6

Calculated energy spectra of ${\mathrm{\Lambda }}_b$, ${\mathrm{\Sigma }}_b$, and ${\mathrm{\Omega }}_b$ for $1/2^{+}$, $3/2^{+}$, $5/2^{+}$, $1/2^{-}$, $3/2^{-}$, and $5/2^{-}$ (solid line) together with experimental data (dashed line). Several thresholds are also shown by dotted lines.

Figure 7

Calculated energy spectra of ${\mathrm{\Xi }}_{cc}$ and ${\mathrm{\Omega }}_{cc}$ for $1/2^{+}$, $3/2^{+}$, $5/2^{+}$, $1/2^{-}$, $3/2^{-}$, and $5/2^{-}$ (solid line) together with experimental data (dashed line).

Figure 8

Calculated energy spectra of ${\mathrm{\Xi }}_{bb}$ and ${\mathrm{\Omega }}_{bb}$ for $1/2^{+}$, $3/2^{+}$, $5/2^{+}$, $1/2^{-}$, $3/2^{-}$, and $5/2^{-}$ (solid line) together with experimental data (dashed line).

Figure 9

Heavy-quark mass dependence of excited energies of the first, second, and third states for $1/2^{-}$ (solid line), $3/2^{-}$ (dashed line), and $5/2^{-}$ (double dotted line) of ${\mathrm{\Lambda }}_Q$ (red) and ${\mathrm{\Sigma }}_Q$ (blue). The bullet denotes a heavy-quark singlet. The pair within a half circle denotes a heavy-quark doublet.

Figure 10

The probability of the $\lambda$ mode (blue line) and the $\rho$ mode (red line) of $1/2^{-}$ for ${\mathrm{\Sigma }}_Q$ (dotted) and ${\mathrm{\Lambda }}_Q$ (solid).

Figure 11

Heavy-quark mass dependence of excited energies of the first, second, and third state for $1/2^{-}$ (red solid line), $3/2^{-}$ (blue dotted line), and $5/2^{-}$ (green dashed line) of ${\mathrm{\Xi }}_{QQ}$.

Figure 12

The probability of the $\lambda$ mode (blue line) and the $\rho$ mode (red line) of $1/2^{-}$ for ${\mathrm{\Xi }}_{QQ}$ (solid) and ${\mathrm{\Omega }}_{QQ}$ (dotted).

Figure 13

The probabilities of $j=0$ (solid line) and $j=1$ (dot-dashed line) for $\mathrm{\Lambda }(1/2^{-})$. Red, blue, and green lines show the first, second, and third state, respectively.

Figure 14

The probabilities of $j=1$ (solid line) and $j=2$ (dot-dashed line) for $\mathrm{\Lambda }(3/2^{-})$. Red, blue, and green lines show the first, second, and third state, respectively.

Figure 15

The probabilities of $j=0$ (solid line) and $j=1$ (dot-dashed line) for $\mathrm{\Sigma }(1/2^{-})$. Red, blue, and green lines show the first, second, and third state, respectively.

Figure 16

The probabilities of $j=1$ (solid line) and $j=2$ (dot-dashed line) for $\mathrm{\Sigma }(3/2^{-})$. Red, blue, and green lines show the first, second, and third state, respectively.

Figure 17

The heavy-quark mass dependences of the probabilities of the $S$-wave (0.0) component (red line) and (1,1) component (blue line) for $\mathrm{\Lambda }(1/2_1^{+})$.

Figure 18

The heavy-quark mass dependences of the probabilities of the $S$-wave (0,0) component (red line) and the (1,1) component (blue line) for $\mathrm{\Lambda }(1/2_2^{+})$.

Figure 19

The heavy-quark mass dependences of the probabilities of the (1,1) component (red line), the (2,0) component (blue line), and the (0,2) component (green line) for $\mathrm{\Lambda }(3/2_1^{+})$.

Figure 20

The heavy-quark mass dependences of the probabilities of the (1,1) component (red line), the (2,0) component (blue line), and the (0,2) component (green line) for $\mathrm{\Lambda }(5/2_1^{+})$.

Figure 21

The heavy-quark mass dependences of the probabilities of the $S$-wave (0,0) component (red line), the (1,1) component (blue line), the (2,0) component (green line), and the (0,2) component (violet line) for $\mathrm{\Sigma }(1/2_1^{+})$.

Figure 22

The heavy-quark mass dependences of the probabilities of the $S$-wave ($l=0$, $L=0$) component (red line), the (1,1) component (blue line), the (2,0) component (green line), and the (0,2) component (violet line) for $\mathrm{\Sigma }(3/2_1^{+})$.