Spectroscopy of triply charmed baryons from lattice QCD

The spectrum of excitations of triply charmed baryons is computed using lattice QCD including dynamical light quark fields. Calculations are performed on anisotropic lattices with temporal and spatial spacings $a_t=0.0351(2)\mathrm{fm}$ and $a_s\sim 0.12\mathrm{fm}$, respectively, and with pion mass of about 390 MeV. The spectrum obtained has baryonic states with well-defined total spin up to $\frac{7}{2}$ and the low-lying states closely resemble the expectation from models with an $SU(6)\times O(3)$ symmetry. Energy splittings between extracted states, including those due to spin-orbit coupling in the heavy quark limit, are computed and compared against data at other quark masses.

Figure 1

Principal correlator fits for six states in irrep $H_g$ that are identified as $J^P=3/2^{+}$. Fits are obtained using Eq. (5). Data points are obtained from $e^{m_n(t-t_0)}{\lambda }_n(t)$ and the lines show the fits and one sigma-deviation according to the fitting form $e^{m_n(t-t_0)}{\lambda }_n(t)=1-A_n+A_n{e}^{-(m_n^{\prime }-m_n)(t-t_0)}$, with $t_0=10$; the grey points are not included in the fits.

Figure 2

A normalized correlation matrix, $C_{ij}/\sqrt{C_{ii}{C}_{jj}}$, at a given time slice at $t/a_t=5$ for the $H_g$ irrep. Operators are ordered such that those subduced from spin $3/2$ appear first followed by spin $5/2$ and then spin $7/2$.

Figure 3

Histograms of the normalized overlap factor, $Z$ of a set of operators onto some of the lower-lying states in each lattice irrep. The overlaps shown are weighted $\frac{Z_i^n}{{\max }_n[Z_i^n]}$, such that the largest normalized value across all states for a given operator is unity. Various operators are depicted in each state however, due to the large basis size, their names are not shown here. Black bars correspond to spin-$1/2$ state, red for spin-$3/2$, green for spin-$5/2$ and blue represents spin-$7/2$ states. Lighter and darker shades at the top of each bar represent the one-sigma statistical uncertainty.

Figure 4

“Matrix” plot of values of overlap factor, $Z_i^n$, of an operator $i$ to a given state $n$, as defined by Eq. (7). $Z_i^n$ are normalized according to $\frac{Z_i^n}{{\max }_n[Z_i^n]}$, so that for a given operator the largest overlap across all states is unity. The normalized magnitudes of various operator overlaps to states in the $G_{1g}$ irrep are shown. Darker pixel indicates larger values of the operator overlaps as shown in the adjacent legend. Various type of operators, for example, nonrelativistic (nr) and relativistic (r) operators, as well as hybrid (h) and nonhybrid (nh) operators are indicated by column labels. In addition, the continuum spins of the operators are shown by $1/2$ and $7/2$. State 0, the ground state, and excited states 1, 3, 4, 10 and 11 are identified with $J^P={\frac{1}{2}}^{+}$ from the overlap to various types of operators according to pixel strength. States 2 and 8 are identified as $J^P={\frac{7}{2}}^{+}$ states with the predominant overlap to nonrelativistic nonhybrid and relativistic nonhybrid operators, respectively.

Figure 5

Same “matrix” plot, as Fig. 4, for the $H_g$ irrep. Here one can associate state 6 with quantum number $J^P={\frac{7}{2}}^{+}$, the states 3, 5 and 10 with $J^P={\frac{5}{2}}^{+}$, and the rest with $J^P={\frac{3}{2}}^{+}$. The states 7 and 9 are predominantly hybrid in nature, while states 4, 8 and 10 are found to have substantial overlap with nonhybrid operators. Pixel legend is the same as in Fig. 4.

Figure 6

A selection of $Z$ values for states conjectured to be identified with $J^P={\frac{5}{2}}^{+}$ (top two plots), ${\frac{5}{2}}^{-}$ (middle two plots), ${\frac{7}{2}}^{+}$ (bottom left) and ${\frac{7}{2}}^{-}$ (bottom right). Operators in consideration, which overlap to these states, are mentioned at the bottom of each plot. $Z$ values obtained for a given operator, but from different irreps, are found to be consistent or very close to each other which helps to identify the spin of a given state. Some operators are scaled so the set can all be shown on a single panel. These rescalings are indicated by numbers in front of the operator labels.

Figure 7

Mass splitting of the ground state of $J^P={\frac{3}{2}}^{+}$ ${\mathrm{\Omega }}_{ccc}$ from $J/\psi$ meson. A factor $3/2$ is multiplied with $J/\psi$ mass to account for the difference in the number of charm quarks in baryons and mesons. This mass splitting mimics the binding energy of the ground state ${\mathrm{\Omega }}_{ccc}$. Results of this mass splitting from this work (red boxes) are compared with those obtained from other lattice calculations [29, 30, 31, 32].

Figure 8

Spin identified spectra of triply charmed baryons with respect to $\frac{3}{2}{m}_{{\eta }_c}$. The boxes with thick borders corresponds to the states with strong overlap with hybrid operators. The states inside the pink ellipses are those with relatively large overlap to nonrelativistic operators.

Figure 9

Energy splittings between states with same $L$ and $S$ values, starting from light to heavy baryons. For ${\mathrm{\Omega }}_{bbb}$, results are with only nonrelativistic operators [44]; for ${\mathrm{\Omega }}_{ccc}$, results from relativistic and nonrelativistic as well as only nonrelativistic operators are shown, and for light and strange baryons results are with relativistic and nonrelativistic operators [40]. These results are obtained from fitting the jackknife ratio of the correlators which helps to get smaller error bar in splittings. The left column is for the states with $D=2,S=\frac{3}{2}$ and $L=2$. The symbol $H_{g,1,S}$ refers to the first embedding of irrep $H_g$ in the totally symmetric Dirac spin combination, while $D_{L=2,S}^{[2]}$ refers to spatial projection operators with two derivatives in a totally symmetric combination, and with orbital angular momentum two. Similarly, the middle column is for the states with $D=2,S=\frac{1}{2}$ and $L=2$. Here irrep is $G_{1g}$ and both Dirac and derivative are in a mixed symmetric combination. In the right column these negative parity states have $D=1,S=\frac{1}{2}$ and $L=1$. Here again irrep is $G_{1g}$ and both Dirac and derivative are in a mixed symmetric combination.

Figure 10

Energy splittings of triply flavored baryons at various quark masses ($u$ to $b\to \mathrm{left}$ to right) from the respective isoscalar vector meson (irrep $T_1^{--}$) ground state. Top four figures are for positive parity states and bottom four are for negative parity states, respectively. A factor $3/2$ is multiplied with isoscalar vector meson masses to account for the difference in the number of charm quarks in baryons and mesons. For charm quark, results are from this work, while for light and strange quark results are from Ref. [40] and those for bottom quark are from Ref. [44]. The zebra-shaded boxes are the results obtained by using only nonrelativistic operators.

Figure 11

Energy splittings of a few triple-flavored baryons (labeled by different symbols and colors) from isoscalar vector meson (irrep $T_1^{--}$) ground state are plotted against the square of the pseudoscalar masses. Top four figures are for positive parity and bottom four are for negative parity baryons, respectively. This energy splitting mimics the binding energy and considering that these plots show the dependence of binding energies of these states as a function of quark mass. We fit this dependence with a form $a+b/m_{ps}$ (please see text for details).