Charmed bottom baryon spectroscopy from lattice QCD

We calculate the masses of baryons containing one, two, or three heavy quarks using lattice QCD. We consider all possible combinations of charm and bottom quarks, and compute a total of 36 different states with $J^P={\frac{1}{2}}^{+}$ and $J^P={\frac{3}{2}}^{+}$. We use domain-wall fermions for the up, down, and strange quarks, a relativistic heavy-quark action for the charm quarks, and nonrelativistic QCD for the bottom quarks. Our analysis includes results from two different lattice spacings and seven different pion masses. We perform extrapolations of the baryon masses to the continuum limit and to the physical pion mass using $\mathrm{SU}(4|2)$ heavy-hadron chiral perturbation theory including $1/m_Q$ and finite-volume effects. For the 14 singly heavy baryons that have already been observed, our results agree with the experimental values within the uncertainties. We compare our predictions for the hitherto unobserved states with other lattice calculations and quark-model studies.

Figure 1

Matrix fit of the ${\mathrm{\Omega }}_{ccb}$ two-point functions using Eq. (37). The data shown here are from the C54 set; the lines show the fit functions in the chosen fit ranges. The correlator $⟨O[\tilde{c},\tilde{c},\tilde{b}]\overline{O}[c,c,b]⟩$, which equals $⟨O[c,c,b]\overline{O}[\tilde{c},\tilde{c},\tilde{b}]⟩$ in the limit of infinite statistics, is not shown for clarity.

Figure 2

Effective-energy plot for the $2\times 2$ matrix of ${\mathrm{\Omega }}_{ccb}$ two-point functions from the C54 set. The effective energy for a correlator $C(t)$ is computed as $a{E}_{\text{eff}}(t+\frac{a}{2})=\ln [C(t)/C(t+a)]$. The lines indicate the time ranges and the energy obtained from the fit shown in Fig. 1.

Figure 3

${\mathrm{\Xi }}_{cc}^{*}$ energies obtained using the four different fit methods for each data set as explained in the main text. Also shown are the method-averaged energies (correlations are taken into account). For the method-averaged energies, the outer error bars include a scale factor in the cases where the average has ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.>1$ (here, for the C14 and C54 data sets).

Figure 4

Coupled two-exponential fit to the correlators $⟨O_5[\tilde{u},\tilde{s},\tilde{c}](t)\overline{O_5}[\tilde{u},\tilde{s},\tilde{c}](0)⟩$, $⟨O_5[\tilde{u},\tilde{s},\tilde{c}](t)\overline{O_5^{\prime }}[\tilde{u},\tilde{s},\tilde{c}](0)⟩$, and $⟨O_5^{\prime }[\tilde{u},\tilde{s},\tilde{c}](t)\overline{O_5^{\prime }}[\tilde{u},\tilde{s},\tilde{c}](0)⟩$ using Eqs. (40), (41), and (42). The data shown here are from the C54 set.

Figure 5

Independent single-exponential fits to the correlators $⟨O_5[\tilde{u},\tilde{s},\tilde{c}](t)\overline{O_5}[\tilde{u},\tilde{s},\tilde{c}](0)⟩$ and $⟨O_5^{\prime }[\tilde{u},\tilde{s},\tilde{c}](t)\overline{O_5^{\prime }}[\tilde{u},\tilde{s},\tilde{c}](0)⟩$ according to Eqs. (44) and (45). The data shown here are from the C54 set.

Figure 6

Chiral and continuum extrapolations for the $\{{\mathrm{\Lambda }}_Q,{\mathrm{\Sigma }}_Q,{\mathrm{\Sigma }}_Q^{*}\}$ baryons. The curves show the fit functions in infinite volume at $m_{\pi }^{(\mathrm{vv})}=m_{\pi }^{(\mathrm{vs})}=m_{\pi }$, for the two different lattice spacings where we have data, and in the continuum limit. For the continuum curves, the shaded bands indicate the $1\sigma$ uncertainty. The lattice data have been shifted to infinite volume (see Table 11 for the values of the shifts); data points at the coarse lattice spacing are plotted with circles, and data points at the fine lattice spacing are plotted with squares. The partially quenched data points, which have $m_{\pi }^{(\mathrm{vv})}<m_{\pi }^{(\mathrm{vs})}$, are included in the plot with open symbols at $m_{\pi }=m_{\pi }^{(\mathrm{vv})}$, even though the fit functions actually have slightly different values for these points. The data sets with the lowest two pion masses (C14 and F23) are excluded here because our treatment of finite-volume effects in $\mathrm{HH}\chi \mathrm{PT}$ breaks down below the ${\mathrm{\Sigma }}_Q^{(*)}\to {\mathrm{\Lambda }}_Q\pi$ strong-decay thresholds. The vertical lines indicate the physical value of the pion mass.

Figure 7

Imaginary parts of the $\{{\mathrm{\Lambda }}_Q,{\mathrm{\Sigma }}_Q,{\mathrm{\Sigma }}_Q^{*}\}$ baryon energies in infinite volume, obtained from $\mathrm{HH}\chi \mathrm{PT}$. The imaginary parts depend strongly on $\mathrm{\Delta }$ and $\mathrm{\Delta }+{\mathrm{\Delta }}_{*}$; the values used here are given in Table 9.

Figure 8

Chiral and continuum extrapolations for the $\{{\mathrm{\Xi }}_Q,{\mathrm{\Xi }}_Q^{\prime },{\mathrm{\Xi }}_Q^{*}\}$ baryons. The details are as explained in the caption of Fig. 6, except that now the curves also correspond to different values of $m_{{\eta }_s}^{(\mathrm{vv})}$ as shown in the legend, and no data points are excluded here. Two of the data points at the coarse lattice spacing have equal pion masses; these points are from the C54 and C53 data sets, which have different valence strange-quark masses.

Figure 9

Chiral and continuum extrapolations for the $\{{\mathrm{\Omega }}_Q,{\mathrm{\Omega }}_Q^{*}\}$ baryons. The curves show the fit functions for the two different lattice spacings where we have data, and in the continuum limit, evaluated at appropriate values of $m_{{\eta }_s}^{(\mathrm{vv})}$ as shown in the legend. For the continuum curves, the shaded bands indicate the $1\sigma$ uncertainty. Data points at the coarse lattice spacing are plotted with circles, and data points at the fine lattice spacing are plotted with squares. The open circles are from the C53 data set with a lower-than-physical valence strange-quark mass.

Figure 10

Chiral and continuum extrapolations for the $\{{\mathrm{\Xi }}_{QQ},{\mathrm{\Xi }}_{QQ}^{*}\}$ baryons. The details of the plots are as explained in the caption of Fig. 6, except that no data sets are excluded here.

Figure 11

Left panel: Chiral and continuum extrapolations for the $\{{\mathrm{\Xi }}_{cb},{\mathrm{\Xi }}_{cb}^{\prime },{\mathrm{\Xi }}_{cb}^{*}\}$ baryons. The details of the plots are as explained in the caption of Fig. 6, except that no data sets are excluded here. Right panel: Chiral and continuum extrapolations for the $\{{\mathrm{\Omega }}_{cb},{\mathrm{\Omega }}_{cb}^{\prime },{\mathrm{\Omega }}_{cb}^{*}\}$ baryons. See the caption of Fig. 9 for explanations.

Figure 12

Chiral and continuum extrapolations for the $\{{\mathrm{\Omega }}_{QQ},{\mathrm{\Omega }}_{QQ}^{*}\}$ baryons. See the caption of Fig. 9 for explanations.

Figure 13

Chiral and continuum extrapolations for the ${\mathrm{\Omega }}_{ccc}$ (left panel) and ${\mathrm{\Omega }}_{bbb}$ (right panel). The curves show the fit functions for the two different lattice spacings where we have data, and in the continuum limit. For the continuum curves, the shaded bands indicate the $1\sigma$ uncertainty. Data points at the coarse lattice spacing are plotted with circles, and data points at the fine lattice spacing are plotted with squares.

Figure 14

Chiral and continuum extrapolations for the $\{{\mathrm{\Omega }}_{ccb},{\mathrm{\Omega }}_{ccb}^{*}\}$ baryons (left panel) and for the $\{{\mathrm{\Omega }}_{cbb},{\mathrm{\Omega }}_{cbb}^{*}\}$ baryons (right panel). See the caption of Fig. 13 for explanations.

Figure 15

Our results for the masses of charmed and/or bottom baryons, compared to the experimental results where available [8, 10, 12]. The masses of baryons containing $n_b$ bottom quarks have been offset by $-n_b·(3000\mathrm{MeV})$ to fit them into this plot. Note that the uncertainties of our results for nearby states are highly correlated, and hyperfine splittings such as $M_{{\mathrm{\Omega }}_b^{*}}-M_{{\mathrm{\Omega }}_b}$ can in fact be resolved with much smaller uncertainties than is apparent from this figure (see Table 19).

Figure 16

Comparison of lattice QCD results for the doubly and triply charmed baryon masses [41, 45, 46, 47, 49, 51]; our results are labeled as “Brown et al., 2014.” Only calculations with dynamical light quarks are included; for the doubly charmed baryons, we further required that the calculations were performed at or extrapolated to the physical pion mass. Results without estimates of systematic uncertainties are labeled “stat. only.” The lattice-spacing values used in the calculations are also given; $a=0$ indicates that the results have been extrapolated to the continuum limit. Reference [49] (Padmanath et al., 2013) gives results for $m_{{\mathrm{\Omega }}_{ccc}}-\frac{3}{2}{m}_{J/\psi }$ for two different values of the Sheikholeslami-Wohlert coefficient; here we took the result with $c_{\mathrm{SW}}=1.35$ and added the experimental value of $\frac{3}{2}{m}_{J/\psi }$ [8]. In the plot of the doubly charmed baryons, the unconfirmed experimental result for the ${\mathrm{\Xi }}_{cc}^{+}$ mass from SELEX [13, 14] is shown with a dashed line. Note that the lattice QCD calculations consistently predict a ${\mathrm{\Xi }}_{cc}$ mass higher than the SELEX result.

Figure 17

Comparison of lattice QCD results for the doubly bottom baryon masses. The only other published unquenched calculation is the one of Ref. [53]. Our results have larger statistical uncertainties, but our calculation was performed with closer-to-physical pion masses and included a combined chiral and continuum extrapolation.

Figure 18

Comparison of our lattice QCD results for the ${\mathrm{\Xi }}_{cb}$, ${\mathrm{\Xi }}_{cb}^{\prime }$, ${\mathrm{\Xi }}_{cb}^{*}$, ${\mathrm{\Omega }}_{cb}$, ${\mathrm{\Omega }}_{cb}^{\prime }$, and ${\mathrm{\Omega }}_{cb}^{*}$ baryon masses with estimates based on continuum methods, including quark models and QCD sum rules [100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113]. From Refs. [101] (Silvestre-Brac, 1996) and [112] (Ghalenovi et al., 2014), we show results for multiple different choices of the interquark potentials. Note that the bag-model calculation of Ref. [104] (He et al., 2004) predicts $m_{{\mathrm{\Xi }}_{cb}^{*}}<m_{{\mathrm{\Xi }}_{cb}^{\prime }}$ and $m_{{\mathrm{\Omega }}_{cb}^{*}}<m_{{\mathrm{\Omega }}_{cb}^{\prime }}$, and the QCD sum-rule calculation of Ref. [111] (Tang et al., 2012) predicts $m_{{\mathrm{\Xi }}_{cb}^{\prime }}<m_{{\mathrm{\Xi }}_{cb}}$ and $m_{{\mathrm{\Omega }}_{cb}^{\prime }}<m_{{\mathrm{\Omega }}_{cb}}$, both of which are rather unusual. The sum-rule calculation of Ref. [109] (Zhang et al., 2008) gives extremely large hyperfine splittings $m_{{\mathrm{\Xi }}_{cb}^{*}}-m_{{\mathrm{\Xi }}_{cb}^{\prime }}\approx 1\mathrm{GeV}$ and $m_{{\mathrm{\Omega }}_{cb}^{*}}-m_{{\mathrm{\Omega }}_{cb}^{\prime }}\approx 0.5\mathrm{GeV}$ [our results for the hyperfine splittings are $m_{{\mathrm{\Xi }}_{cb}^{*}}-m_{{\mathrm{\Xi }}_{cb}^{\prime }}=26.7(3.3)(8.4)\mathrm{MeV}$, $m_{{\mathrm{\Omega }}_{cb}^{*}}-m_{{\mathrm{\Omega }}_{cb}^{\prime }}=27.4(2.0)(6.7)\mathrm{MeV}$]; the ${\mathrm{\Xi }}_{cb}^{*}$ mass from Ref. [109] is beyond the upper limit of the plot.

Figure 19

Comparison of our lattice QCD results for the masses of triply heavy charm-bottom baryons with estimates based on continuum methods, including quark models, QCD sum rules, and perturbative QCD [6, 101, 106, 107, 110, 112, 114, 115, 116, 117, 118, 119, 120]. From Refs. [101] (Silvestre-Brac, 1996) and [112] (Ghalenovi et al., 2014), we show results for multiple different choices of the interquark potentials. From Ref. [6], which used the static three-quark potential from perturbative QCD, we show the next-to-next-leading-order results from Table 16. Not shown in this plot are the results of the QCD sum-rule calculation of Ref. [121], which are far lower than all other results.