$D$-wave heavy baryons of the $SU(3)$ flavor $6_F$ representation

We use the method of QCD sum rules to study the $D$-wave charmed and bottom baryons of the $SU(3)$ flavor ${\mathbf{6}}_F$ and calculate their masses up to the order $\mathcal{O}(1/m_Q)$ with $m_Q$ the heavy-quark mass. Our results suggest that the ${\mathrm{\Xi }}_c(3123)$ can be well interpreted as a $D$-wave ${\mathrm{\Xi }}_c^{\prime }({\mathbf{6}}_F)$ state, and it probably has a partner state close to it. Our results also suggest that there may exist as many as four $D$-wave ${\mathrm{\Omega }}_c$ states in the energy region 3.3–3.5 GeV, and we propose to search for them in the future LHCb and BelleII experiments.

Figure 1

The notations for the $D$-wave charmed baryons of the $SU(3)$ flavor ${\mathbf{6}}_F$. See Fig. 1 of Ref. [10] and discussions therein for details.

Figure 2

Variations of ${\overline{\mathrm{\Lambda }}}_{{\mathrm{\Sigma }}_c,1,1,\rho \rho }$ (left), $K_{{\mathrm{\Sigma }}_c,1,1,\rho \rho }$ (middle), and ${\mathrm{\Sigma }}_{{\mathrm{\Sigma }}_c,1,1,\rho \rho }$ (right) with respect to the Borel mass $T$, calculated using the charmed baryon doublet $[{\mathrm{\Sigma }}_c({\mathbf{6}}_F),1,1,\rho \rho ]$. The short-dashed, solid, and long-dashed curves are obtained by fixing ${\omega }_c=3.0$, 3.2, and 3.4 GeV, respectively.

Figure 3

Variations of $m_{{\mathrm{\Sigma }}_c(1/2^{+})}$ with respect to the threshold value ${\omega }_c$ (left) and the Borel mass $T$ (right), calculated using the charmed baryon doublet $[{\mathrm{\Sigma }}_c({\mathbf{6}}_F),1,1,\rho \rho ]$. The shady band in the left panel is obtained by changing $T$ inside Borel windows. There exist nonvanishing working regions of $T$ as long as ${\omega }_c\ge 3.0\mathrm{GeV}$, while the results for ${\omega }_c<3.0\mathrm{GeV}$ are also shown, for which cases we choose the Borel mass $T$ when the PC defined in Eq. (12) is around 10%. In the right panel, the short-dashed, solid, and long-dashed curves are obtained by setting ${\omega }_c=3.0$, 3.2, and 3.4 GeV, respectively.

Figure 4

Variations of $m_{{\mathrm{\Xi }}_c^{\prime }(1/2^{+})}$ (left) and $m_{{\mathrm{\Omega }}_c(1/2^{+})}$ (right) with respect to the Borel mass $T$, calculated using the charmed baryon multiplets $[{\mathrm{\Xi }}_c^{\prime }({\mathbf{6}}_F),1,1,\rho \rho ]$ and $[{\mathrm{\Omega }}_c({\mathbf{6}}_F),1,1,\rho \rho ]$. In the left figure, the short-dashed, solid, and long-dashed curves are obtained by setting ${\omega }_c=3.5$, 3.7, and 3.9 GeV, respectively. In the left figure, the short-dashed, solid, and long-dashed curves are obtained by setting ${\omega }_c=4.0$, 4.2, and 4.4 GeV, respectively.

Figure 5

Variations of $m_{[{\mathrm{\Sigma }}_c,3,1,\lambda \lambda ]}$ (left), $m_{[{\mathrm{\Xi }}_c^{\prime },3,1,\lambda \lambda ]}$ (middle), and $m_{[{\mathrm{\Omega }}_c,3,1,\lambda \lambda ]}$ (right) with respect to the threshold value ${\omega }_c$, calculated using the charmed baryon doublet $[{\mathbf{6}}_F,3,1,\lambda \lambda ]$. The shady bands in these figures are obtained by changing $T$ inside Borel windows. There exist Borel windows as long as ${\omega }_c([{\mathrm{\Sigma }}_c,3,1,\lambda \lambda ])\ge 3.1\mathrm{GeV}$ (left), ${\omega }_c([{\mathrm{\Xi }}_c^{\prime },3,1,\lambda \lambda ])\ge 3.2\mathrm{GeV}$ (left), and ${\omega }_c([{\mathrm{\Omega }}_c,3,1,\lambda \lambda ])\ge 3.5\mathrm{GeV}$ (right). Accordingly, we choose ${\omega }_c$ to be around 3.1, 3.3, and 3.5 GeV in the left, middle, and right panels, respectively.

Figure 6

Variations of $m_{{\mathrm{\Sigma }}_c(3/2^{+})}$ (left), $m_{{\mathrm{\Xi }}_c^{\prime }(3/2^{+})}$ (middle), and $m_{{\mathrm{\Omega }}_c(5/2^{+})}$ (right) with respect to the threshold value ${\omega }_c$, calculated using the charmed baryon multiplet $[{\mathbf{6}}_F,2,0,\rho \lambda ]$. The shady bands in these figures are obtained by changing $T$ inside Borel windows. There exist Borel windows as long as ${\omega }_c([{\mathrm{\Sigma }}_c,2,0,\rho \lambda ])\ge 2.8\mathrm{GeV}$ (left), ${\omega }_c([{\mathrm{\Xi }}_c^{\prime },2,0,\rho \lambda ])\ge 3.6\mathrm{GeV}$ (middle), and ${\omega }_c([{\mathrm{\Omega }}_c,2,0,\rho \lambda ])\ge 4.0\mathrm{GeV}$ (right). Accordingly, we choose ${\omega }_c$ to be around 3.0, 3.7, and 4.0 GeV in the left, middle, and right panels, respectively.