$P$-wave charmed baryons from QCD sum rules

We study the $P$-wave charmed baryons using the method of QCD sum rule in the framework of heavy quark effective theory. We consider systematically all possible baryon currents with a derivative for internal $\rho$- and $\lambda$-mode excitations. We have found a good working window for the currents corresponding to the $\rho$-mode excitations for ${\mathrm{\Lambda }}_c(2595)$, ${\mathrm{\Lambda }}_c(2625)$, ${\mathrm{\Xi }}_c(2790)$, and ${\mathrm{\Xi }}_c(2815)$ that complete two $SU(3)$ ${\overline{\mathbf{3}}}_F$ multiplets of $J^P=1/2^{-}$ and $3/2^{-}$, while the currents corresponding to the $\lambda$-mode excitations seem also consistent with the data. Our results also suggest that there are two ${\mathrm{\Sigma }}_c(2800)$ states of $J^P=1/2^{-}$ and $3/2^{-}$ whose mass splitting is $14\pm 7\mathrm{MeV}$, and two ${\mathrm{\Xi }}_c(2980)$ states whose mass splitting is $12\pm 7\mathrm{MeV}$. They have two ${\mathrm{\Omega }}_c$ partners of $J^P=1/2^{-}$ and $3/2^{-}$, whose masses are around $3.25\pm 0.20\mathrm{GeV}$ with mass splitting $10\pm 6\mathrm{MeV}$. All of them together complete two $SU(3)$ ${\mathbf{6}}_F$ multiplets of $J^P=1/2^{-}$ and $3/2^{-}$. They may also have $J^P=5/2^{-}$ partners. ${\mathrm{\Xi }}_c(3080)$ may be one of them, and the other two are ${\mathrm{\Sigma }}_c(5/2^{-})$ and ${\mathrm{\Omega }}_c(5/2^{-})$, whose masses are $85\pm 23\mathrm{MeV}$ and $50\pm 27\mathrm{MeV}$ larger.

Figure 1

The $SU(3)$ flavor multiplets of charmed baryons.

Figure 2

The notations for $P$-wave charmed baryons. ${\mathbf{6}}_F$ ($\mathbf{S}$) and ${\overline{\mathbf{3}}}_F$ ($\mathbf{A}$) denote the $SU(3)$ flavor representations. ${\overline{\mathbf{3}}}_C$ ($\mathbf{A}$) denotes the $SU(3)$ color representation. $s_l$ is the spin angular momentum of the two light quarks, and $j_l=l_{\lambda }\otimes l_{\rho }\otimes s_l$ is the total angular momentum of the two light quarks.

Figure 3

The variations of CVG and PC, as defined in Eqs. (32) and (34), with respect to the Borel mass $T$, when $J_{1/2,-,{\mathrm{\Xi }}_c,1,1,\rho }$ is used. We obtain the short-dashed, solid, and long-dashed curves by fixing ${\omega }_c=3.4$, 3.6, and 3.8 GeV, respectively.

Figure 4

The variations of ${\overline{\mathrm{\Lambda }}}_{{\mathrm{\Xi }}_c,1,1,\rho }$ (left) and $f_{{\mathrm{\Xi }}_c,1,1,\rho }$ (right) with respect to the Borel mass $T$, when $J_{1/2,-,{\mathrm{\Xi }}_c,1,1,\rho }$ is used. We use the working region $0.54\mathrm{GeV}<T<0.64\mathrm{GeV}$, and obtain the short-dashed, solid, and long-dashed curves by fixing ${\omega }_c=3.4$, 3.6, and 3.8 GeV, respectively.

Figure 5

The variations of $K_{{\mathrm{\Xi }}_c,1,1,\rho }$ (left) and ${\mathrm{\Sigma }}_{{\mathrm{\Xi }}_c,1,1,\rho }$ (right) with respect to the Borel mass $T$, when $J_{1/2,-,{\mathrm{\Xi }}_c,1,1,\rho }$ is used. We use the working region $0.54\mathrm{GeV}<T<0.64\mathrm{GeV}$, and obtain the short-dashed, solid, and long-dashed curves by fixing ${\omega }_c=3.4$, 3.6, and 3.8 GeV, respectively.

Figure 6

The variations of ${\overline{\mathrm{\Lambda }}}_{{\mathrm{\Xi }}_c^{\prime },1,0,\rho }$, $f_{{\mathrm{\Xi }}_c^{\prime },1,0,\rho }$, $K_{{\mathrm{\Xi }}_c^{\prime },1,0,\rho }$, and ${\mathrm{\Sigma }}_{{\mathrm{\Xi }}_c^{\prime },1,0,\rho }$ with respect to the Borel mass $T$, when $J_{1/2,-,{\mathrm{\Xi }}_c^{\prime },1,0,\rho }$ is used. We use the working region $0.52\mathrm{GeV}<T<0.70\mathrm{GeV}$, and obtain the short-dashed, solid, and long-dashed curves by fixing ${\omega }_c=3.7$, 3.9, and 4.1 GeV, respectively.

Figure 7

The variations of ${\overline{\mathrm{\Lambda }}}_{{\mathrm{\Xi }}_c^{\prime },2,1,\lambda }$, $f_{{\mathrm{\Xi }}_c^{\prime },2,1,\lambda }$, $K_{{\mathrm{\Xi }}_c^{\prime },2,1,\lambda }$, and ${\mathrm{\Sigma }}_{{\mathrm{\Xi }}_c^{\prime },2,1,\lambda }$ with respect to the Borel mass $T$, when $J_{3/2,-,{\mathrm{\Xi }}_c^{\prime },2,1,\lambda }$ is used. We work in the region $0.53\mathrm{GeV}<T<0.64\mathrm{GeV}$, and obtain the short-dashed, solid, and long-dashed curves by fixing ${\omega }_c=3.3$, 3.5, and 3.7 GeV, respectively.