QCD sum rule calculation for $P$-wave bottom baryons

We study the $P$-wave bottom baryons using the method of QCD sum rule and heavy quark effective theory. Our results suggest that ${\mathrm{\Lambda }}_b(5912{)}^0$ and ${\mathrm{\Lambda }}_b(5920{)}^0$ can be well described by the baryon doublet $[{\overline{\mathbf{3}}}_F,1,1,\rho ]$, and they belong to the $SU(3)$ ${\overline{\mathbf{3}}}_F$ multiplets of $J^P=1/2^{-}$ and $3/2^{-}$. Their $SU(3)$ flavor partners, ${\mathrm{\Xi }}_b(1/2^{-})$ and ${\mathrm{\Xi }}_b(3/2^{-})$, have masses $6.06\pm 0.13\mathrm{GeV}$ and $6.07\pm 0.13\mathrm{GeV}$, respectively, with mass splitting $9\pm 4\mathrm{MeV}$. The results obtained using baryon doublet $[{\overline{\mathbf{3}}}_F,1,0,\lambda ]$ are similar and also consistent with the experimental data. We also study the $SU(3)$ ${\mathbf{6}}_F$ multiplets by using the baryon multiplets $[{\mathbf{6}}_F,0,1,\lambda ]$, $[{\mathbf{6}}_F,1,0,\rho ]$ and $[{\mathbf{6}}_F,2,1,\lambda ]$, and our results suggest that the $P$-wave bottom baryons ${\mathrm{\Sigma }}_b$, ${\mathrm{\Xi }}_b^{\prime }$ and ${\mathrm{\Omega }}_b$ have (averaged) masses of about 6.0 GeV, 6.2 GeV and 6.4 GeV, respectively.

Figure 1

The notations for $P$-wave bottom baryons. See Fig. 2 of Ref. [58] and discussions therein for details.

Figure 2

Variations of $m_{{\mathrm{\Lambda }}_b(1/2^{-})}$ with respect to the Borel mass $T$ (top) and the threshold value ${\omega }_c$ (bottom), calculated using the bottom baryon doublet $[{\mathrm{\Lambda }}_b({\overline{\mathbf{3}}}_F),1,1,\rho ]$. In the top panel we take ${\omega }_c=3.1\mathrm{GeV}$, and the Borel window is $0.538\mathrm{Ge}\mathrm{V}<T<0.588\mathrm{GeV}$. In the bottom panel, the upper and lower bands are obtained by using $T_{\min }$ and $T_{\max }$, respectively. There are nonvanishing Borel windows as long as ${\omega }_c\ge 2.85\mathrm{GeV}$, but they quickly vanish around this point. The results for ${\omega }_c\le 2.85\mathrm{GeV}$ are also shown. For such cases we choose the Borel mass $T$ when the PC [pole contribution, defined in Eq. (34) of Ref. 58] is around 20%, and so this curve connects to the lower band obtained by using $T_{\max }$.

Figure 3

Variations of $\mathrm{\Delta }{m}_{[{\mathrm{\Lambda }}_b,1,1,\rho ]}$ with respect to the Borel mass $T$ (top) and the threshold value ${\omega }_c$ (bottom), calculated using the bottom baryon doublet $[{\mathrm{\Lambda }}_b({\overline{\mathbf{3}}}_F),1,1,\rho ]$. In the top panel we take ${\omega }_c=3.1\mathrm{GeV}$, and the Borel window is $0.538\mathrm{Ge}\mathrm{V}<T<0.588\mathrm{GeV}$.

Figure 4

Variations of $m_{{\mathrm{\Lambda }}_b(1/2^{-})}$ with respect to the Borel mass $T$ (top) and the threshold value ${\omega }_c$ (bottom), calculated using the bottom baryon doublet $[{\mathrm{\Lambda }}_b({\overline{\mathbf{3}}}_F),1,0,\lambda ]$. In the top panel we take ${\omega }_c=3.6\mathrm{GeV}$, and the Borel window is $0.597\mathrm{GeV}<T<0.684\mathrm{GeV}$ and 3.4 GeV.

Figure 5

Variations of $m_{{\mathrm{\Sigma }}_b(1/2^{-})}$ with respect to the Borel mass $T$ (top) and the threshold value ${\omega }_c$ (bottom), calculated using the bottom baryon singlet $[{\mathrm{\Sigma }}_b({\mathbf{6}}_F),0,1,\lambda ]$. In the top panel we take ${\omega }_c=3.3\mathrm{GeV}$, and the Borel window is $0.527\mathrm{GeV}<T<0.628\mathrm{GeV}$.

Figure 6

Variations of $m_{{\mathrm{\Sigma }}_b(1/2^{-})}$ with respect to the Borel mass $T$ (top) and the threshold value ${\omega }_c$ (bottom), calculated using the bottom baryon doublet $[{\mathrm{\Sigma }}_b({\mathbf{6}}_F),1,0,\rho ]$. In the top panel we take ${\omega }_c=3.3\mathrm{GeV}$, and the Borel window is $0.534\mathrm{GeV}<T<0.624\mathrm{GeV}$.

Figure 7

Variations of $m_{{\mathrm{\Sigma }}_b(3/2^{-})}$ with respect to the Borel mass $T$ (top) and the threshold value ${\omega }_c$ (bottom), calculated using the bottom baryon doublet $[{\mathrm{\Sigma }}_b({\mathbf{6}}_F),2,1,\lambda ]$. In the top panel we take ${\omega }_c=3.3\mathrm{GeV}$, and the Borel window is $0.537\mathrm{GeV}<T<0.620\mathrm{GeV}$.