Strong decays of charmed baryons

TL;DR

This paper analyzes the strong decay patterns of various excited charmed baryons using the $^3P_0$ model, comparing predictions with experimental data to infer their internal structures and quantum numbers.

Contribution

It applies the $^3P_0$ model to study decay patterns of excited charmed baryons and interprets recent experimental observations.

Findings

Decay patterns match experimental data for certain states.

Possible quantum numbers and structures are proposed for observed baryons.

Provides theoretical support for identifying charmed baryon states.

Abstract

There has been important experimental progress in the sector of heavy baryons in the past several years. We study the strong decays of the S-wave, P-wave, D-wave and radially excited charmed baryons using the $^3P_0$ model. After comparing the calculated decay pattern and total width with the available data, we discuss the possible internal structure and quantum numbers of those charmed baryons observed recently.

Figures

Tables

Table 1. Table 1: A summary of recently observed charmed baryons by Babar and Belle collaborations.

State	Mass and Width (MeV)	Decay channels in experiments	Other information
	$2881.5\pm 0.3$, cleo-2880	${\Lambda }_c{\pi }^{+}{\pi }^{-}$
	$2881.9\pm 0.1\pm 0.5$ , $5.8\pm 1.5\pm 1.1$ babar-2880	$D^{0}p$	$J^P$ favors ${\frac{5}{2}}^{+}$ belle-2880 ,
${\Lambda }_c{\left(2880\right)}^{+}$	$2881.2\pm {0.2}_{-0.3}^{+0.4}$, ${5.5}_{-0.5}^{+0.7}\pm 0.4$ belle-2880	${\Sigma }_c^{\star 0,++}\left(2520\right){\pi }^{+,-}$	$\frac{\Gamma \left({\Sigma }_c^{\star }\left(2520\right){\pi }^{\pm }\right)}{\Gamma \left({\Sigma }_c\left(2455\right){\pi }^{\pm }\right)}=0.225\pm 0.062\pm 0.025$ belle-2880
	$2939.\pm 1.3\pm 1.0$, $17.5\pm 5.2\pm 5.9$ babar-2880	$D^{0}p$
${\Lambda }_c{\left(2940\right)}^{+}$	$2937.9\pm {1.0}_{-0.4}^{+1.8}$, $10\pm 4\pm 5$ belle-2880	${\Sigma }_c{\left(2455\right)}^{0,++}{\pi }^{+,-}$	-
	$2967.1\pm 1.9\pm 1.0$, $23.6\pm 2.8\pm 1.3$ babar-2980-3077	${\Lambda }_c^{+}{K}^{-}{\pi }^{+}$
${\Xi }_c{\left(2980\right)}^{+}$	$2978.5\pm 2.1\pm 2.0$, $43.5\pm 7.5\pm 7.0$ belle-2980-3077	${\Lambda }_c^{+}{K}^{-}{\pi }^{+}$	-
${\Xi }_c{\left(2980\right)}^0$	$2977.1\pm 8.8\pm 3.5$, $43.5$ belle-2980-3077	${\Lambda }_c^{+}{K}_S^0{\pi }^{-}$	-
	$3076.4\pm 0.7\pm 0.3$, $6.2\pm 1.6\pm 0.5$ babar-2980-3077	${\Lambda }_c^{+}{K}^{-}{\pi }^{+}$
${\Xi }_c{\left(3077\right)}^{+}$	$3076.7\pm 0.9\pm 0.5$, $6.2\pm 1.2\pm 0.8$ belle-2980-3077	${\Lambda }_c^{+}{K}^{-}{\pi }^{+}$	-
${\Xi }_c{\left(3077\right)}^0$	$3082.8\pm 1.8\pm 1.5$, $5.2\pm 3.1\pm 1.8$ belle-2980-3077	${\Lambda }_c^{+}{K}_S^0{\pi }^{-}$	-
${\Omega }_c{\left(2768\right)}^0$	$2768.3\pm 3.0$ babar-omega	${\Omega }_c^0\gamma$	$J^P={\frac{3}{2}}^{+}$

Table 2. Table 2: The strong decay widths of S-wave charmed baryons Σ c + + , + , 0 ​ ( 2455 ) subscript superscript Σ absent 0 𝑐 2455 \Sigma^{++,+,0}_{c}(2455) , Σ c ∗ ⁣ + + , + , 0 ​ ( 2520 ) subscript superscript Σ absent 0 𝑐 2520 \Sigma^{*++,+,0}_{c}(2520) and Ξ c ∗ + , 0 ​ ( 2645 ) subscript superscript Ξ absent 0 𝑐 2645 \Xi^{*+,0}_{c}(2645) . Here all results are in units of MeV.

	$J^P$	Channel	Width	Total width (Exp) PDG
${\Sigma }_c^{++}\left(2455\right)$	${\frac{1}{2}}^{+}$	${\Lambda }_c^{+}{\pi }^{+}$	$1.24$	$2.23\pm 0.30$
${\Sigma }_c^{+}\left(2455\right)$	${\frac{1}{2}}^{+}$	${\Lambda }_c^{+}{\pi }^0$	$1.40$
${\Sigma }_c^0\left(2455\right)$	${\frac{1}{2}}^{+}$	${\Lambda }_c^{+}{\pi }^{-}$	$1.24$	$2.2\pm 0.40$
${\Sigma }_c^{\ast ++}\left(2520\right)$	${\frac{3}{2}}^{+}$	${\Lambda }_c^{+}{\pi }^{+}$	$11.9$	$14.9\pm 1.9$
${\Sigma }_c^{\ast +}\left(2520\right)$	${\frac{3}{2}}^{+}$	${\Lambda }_c^{+}{\pi }^0$	$12.1$
${\Sigma }_c^{\ast 0}\left(2520\right)$	${\frac{3}{2}}^{+}$	${\Lambda }_c^{+}{\pi }^{-}$	$11.9$	$16.1\pm 2.1$
${\Xi }_c^{\ast +}\left(2645\right)$	${\frac{3}{2}}^{+}$	${\Xi }_c^{+}{\pi }^0$	$0.64$
${\Xi }_c^{\ast +}\left(2645\right)$	${\frac{3}{2}}^{+}$	${\Xi }_c^0{\pi }^{+}$	$0.49$
${\Xi }_c^{\ast 0}\left(2645\right)$	${\frac{3}{2}}^{+}$	${\Xi }_c^{+}{\pi }^{-}$	$0.54$
${\Xi }_c^{\ast 0}\left(2645\right)$	${\frac{3}{2}}^{+}$	${\Xi }_c^0{\pi }^0$	$0.54$

Table 3. Table 3: The strong decay widths of S-wave bottom baryons Σ b subscript Σ 𝑏 \Sigma_{b} , Σ b ∗ subscript superscript Σ 𝑏 \Sigma^{*}_{b} , Ξ b ′ subscript superscript Ξ ′ 𝑏 \Xi^{\prime}_{b} and Ξ b ∗ subscript superscript Ξ 𝑏 \Xi^{*}_{b} . Here all results are in units of MeV.

	$J^P$	Channel	Width	Experimental results CDF
${\Sigma }_b^{+}$	${\frac{1}{2}}^{+}$	${\Lambda }_b^0{\pi }^{+}$	$3.5$
${\Sigma }_b^{-}$	${\frac{1}{2}}^{+}$	${\Lambda }_b^0{\pi }^{-}$	$4.7$	$\sim 8$
${\Sigma }_b^{\ast +}$	${\frac{3}{2}}^{+}$	${\Lambda }_b^0{\pi }^{+}$	$7.5$
${\Sigma }_b^{\ast -}$	${\frac{3}{2}}^{+}$	${\Lambda }_b^0{\pi }^{-}$	$9.2$	$\sim 15$
${\Xi }_b^{\prime }$	${\frac{1}{2}}^{+}$	${\Xi }_b\pi$	$0.10$	-
${\Xi }_b^{\ast }$	${\frac{3}{2}}^{+}$	${\Xi }_b\pi$	$0.85$	-

Table 4. Table 4: The decay widths of P-wave charmed baryons Λ c + ​ ( 2593 , 2625 ) subscript superscript Λ 𝑐 2593 2625 \Lambda^{+}_{c}(2593,2625) and Ξ c + , 0 ​ ( 2790 , 2815 ) superscript subscript Ξ 𝑐 0 2790 2815 \Xi_{c}^{+,0}(2790,2815) with the fixed structure and quantum number assignments. Here all results are in units of MeV.

	Assignment	Channel	$\Gamma$	${\Gamma }_{Exp}$ PDG
		${\Sigma }_c^{++}{\pi }^{-}$	$3.4$
		${\Sigma }_c^{+}{\pi }^0$	$6.4$
${\Lambda }_c^{+}\left(2593\right)$	${\Lambda }_{c1}\left({\frac{1}{2}}^{-}\right)$	${\Sigma }_c^0{\pi }^{+}$	$3.4$	${3.6}_{-1.3}^{+2.0}$
		${\Sigma }_c^{++}{\pi }^{-}$	$1.9\times {10}^{-3}$
		${\Sigma }_c^{+}{\pi }^0$	$2.6\times {10}^{-3}$
${\Lambda }_c^{+}\left(2625\right)$	${\Lambda }_{c1}\left({\frac{3}{2}}^{-}\right)$	${\Sigma }_c^0{\pi }^{+}$	$1.9\times {10}^{-3}$
		${\Xi }_c^{\prime +}{\pi }^0$	$5.0$
${\Xi }_c^{+}\left(2790\right)$	${\Xi }_{c1}\left({\frac{1}{2}}^{-}\right)$	${\Xi }_c^{\prime 0}{\pi }^{+}$	$4.9$
		${\Xi }_c^{\prime +}{\pi }^{-}$	$5.2$
${\Xi }_c^0\left(2790\right)$	${\Xi }_{c1}\left({\frac{1}{2}}^{-}\right)$	${\Xi }_c^{\prime 0}{\pi }^0$	$5.1$
		${\Xi }_c^{\star +}{\pi }^0$	$2.7$
${\Xi }_c^{+}\left(2815\right)$	${\Xi }_{c1}\left({\frac{3}{2}}^{-}\right)$	${\Xi }_c^{\star 0}{\pi }^{+}$	$2.6$
		${\Xi }_c^{\star +}{\pi }^{-}$	$2.7$
${\Xi }_c^0\left(2815\right)$	${\Xi }_{c1}\left({\frac{3}{2}}^{-}\right)$	${\Xi }_c^{\star 0}{\pi }^0$	$2.8$

Table 5. Table 5: The decay widths of Σ c + + ​ ( 2800 ) subscript superscript Σ absent 𝑐 2800 \Sigma^{++}_{c}(2800) in different P-wave charmed baryons assignments. ℛ = Σ c + , + + ​ π + , 0 / Σ c ⋆ + , + + ​ π + , 0 ℛ subscript superscript Σ absent 𝑐 superscript 𝜋 0 subscript superscript Σ ⋆ absent absent 𝑐 superscript 𝜋 0 \mathcal{R}={\Sigma^{+,++}_{c}\pi^{+,0}}/{\Sigma^{\star+,++}_{c}\pi^{+,0}} . The total width of Σ c + + ​ ( 2800 ) subscript superscript Σ absent 𝑐 2800 \Sigma^{++}_{c}(2800) is 75 − 17 + 22 subscript superscript 75 22 17 75^{+22}_{-17} MeV PDG . Here all results are in units of MeV.

Assignment	${\Lambda }_c^{+}{\pi }^{+}$	${\Sigma }_c^{+,++}{\pi }^{+,0}$	${\Sigma }_c^{\star +,++}{\pi }^{+,0}$	$ℛ$
${\Sigma }_{c0}\left({\frac{1}{2}}^{-}\right)$	$307$	$0.0$	$0.0$	-
${\Sigma }_{c1}\left({\frac{1}{2}}^{-}\right)$	$0.0$	$296$	$0.4$	740
${\Sigma }_{c1}\left({\frac{3}{2}}^{-}\right)$	$0.0$	$0.7$	$220$	$3\times {10}^{-3}$
${\Sigma }_{c2}\left({\frac{3}{2}}^{-}\right)$	$8.1$	$1.3$	$0.3$	4.3
${\Sigma }_{c2}\left({\frac{5}{2}}^{-}\right)$	$8.1$	$0.6$	$0.5$	1.2
${\tilde{\Sigma }}_{c1}\left({\frac{1}{2}}^{-}\right)$	$0.0$	$75$	$69$	1.1
${\tilde{\Sigma }}_{c1}\left({\frac{3}{2}}^{-}\right)$	$0.0$	$75$	$69$	1.1

Table 6. Table 6: The decay widths of Λ c + ​ ( 2880 ) subscript superscript Λ 𝑐 2880 \Lambda^{+}_{c}(2880) with different D-wave assignments. All results are in units of MeV.

Assignment	${\Sigma }_c^{0,+,++}{\pi }^{+,0,-}$	${\Sigma }_c^{\star 0,+,++}{\pi }^{+,0,-}$	$\frac{\Gamma \left({\Sigma }_c^{\star }{\pi }^{\pm }\right)}{\Gamma \left({\Sigma }_c{\pi }^{\pm }\right)}$	$D^{0}p$	Remark
${\Lambda }_{c2}\left({\frac{3}{2}}^{+}\right)$	$7.8$	$0.9$	$0.11$	$0.0$	$\times$
${\Lambda }_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.06$	$5.34$	$89$	$0.0$	$\times$
${\hat{\Lambda }}_{c2}\left({\frac{3}{2}}^{+}\right)$	$78.3$	$59.1$	$0.75$	$0.0$	$\times$
${\hat{\Lambda }}_{c2}\left({\frac{5}{2}}^{+}\right)$	$78.3$	$59.1$	$0.75$	$0.0$	$\times$
${\check{\Lambda }}_{c1}^0\left({\frac{1}{2}}^{+}\right)$	$0.9$	$2.3$	$2.6$	$2.3$	$\times$
${\check{\Lambda }}_{c1}^0\left({\frac{3}{2}}^{+}\right)$	$0.22$	$6.0$	$27$	$2.3$	$\times$
${\check{\Lambda }}_{c0}^1\left({\frac{1}{2}}^{+}\right)$	$132$	$144$	$1.1$	$0.0$	$\times$
${\check{\Lambda }}_{c1}^1\left({\frac{1}{2}}^{+}\right)$	$66.3$	$18.0$	$0.27$	$150$	$\times$
${\check{\Lambda }}_{c1}^1\left({\frac{3}{2}}^{+}\right)$	$16.5$	$45.0$	$2.7$	$150$	$\times$
${\check{\Lambda }}_{c2}^1\left({\frac{3}{2}}^{+}\right)$	$82.8$	$9.0$	$0.10$	$0.0$	$\times$
${\check{\Lambda }}_{c2}^1\left({\frac{5}{2}}^{+}\right)$	$0.0$	$54.1$	$-$	$0.0$	$\times$
${\check{\Lambda }}_{c1}^2\left({\frac{1}{2}}^{+}\right)$	$25.7$	$8.1$	$0.32$	$64$	$\times$
${\check{\Lambda }}_{c1}^2\left({\frac{3}{2}}^{+}\right)$	$6.5$	$20.4$	$3.1$	$64$	$\times$
${\check{\Lambda }}_{c2}^2\left({\frac{3}{2}}^{+}\right)$	$57.9$	$14.2$	$0.24$	$0.0$	$\times$
${\check{\Lambda }}_{c2}^2\left({\frac{5}{2}}^{+}\right)$	$9.4$	$47.1$	$5.0$	$0.0$	$\times$
${\check{\Lambda }}_{c3}^2\left({\frac{5}{2}}^{+}\right)$	$10.8$	$5.5$	$0.51$	$12$
${\check{\Lambda }}_{c3}^2\left({\frac{7}{2}}^{+}\right)$	$6.1$	$7.4$	$1.2$	$12$	$\times$

Table 7. Table 7: The decay widths of Λ c + ​ ( 2940 ) subscript superscript Λ 𝑐 2940 \Lambda^{+}_{c}(2940) with different D-wave assignments. Here all results are in units of MeV.

Assignment	${\Sigma }_c^{0,+,++}{\pi }^{+,0,-}$	${\Sigma }_c^{\star 0,+,++}{\pi }^{+,0,-}$	$\frac{\Gamma \left({\Sigma }_c^{\star }{\pi }^{\pm }\right)}{\Gamma \left({\Sigma }_c{\pi }^{\pm }\right)}$	$D^{0}p$	Remark
${\Lambda }_{c2}\left({\frac{3}{2}}^{+}\right)$	$11.7$	$9.1$	$0.77$	$0.0$	$\times$
${\Lambda }_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.2$	$9.1$	$46$	$0.0$	$\times$
${\hat{\Lambda }}_{c2}\left({\frac{3}{2}}^{+}\right)$	$170$	$150$	$0.88$	$0.0$	$\times$
${\hat{\Lambda }}_{c2}\left({\frac{5}{2}}^{+}\right)$	$170$	$150$	$0.88$	$0.0$	$\times$
${\check{\Lambda }}_{c1}^0\left({\frac{1}{2}}^{+}\right)$	$2.2$	$0.5$	$0.23$	$11$
${\check{\Lambda }}_{c1}^0\left({\frac{3}{2}}^{+}\right)$	$0.6$	$1.4$	$2.3$	$11$
${\check{\Lambda }}_{c0}^1\left({\frac{1}{2}}^{+}\right)$	$212$	$259$	$1.2$	$0.0$	$\times$
${\check{\Lambda }}_{c1}^1\left({\frac{1}{2}}^{+}\right)$	$106$	$32.4$	$0.31$	$340$	$\times$
${\check{\Lambda }}_{c1}^1\left({\frac{3}{2}}^{+}\right)$	$26.5$	$81.0$	$3.1$	$340$	$\times$
${\check{\Lambda }}_{c2}^1\left({\frac{3}{2}}^{+}\right)$	$142$	$16.2$	$0.11$	$0.0$	$\times$
${\check{\Lambda }}_{c2}^1\left({\frac{5}{2}}^{+}\right)$	$0.0$	$97.0$	$-$	$0.0$	$\times$
${\check{\Lambda }}_{c1}^2\left({\frac{1}{2}}^{+}\right)$	$34.5$	$12.6$	$0.37$	$95$	$\times$
${\check{\Lambda }}_{c1}^2\left({\frac{3}{2}}^{+}\right)$	$8.6$	$31.7$	$3.7$	$95$	$\times$
${\check{\Lambda }}_{c2}^2\left({\frac{3}{2}}^{+}\right)$	$77.7$	$27.7$	$0.36$	$0.0$	$\times$
${\check{\Lambda }}_{c2}^2\left({\frac{5}{2}}^{+}\right)$	$19.5$	$75.6$	$3.9$	$0.0$	$\times$
${\check{\Lambda }}_{c3}^2\left({\frac{5}{2}}^{+}\right)$	$22.2$	$12.9$	$0.58$	$49$	$\times$
${\check{\Lambda }}_{c3}^2\left({\frac{7}{2}}^{+}\right)$	$12.4$	$17.5$	$1.4$	$49$	$\times$

Table 8. Table 8: The decay widths of Ξ c + ​ ( 2980 ) subscript superscript Ξ 𝑐 2980 \Xi^{+}_{c}(2980) with different D-wave assignments. Here all results are in units of MeV.

Assignment	${\Xi }_c^0{\pi }^{+}$	${\Xi }_c^{\prime 0}{\pi }^{+}$	${\Xi }_c^{\star 0}{\pi }^{+}$	${\Sigma }_c^{++}{k}^{-}$	${\Lambda }_c^{+}{\overline{k}}^0$	Remark
${\Xi }_{c2}\left({\frac{3}{2}}^{+}\right)$	$0.0$	$1.1$	$0.11$	$0.37$	$0.0$	$\times$
${\Xi }_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.0$	$0.12\times {10}^{-2}$	$0.67$	$0.11\times {10}^{-3}$	$0.0$	$\times$
${\Xi }_{c1}^{\prime }\left({\frac{1}{2}}^{+}\right)$	$4.4$	$0.72$	$0.18$	$0.25$	$5.3$
${\Xi }_{c1}^{\prime }\left({\frac{3}{2}}^{+}\right)$	$4.4$	$0.18$	$0.46$	$0.062$	$5.3$
${\Xi }_{c2}^{\prime }\left({\frac{3}{2}}^{+}\right)$	$0.0$	$0.16$	$0.17$	$0.56$	$0.0$	$\times$
${\Xi }_{c2}^{\prime }\left({\frac{5}{2}}^{+}\right)$	$0.0$	$0.47\times {10}^{-2}$	$1.0$	$0.71\times {10}^{-4}$	$0.0$	$\times$
${\Xi }_{c3}^{\prime }\left({\frac{5}{2}}^{+}\right)$	$0.054$	$0.53\times {10}^{-2}$	$0.14\times {10}^{-2}$	$0.82\times {10}^{-4}$	$0.053$	$\times$
${\Xi }_{c3}^{\prime }\left({\frac{7}{2}}^{+}\right)$	$0.054$	$0.30\times {10}^{-2}$	$0.19\times {10}^{-2}$	$0.46\times {10}^{-4}$	$0.053$	$\times$
${\hat{\Xi }}_{c2}\left({\frac{3}{2}}^{+}\right)$	$0.0$	$9.5$	$6.1$	$0.61$	$0.0$
${\hat{\Xi }}_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.0$	$9.5$	$6.1$	$0.61$	$0.0$
${\widehat{{\Xi }^{\prime }}}_{c1}\left({\frac{1}{2}}^{+}\right)$	$74$	$6.3$	$1.0$	$0.40$	$78$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c1}\left({\frac{3}{2}}^{+}\right)$	$74$	$1.6$	$2.5$	$0.10$	$78$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c2}\left({\frac{3}{2}}^{+}\right)$	$0.0$	$14$	$4.5$	$0.91$	$0.0$
${\widehat{{\Xi }^{\prime }}}_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.0$	$6.3$	$7.1$	$0.40$	$0.0$
${\widehat{{\Xi }^{\prime }}}_{c3}\left({\frac{5}{2}}^{+}\right)$	$48$	$7.2$	$2.9$	$0.46$	$50$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c3}\left({\frac{7}{2}}^{+}\right)$	$48$	$4.1$	$3.9$	$0.26$	$50$	$\times$
${\check{{\Xi }^{\prime }}}_{c0}^0\left({\frac{1}{2}}^{+}\right)$	$0.0$	$0.30$	$1.4$	$1.3$	$0.0$	$\times$
${\check{\Xi }}_{c1}^0\left({\frac{1}{2}}^{+}\right)$	$1.0$	$0.40$	$0.46$	$1.7$	$0.46$	$\times$
${\check{\Xi }}_{c1}^0\left({\frac{3}{2}}^{+}\right)$	$1.0$	$0.10$	$1.2$	$0.43$	$0.46$	$\times$
${\check{{\Xi }^{\prime }}}_{c1}^1\left({\frac{1}{2}}^{+}\right)$	$0.0$	$18$	$4.4$	$5.5$	$0.0$
${\check{{\Xi }^{\prime }}}_{c1}^1\left({\frac{3}{2}}^{+}\right)$	$0.0$	$4.5$	$11$	$1.4$	$0.0$
${\check{\Xi }}_{c0}^1\left({\frac{1}{2}}^{+}\right)$	$0.0$	$18$	$18$	$5.5$	$0.0$
${\check{\Xi }}_{c1}^1\left({\frac{1}{2}}^{+}\right)$	$62$	$9.1$	$2.2$	$2.8$	$72$	$\times$
${\check{\Xi }}_{c1}^1\left({\frac{3}{2}}^{+}\right)$	$62$	$2.3$	$5.5$	$0.69$	$72$	$\times$
${\check{\Xi }}_{c2}^1\left({\frac{3}{2}}^{+}\right)$	$0.0$	$11$	$1.1$	$0.34$	$0.0$	$\times$
${\check{\Xi }}_{c2}^1\left({\frac{5}{2}}^{+}\right)$	$0.0$	$0.0$	$6.6$	$0.0$	$0.0$	$\times$
${\check{{\Xi }^{\prime }}}_{c2}^2\left({\frac{3}{2}}^{+}\right)$	$0.0$	$5.6$	$1.8$	$2.4$	$0.0$
${\check{{\Xi }^{\prime }}}_{c2}^2\left({\frac{5}{2}}^{+}\right)$	$0.0$	$1.7$	$4.32$	$0.24$	$0.0$
${\check{\Xi }}_{c1}^2\left({\frac{1}{2}}^{+}\right)$	$19$	$3.7$	$1.1$	$1.6$	$23$
${\check{\Xi }}_{c1}^2\left({\frac{3}{2}}^{+}\right)$	$19$	$0.93$	$2.6$	$0.40$	$23$
${\check{\Xi }}_{c2}^2\left({\frac{3}{2}}^{+}\right)$	$0.0$	$8.4$	$1.7$	$0.36$	$0.0$
${\check{\Xi }}_{c2}^2\left({\frac{5}{2}}^{+}\right)$	$0.0$	$1.2$	$6.0$	$0.16$	$0.0$
${\check{\Xi }}_{c3}^2\left({\frac{5}{2}}^{+}\right)$	$8.1$	$1.3$	$60$	$0.19$	$8.7$	$\times$
${\check{\Xi }}_{c3}^2\left({\frac{7}{2}}^{+}\right)$	$8.1$	$0.75$	$0.81$	$0.10$	$8.7$

Table 9. Table 9: The decay widths of Ξ c + ​ ( 3077 ) subscript superscript Ξ 𝑐 3077 \Xi^{+}_{c}(3077) with different D-wave assignments. Here all results are in units of MeV.

Assignment	${\Xi }_c^0{\pi }^{+}$	${\Xi }_c^{\prime 0}{\pi }^{+}$	${\Xi }_c^{\star 0}{\pi }^{+}$	${\Sigma }_c^{++}{k}^{-}$	${\Sigma }_c^{++}{k}^{-}$	${\Lambda }_c^{+}{\overline{k}}^0$	$D^{+}\Lambda$	Remark
${\Xi }_{c2}\left({\frac{3}{2}}^{+}\right)$	$0.0$	$2.1$	$0.30$	$0.73$	$0.054$	$0.0$	$0.0$
${\Xi }_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.0$	$0.037$	$1.7$	$0.42\times {10}^{-2}$	$0.32$	$0.0$	$0.0$
${\Xi }_{c1}^{\prime }\left({\frac{1}{2}}^{+}\right)$	$7.0$	$1.4$	$0.46$	$0.49$	$0.089$	$4.4$	$3.2$
${\Xi }_{c1}^{\prime }\left({\frac{3}{2}}^{+}\right)$	$7.0$	$0.36$	$1.1$	$0.12$	$0.22$	$4.4$	$3.2$
${\Xi }_{c2}^{\prime }\left({\frac{3}{2}}^{+}\right)$	$0.0$	$3.2$	$0.43$	1.1	$0.081$	$0.0$	$0.0$
${\Xi }_{c2}^{\prime }\left({\frac{5}{2}}^{+}\right)$	$0.0$	$0.025\times {10}^{-2}$	$2.5$	$0.28\times {10}^{-2}$	$0.48$	$0.0$	$0.0$	$\times$
${\Xi }_{c3}^{\prime }\left({\frac{5}{2}}^{+}\right)$	$0.19$	$0.029$	$0.012$	$0.32\times {10}^{-2}$	$0.32\times {10}^{-3}$	$0.12$	$0.026$	$\times$
${\Xi }_{c3}^{\prime }\left({\frac{7}{2}}^{+}\right)$	$0.19$	$0.016\times {10}^{-3}$	$0.016$	$0.18\times {10}^{-2}$	$0.44\times {10}^{-3}$	$0.12$	$0.026$	$\times$
${\hat{\Xi }}_{c2}\left({\frac{3}{2}}^{+}\right)$	$0.0$	$34$	$29$	$6.0$	$2.0$	$0.0$	$0.0$	$\times$
${\hat{\Xi }}_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.0$	$34$	$29$	$6.0$	$2.0$	$0.0$	$0.0$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c1}\left({\frac{1}{2}}^{+}\right)$	$201$	$23$	$4.8$	$4.0$	$0.33$	$130$	$38$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c1}\left({\frac{3}{2}}^{+}\right)$	$201$	$5.7$	$12$	$1.0$	$0.83$	$130$	$38$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c2}\left({\frac{3}{2}}^{+}\right)$	$0.0$	$51$	$22$	$8.9$	$1.5$	$0.0$	$0.0$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c2}\left({\frac{5}{2}}^{+}\right)$	$0.0$	$23$	$34$	$4.0$	$2.3$	$0.0$	$0.0$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c3}\left({\frac{5}{2}}^{+}\right)$	$129$	$26$	$14$	$4.5$	$0.94$	$84$	$25$	$\times$
${\widehat{{\Xi }^{\prime }}}_{c3}\left({\frac{7}{2}}^{+}\right)$	$129$	$15$	$19$	$2.6$	$0.13$	$84$	$25$	$\times$
${\check{{\Xi }^{\prime }}}_{c0}^0\left({\frac{1}{2}}^{+}\right)$	$0.0$	$0.69$	$0.13$	$0.29$	$1.2$	$0.0$	$0.0$
${\check{\Xi }}_{c1}^0\left({\frac{1}{2}}^{+}\right)$	$15$	$0.92$	$0.044$	$0.39$	$0.38$	$11$	$0.64\times {10}^{-3}$	$\times$
${\check{\Xi }}_{c1}^0\left({\frac{3}{2}}^{+}\right)$	$15$	$0.23$	$0.11$	$0.096$	$0.96$	$11$	$0.64\times {10}^{-3}$	$\times$
${\check{{\Xi }^{\prime }}}_{c1}^1\left({\frac{1}{2}}^{+}\right)$	$0.0$	$39$	$12$	$12$	$0.21$	$0.0$	$0.0$	$\times$
${\check{{\Xi }^{\prime }}}_{c1}^1\left({\frac{3}{2}}^{+}\right)$	$0.0$	$9.9$	$30$	$3.0$	$5.2$	$0.0$	$0.0$	$\times$
${\check{\Xi }}_{c0}^1\left({\frac{1}{2}}^{+}\right)$	$0.0$	$39$	$47$	$12$	$8.3$	$0.0$	$0.0$	$\times$
${\check{\Xi }}_{c1}^1\left({\frac{1}{2}}^{+}\right)$	$110$	$20$	$5.9$	$6.1$	$1.0$	$69$	$42$	$\times$
${\check{\Xi }}_{c1}^1\left({\frac{3}{2}}^{+}\right)$	$110$	$5.0$	$15$	$1.5$	$2.6$	$69$	$42$	$\times$
${\check{\Xi }}_{c2}^1\left({\frac{3}{2}}^{+}\right)$	$0.0$	$25$	$3.0$	$7.6$	$0.52$	$0.0$	$0.0$	$\times$
${\check{\Xi }}_{c2}^1\left({\frac{5}{2}}^{+}\right)$	$0.0$	$0.0$	$18$	$0.0$	$3.1$	$0.0$	$0.0$	$\times$
${\check{{\Xi }^{\prime }}}_{c2}^2\left({\frac{3}{2}}^{+}\right)$	$0.0$	$9.2$	$6.0$	$3.9$	$0.75$	$0.0$	$0.0$
${\check{{\Xi }^{\prime }}}_{c2}^2\left({\frac{5}{2}}^{+}\right)$	$0.0$	$5.8$	$10$	$1.1$	$2.1$	$0.0$	$0.0$
${\check{\Xi }}_{c1}^2\left({\frac{1}{2}}^{+}\right)$	$22$	$6.1$	$2.3$	$2.6$	$0.54$	$14$	$15$	$\times$
${\check{\Xi }}_{c1}^2\left({\frac{3}{2}}^{+}\right)$	$22$	$1.5$	$5.6$	$0.64$	$1.3$	$14$	$15$	$\times$
${\check{\Xi }}_{c2}^2\left({\frac{3}{2}}^{+}\right)$	$0.0$	$14$	$5.2$	$5.8$	$0.77$	$0.0$	$0.0$	$\times$
${\check{\Xi }}_{c2}^2\left({\frac{5}{2}}^{+}\right)$	$0.0$	$3.9$	$14$	$0.74$	$3.0$	$0.0$	$0.0$	$\times$
${\check{\Xi }}_{c3}^2\left({\frac{5}{2}}^{+}\right)$	$21$	$4.4$	$2.5$	$0.85$	$0.23$	$14$	$4.3$	$\times$
${\check{\Xi }}_{c3}^2\left({\frac{7}{2}}^{+}\right)$	$21$	$2.5$	$3.4$	$0.48$	$0.31$	$14$	$4.3$	$\times$

Equations

$$T$$

$$|A(n_A\text{mbox}{}^{2S_A+1}{L}_A\text{mbox}{}_{J_A{M}_{J_A}})({\text{P}}_A)⟩$$

$$|B(n_B\text{mbox}{}^{2S_B+1}{L}_B\text{mbox}{}_{J_B{M}_{J_B}})({\text{P}}_B)⟩$$

$$⟨A({\text{P}}_A)|A({\text{P}}_A^{\prime })⟩$$

$$⟨B({\text{P}}_B)|B({\text{P}}_B^{\prime })⟩$$

$$⟨f|S|i⟩$$

$${\mathcal{M}}^{M_{J_A}{M}_{J_B}{M}_{J_C}}(A\to BC)$$

$$I_{M_{L_B},M_{L_C}}^{M_{L_A},m}(\text{p})$$

$$\displaystyle\Gamma=\pi^{2}\frac{{|\textbf{p}|}}{M_{A}^{2}}\frac{s}{2J_{A}+1}\sum_{M_{J_{A}},M_{J_{B}},M_{J_{C}}}\big{|}\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}\big{|}^{2},$$

$$\mathcal{A}(q_1,q_2,c_3)+\mathcal{P}(q_4,{\bar{q}}_5)\to \mathcal{B}(q_2,q_4,c_3)+\mathcal{C}(q_1,{\bar{q}}_5),$$

$$\mathcal{A}(q_1,q_2,c_3)+\mathcal{P}(q_4,{\bar{q}}_5)\to \mathcal{B}(q_1,q_4,c_3)+\mathcal{C}(q_2,{\bar{q}}_5),$$

$$\mathcal{A}(q_1,q_2,c_3)+\mathcal{P}(q_4,{\bar{q}}_5)\to \mathcal{B}(q_1,q_2,q_4)+\mathcal{C}(c_3,{\bar{q}}_5)$$

$$M^{M_{J_A}{M}_{J_B}{M}_{J_C}}$$

$$=-2\gamma \sqrt{8E_A{E}_B{E}_C}\sum _{M_{{\rho }_A}}\sum _{M_{L_A}}\sum _{M_{{\rho }_B}}\sum _{M_{L_B}}\sum _{m_1,m_3,m_4,m}$$

$$\times ⟨J_{12}{M}_{12};s_3{m}_3|J_A{M}_{J_A}⟩⟨l_{\rho A}{m}_{\rho A};l_{\lambda A}{m}_{\lambda A}|L_A{M}_{L_A}⟩⟨L_A{M}_{L_A};S_{12}{m}_{12}|J_{12}{M}_{12}⟩$$

$$\times ⟨s_1{m}_1;s_2{m}_2|S_{12}{m}_{12}⟩⟨J_{14}{M}_{14};s_3{m}_3|J_B{M}_{J_B}⟩⟨l_{\rho B}{m}_{\rho B};l_{\lambda B}{m}_{\lambda B}|L_B{M}_{L_B}⟩$$

$$\times ⟨L_B{M}_{L_B};S_{14}{m}_{14}|J_{14}{M}_{14}⟩⟨s_1{m}_1;s_4{m}_4|S_{14}{m}_{14}⟩⟨1m;1-m|00⟩⟨s_4{m}_4;s_5{m}_5|1-m⟩$$

$$\times ⟨L_C{M}_{L_C};S_C{M}_C|J_C{M}_{J_C}⟩⟨s_2{m}_2;s_5{m}_5|S_C{M}_C⟩\times ⟨{\varphi }_B^{1,4,3}{\varphi }_C^{2,5}|{\varphi }_0^{4,5}{\varphi }_A^{1,2,3}⟩\times I_{M_{L_B},M_{L_C}}^{M_{L_A},m}(\mathbf{p}),$$

$$I_{M_{L_B},M_{L_C}}^{M_{L_A},m}(\mathbf{p})$$

$$={\delta }^3({\mathbf{P}}_B-{\mathbf{P}}_C)\int {\mathrm{d}}^3{\mathbf{p}}_1{\mathrm{d}}^3{\mathbf{p}}_2{\psi }_B^{\ast }(l_{\rho B},m_{\rho B},l_{\lambda B},m_{\lambda B})$$

$$\displaystyle\times\psi^{\ast}_{C}(L_{C}\,M_{L_{C}})\,{\cal Y}^{m}_{1}\big{(}\frac{{\mathbf{p}}_{4}-{\mathbf{p}}_{5}}{2}\big{)}\psi_{A}(l_{\rho A},m_{\rho A},l_{\lambda A},m_{\lambda A}).$$

$$I_{M_{L_B},M_{L_C}}^{M_{L_A},m}(\mathbf{p})$$

$$={\delta }^3({\mathbf{P}}_B-{\mathbf{P}}_C)\Pi (l_{\rho A},m_{\rho A},l_{\lambda A},m_{\lambda A},m),$$

$$\displaystyle\psi(0,0,0,0)=3^{3/4}\;(\frac{1}{\pi\alpha^{2}_{\rho}})^{\frac{3}{4}}\,(\frac{1}{\pi\alpha^{2}_{\lambda}})^{\frac{3}{4}}\,\exp\Big{[}{-\frac{{\mathbf{p}}^{2}_{\rho}}{2\alpha^{2}_{\rho}}-\frac{{\mathbf{p}}^{2}_{\lambda}}{2\alpha^{2}_{\lambda}}}\Big{]}.$$

$$\psi (1,m,0,0)$$

$$\psi (0,0,1,m)$$

$$\psi (2,m,0,0)$$

$$\psi (0,0,2,m)$$

$$\psi (1,m,1,m^{\prime })$$

$$\displaystyle\psi(0,0)=\Big{(}\frac{{R}^{2}}{\pi}\Big{)}^{{3}/{4}}\exp\Big{[}-\frac{R^{2}({\mathbf{p}}_{2}-{\mathbf{p}}_{5})^{2}}{8}\Big{]}.$$

$$\psi (1,0)$$

$$\displaystyle=3^{3/4}\sqrt{\frac{2}{3}}\;\Big{(}\frac{1}{\pi^{2}\alpha_{\rho}\alpha_{\lambda}}\Big{)}^{\frac{3}{2}}\Big{[}\frac{3}{2}-\frac{{\mathbf{p}}^{2}_{\rho}}{\alpha^{2}_{\rho}}\Big{]}\exp\Big{[}{-\frac{{\mathbf{p}}^{2}_{\rho}}{2\alpha^{2}_{\rho}}-\frac{{\mathbf{p}}^{2}_{\lambda}}{2\alpha^{2}_{\lambda}}}\Big{]},$$

$$\psi (0,1)$$

$$\displaystyle=3^{3/4}\sqrt{\frac{2}{3}}\;\Big{(}\frac{1}{\pi^{2}\alpha_{\rho}\alpha_{\lambda}}\Big{)}^{\frac{3}{2}}\Big{[}\frac{3}{2}-\frac{{\mathbf{p}}^{2}_{\lambda}}{\alpha^{2}_{\lambda}}\Big{]}\exp\Big{[}{-\frac{{\mathbf{p}}^{2}_{\rho}}{2\alpha^{2}_{\lambda}}-\frac{{\mathbf{p}}^{2}_{\lambda}}{2\alpha^{2}_{\lambda}}}\Big{]},$$

$$\Pi (0,0,0,0,0)=\beta |\text{p}|{\Delta }_{0,0}.$$

$$\Pi (0,0,1,0,0)$$

$$\Pi (0,0,1,1,-1)$$

$$\Pi (1,0,0,0,0)$$

$$\Pi (1,1,0,0,-1)$$

$$\Pi (0,0,2,0,0)$$

Full text

I Introduction

Babar and Belle collaborations observed several excited charmed baryons: $\Lambda_{c}(2880,2940)^{+}$, $\Xi_{c}(2980,3077)^{+,0}$ and $\Omega_{c}(2768)^{0}$ last year babar-2880 ; belle-2880 ; babar-2980-3077 ; belle-2980-3077 ; babar-omega , which inspired several investigations of these states in literature rosner ; xiang ; cheng ; valcarce . We collect the experimental information of these recently observed hadrons in Table 1. Their quantum numbers have not been determined except $\Lambda_{c}(2880)^{+}$. In order to understand their structures using the present experimental information, we study the strong decay pattern of the excited charmed baryons systematically in this work. In the past decades, there has been some research work on heavy baryons cheng ; charmed baryons ; review .

The quantum numbers and decay widths of S-wave and some P-wave charmed baryons are known PDG . We first systematically analyze their strong decays in the framework of the ${}^{3}P_{0}$ strong decay model. Accordingly one can extract the parameters and estimate the accuracy of the ${}^{3}P_{0}$ model when it’s applied in the charmed baryon system. Then we go one step further and extend the same formalism to study the decay patterns of these new charmed baryons $\Lambda_{c}(2880,2940)^{+}$, $\Xi(2980,3077)^{+,0}$ under different assignments of their quantum numbers. After comparing the theoretical results with the available experimental data, we can learn their favorable quantum numbers and assignments in the quark model.

Very recently CDF collaboration reported four particles CDF ; CDF-1 , which are consistent with $\Sigma_{b}^{\pm}$ and $\Sigma_{b}^{*\pm}$ predicted in the quark model theory . Their masses are $M_{\Sigma_{b}^{+}}=5808^{+2.0}_{-2.3}\pm 1.7\;{\rm MeV}$, $M_{\Sigma_{b}^{-}}=5816^{+1.0}_{-1.0}\pm 1.7{\rm MeV}$, $M_{\Sigma_{b}^{*+}}=5829^{+1.6}_{-1.8}\pm 1.7$, $M_{\Sigma_{b}^{*-}}=5837^{+2.1}_{-1.9}\pm 1.7{\rm MeV}$. The mass splitting between $\Sigma_{b}$ and $\Sigma_{b}^{*}$ was discussed in Refs. rosner-1 ; lipkin while the strong decays of $\Sigma_{b}^{\pm(*)}$ were studied in Ref. hwang . As a byproduct, we also calculate the strong decays of $\Sigma_{b}^{(*)\pm}$ and other S-wave bottom baryons in this work.

This paper is organized as follows. We give a short theoretical review of S-wave, P-wave and D-wave charmed baryons and introduce our notations for them in Section II. Then we give a brief review of ${}^{3}P_{0}$ model in Section III. We present the strong decay amplitudes of charmed baryons in Section IV. Section V is the numerical results. The last section is our discussion and conclusion. Some lengthy formulae are collected in the Appendix.

II The notations and conventions of charmed baryon

We first introduce our notations for the excited charmed baryons. Inside a charmed baryon there are one charm quark and two light quarks ($u$, $d$ or $s$). It belongs to either the symmetric $6_{F}$ or antisymmetric $\bar{3}_{F}$ flavor representation (see Fig. 1). For the S-wave charmed baryons, the total color-flavor-spin wave function and color wave function must be symmetric and antisymmetric respectively. Hence the spin of the two light quarks is S=1 for $6_{F}$ or S=0 for $\bar{3}_{F}$. The angular momentum and parity of the S-wave charmed baryons are $J^{P}=\frac{1}{2}^{+}$ or $\frac{3}{2}^{+}$ for $6_{F}$ and $J^{P}=\frac{1}{2}^{+}$ for $\bar{3}_{F}$. The names of S-wave charmed baryons are listed in Fig. 1, where we use the star to denote $\frac{3}{2}^{+}$ baryons and the prime to denote the $J^{P}=\frac{1}{2}^{+}$ baryons in the ${6}_{F}$ representation.

In Fig. 2 we introduce our notations and conventions for the P-wave charmed baryons. $l_{\rho}$ is the orbital angular momentum between the two light quarks while $l_{\lambda}$ denotes the orbital angular momentum between the charm quark and the two light quark system. We use the prime to label the $\Xi_{cJ_{l}}$ baryons in the $6_{F}$ representation and the tilde to discriminate the baryons with $l_{\rho}=1$ from that with $l_{\lambda}=1$.

The notation for D-wave charmed baryons is more complicated (see Fig. 3). Besides the prime, $l_{\rho}$ and $l_{\lambda}$ defined above, we use the hat and check to denote the charmed baryons with $l_{\rho}=2$ and $l_{\rho}=1$ respectively. For the baryons with $l_{\rho}=1$ and $l_{\lambda}=1$, we use the superscript $L$ to denote the different total angular momentum in $\check{\Lambda}_{cJ_{l}}^{L}$, $\check{\Sigma}_{cJ_{l}}^{L}$ and $\check{\Xi}_{cJ_{l}}^{L}$.

III The ${}^{3}P_{0}$ model

The ${}^{3}P_{0}$ model was first proposed by Micu Micu and further developed by Yaouanc et al. later yaouanc ; yaouanc-1 ; yaouanc-book . Now this model is widely used to study the strong decays of hadrons qpc-1 ; qpc-2 ; qpc-90 ; ackleh ; Zou ; liu ; lujie ; baryon-decay .

According to this model, a pair of quarks with $J^{PC}=0^{++}$ is created from the vacuum when a hadron decays, which is shown in Fig. 4 for the baryon decay process $A\to B+C$. The new $q\bar{q}$ pair created from the vacuum together with the $qqq$ within the the initial baryon regroup into the outgoing meson and baryon via the quark rearrangement process. In the non-relativistic limit, the transition operator is written as

[TABLE]

where $i$ and $j$ are the color indices of the created quark and anti-quark. $\varphi^{45}_{0}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$ and $\omega_{0}^{45}=\delta_{ij}$ for the flavor and color singlet respectively. $\chi_{{1,-m}}^{45}$ is for the spin triplet state. $\mathcal{Y}_{1}^{m}(\mathbf{k})\equiv|\mathbf{k}|Y_{1}^{m}(\theta_{k},\phi_{k})$ is a solid harmonic polynomial corresponding to the p-wave quark pair. $\gamma$ is a dimensionless constant related to the strength of the quark pair creation from the vacuum, which was extracted by fitting to data. The hadron and meson state are defined as respectively according to the definition of mock state mockmeson

[TABLE]

[TABLE]

and satisfy the normalization condition

[TABLE]

The subscripts $1,\;2,\;3$ denote the quarks of parent hadron A. $a$ and $b$ refer to the quark and antiquark within the meson B respectively. ${\textbf{k}}_{i}(i=1,2,3)$ are the momentum of quarks in hadron A. ${\textbf{k}}_{a}$ and ${\textbf{k}}_{b}$ are the momentum of the quark and antiquark in meson B. ${\textbf{P}}_{A(B)}$ represents the momentum of state A(B). $S_{A(B)}$ and $J_{A(B)}$ denote the total spin and the total angular momentum of state A(B).

The S-matrix is defined as

[TABLE]

The helicity amplitude of the process $A\rightarrow B+C$ in the center of mass frame of meson A is

[TABLE]

where the spatial integral $I^{M_{L_{A}},m}_{M_{L_{B}},M_{L_{C}}}({\textbf{p}})$ is defined as

[TABLE]

$\langle\chi^{235}_{S_{C}M_{S_{C}}}\chi^{14}_{S_{B}M_{S_{B}}}|\chi^{123}_{S_{A}M_{S_{A}}}\chi^{45}_{1-\!m}\rangle$ and $\langle\varphi^{235}_{C}\varphi^{14}_{B}|\varphi^{123}_{A}\varphi^{45}_{0}\rangle$ denote the spin and flavor matrix element respectively.

The decay width of the process $A\to B+C$ is

[TABLE]

where $|\textbf{p}|$ is the momentum of the daughter baryon in the parent’s center of mass frame. $s=1/(1+\delta_{BC})$ is a statistical factor which is needed if $B$ and $C$ are identical particles.

IV The strong decays of charmed baryon

According to the ${}^{3}P_{0}$ model, the decay occurs through the recombination of the five quarks from the initial charmed baryon and the created quark pair. So there are three ways of regrouping:

[TABLE]

where $q_{i}$ and $c_{3}$ denote the light quark and charm quark respectively.

When the excited charmed baryon decays into a charmed baryon plus a light meson as shown in Eq. (12) and (13), the total decay amplitude reads

[TABLE]

where the pre-factor 2 in front of $\gamma$ arises from the fact that the amplitude from the Eq. (12) is the same as that from Eq. (13).

The overlap integral in the momentum space is

[TABLE]

Since all hadrons in the final states are S-wave in this work, eq. (IV) can be further expressed as

[TABLE]

where we have used the harmonic oscillator wave functions for both the meson and baryon. The expressions of $\Pi(l_{\rho A},m_{\rho A},l_{\lambda A},m_{\lambda A},m)$ for the decays of S-wave, P-wave and D-wave charmed baryons are collected in the Appendix. We also move the lengthy expressions of momentum space integration of S-wave, P-wave and D-wave charmed baryons to the Appendix.

V Numerical results

The decay widths of charmed baryons from the ${}^{3}P_{0}$ model involve several parameters: the strength of quark pair creation from vacuum $\gamma$, the R value in the harmonic oscillator wave function of meson and the $\alpha_{\rho,\lambda}$ in the baryon wave functions. We follow the convention of Ref. Godfrey and take $\gamma=13.4$, which is considered as a universal parameter in the ${}^{3}P_{0}$ model. The R value of $\pi$ and $K$ mesons is $2.1$ GeV*-1* Godfrey while it’s $R=2.3$ GeV*-1* for the $D$ meson parameter-2 . $\alpha_{\rho}=\alpha_{\lambda}=0.5$ GeV for the proton and $\Lambda$ baryon-decay . For S-wave charmed baryons, the parameters $\alpha_{\rho}$ and $\alpha_{\lambda}$ in the harmonic oscillator wave functions can be fixed to reproduce the mass splitting through the contact term in the potential model potential . Their values are $\alpha_{\rho}=0.6$ GeV and $\alpha_{\lambda}=0.6$ GeV. For P-wave and D-wave charmed baryons, $\alpha_{\rho}$ and $\alpha_{\lambda}$ are expected to lie in the range $0.5\sim 0.7$ GeV. In the following, our numerical results are obtained with the typical values $\alpha_{\rho}=\alpha_{\lambda}=0.6$ GeV.

The strong decay widths of the S-wave charmed baryons $\Sigma^{++,+,0}_{c}(2455)$, $\Sigma^{*++,+,0}_{c}(2520)$ and $\Xi^{*+,0}_{c}(2645)$ are listed in Table 2. Accordingly the decay widths of S-wave bottomed baryons are presented in Table 3. Because $\Xi_{b}$, $\Xi_{b}^{{}^{\prime}}$ and $\Xi_{b}^{*}$ have not been observed so far, their masses are taken from the theoretical estimate in Ref. E.Jenkins , which are $m_{\Xi_{b}}=5805.7$ MeV, $m_{\Xi_{b}^{{}^{\prime}}}=5950$ MeV and $m_{\Xi_{b}^{*}}=5966.1$ MeV.

The quantum number and internal structure of the following P-wave charmed baryons $\Lambda^{+}_{c}(2593)$, $\Lambda^{+}_{c}(2625)$, $\Xi_{c}^{+,0}(2790)$ and $\Xi_{c}^{+,0}(2815)$ are relatively known experimentally PDG . Their strong decay modes and widths from the ${}^{3}P_{0}$ model are collected in Table 4. The quantum number of $\Sigma^{++}_{c}(2800)$ is still unknown PDG . Thus under different P-wave assignments of $\Sigma^{++}_{c}(2800)$, we present the strong decay widths of its possible decay modes in Table 5. In the heavy quark limit, the process $\Sigma^{++}_{c}(2800)\to\Lambda_{c}^{+}\pi^{+}$ is forbidden if $\Sigma^{++}_{c}(2800)$ is assigned as $\Sigma_{c1}(\frac{1}{2}^{-})$, $\Sigma_{c1}(\frac{3}{2}^{-})$, $\tilde{\Sigma}_{c1}(\frac{1}{2}^{-})$ and $\tilde{\Sigma}_{c1}(\frac{3}{2}^{-})$, which is observed in our calculation as can be seen from Table 5.

$\Lambda_{c}(2880)^{+}$ and $\Lambda_{c}(2940)^{+}$ are observed in the invariant mass spectrum of $D^{0}p$ babar-2880 . The first radial excitation of $\Lambda_{c}$ does not decay into $D^{0}p$ from the ${}^{3}P_{0}$ model. Hence the possibility of $\Lambda_{c}(2880)^{+}$ and $\Lambda_{c}(2940)^{+}$ being a radial excitation is excluded. We calculate their strong decays assuming they are D-wave charmed baryons. The results are shown in Table 6 and 7.

With positive parity, $\Xi(2980)^{+,0}$ and $\Xi(3077)^{+,0}$ can be either the first radially excited charmed baryons or the D-wave charmed baryons. With different assumptions of their quantum numbers we present their strong decay widths in Table 8, 9 and Fig. 8.

The numerical results depend on the parameters $\alpha_{\rho}$ and $\alpha_{\lambda}$ in the harmonic oscillator wave functions of the charmed baryons. We illustrate such a dependence in Figs. 5, 6 and 7 using several typical decay channels: $\Sigma_{c}^{++}(2455)\to\Lambda_{c}^{+}\pi^{+}$, $\Lambda_{c}^{+}(2593)\to\Sigma_{c}^{++}(2455)\pi^{-}$ and $\Lambda_{c}^{+}(2880)\to\Sigma_{c}^{*++}(2520)\pi^{-}$, where $\Sigma_{c}^{++}(2455)$, $\Lambda_{c}^{+}(2593)$ and $\Lambda_{c}^{+}(2880)$ are S-wave, P-wave and D-wave baryons respectively.

VI Discussion and conclusion

At present it is still too difficult to calculate the strong decay widths of hadrons from the first principles of QCD. For this purpose, some phenomenological strong decay models were proposed such as the ${}^{3}P_{0}$ model, flux tube model, QCD sum rule, lattice QCD etc, among which only the first two approaches can be applied to the strong decays of excited hadrons. To a large extent, the predictions from the ${}^{3}P_{0}$ and flux tube models roughly agree with each other.

The ${}^{3}P_{0}$ model possesses inherent uncertainties yaouanc ; Godfrey ; liu . In certain cases, the result from the ${}^{3}P_{0}$ model may be a factor of $2\sim 3$ off the experimental width. The uncertainty source of the ${}^{3}P_{0}$ model arises from the strength of the quark pair creation from the vacuum $\gamma$, the approximation of non-relativity, and assuming the simple harmonic oscillator radial wave functions for the hadrons. Even with the above uncertainty, the ${}^{3}P_{0}$ model is still the most systematic, effective and widely used framework to study the hadron strong decays.

In this work, we have calculated the strong decay widths of charmed baryons using the ${}^{3}P_{0}$ model. Our numerical results do not strongly depend on the parameters $\alpha_{\rho}$ and $\alpha_{\lambda}$ as shown in Figs. 5, 6 and 7. Thus the following qualitative features and conclusions remain essentially unchanged with reasonable variations of $\alpha_{\rho}$ and $\alpha_{\lambda}$.

Our results for the S-wave charmed baryons $\Sigma^{++,+,0}_{c}(2455)$, $\Sigma^{*++,+,0}_{c}(2520)$ and $\Xi^{*+,0}_{c}(2645)$ are roughly consistent with experimental data within the inherent uncertainty of the ${}^{3}P_{0}$ model. As a byproduct, we have also calculated the strong decays of $\Sigma_{b}^{\pm}$ and $\Sigma_{b}^{*\pm}$ observed by CDF Collaboration recently. The numerical results are consistent with the experimental values too.

The decay width of P-wave baryon $\Lambda^{+}_{c}(2593)$ is three times larger than the experimental value. With the large experimental uncertainty and the inherent theoretical uncertainty of the the ${}^{3}P_{0}$ model, such a deviation is still acceptable. The decay widths of $\Lambda^{+}_{c}(2625)$ and $\Xi_{c}^{+,0}(2790,2815)$ are compatible with the experimental upper bound. By comparing our results with the experimental total width, we tend to exclude the $\Sigma_{c0}(\frac{1}{2}^{-})$ assignment for $\Sigma_{c}^{++}(2800)$. Since the $\Sigma_{c}^{++}(2800)$ is observed in $\Lambda_{c}^{+}\pi^{+}$ channel belle-2800 , there are only two assignments left for $\Sigma_{c}^{++}(2800)$, i.e. $\Sigma_{c2}(\frac{3}{2}^{-})$ or $\Sigma_{c2}(\frac{5}{2}^{-})$. More experimental information such as the ratio ${\Gamma[\Sigma_{c}^{++}(2800)\to\Sigma^{+,++}_{c}\pi^{+,0}]\over\Gamma[\Sigma_{c}^{++}(2800)\to\Sigma^{\star+,++}_{c}\pi^{+,0}]}$ will be helpful in the determination of the quantum number of $\Sigma_{c}^{++}(2800)$.

We have also calculated the strong decay widths of newly observed $\Lambda_{c}(2880,2940)^{+}$, $\Xi(2980,3077)^{+,0}$ assuming they are candidates of D-wave charmed baryons. We find that the only possible assignment of $\Lambda_{c}(2880)^{+}$ is $\check{\Lambda}^{2}_{c3}(\frac{5}{2}^{+})$ after considering both its total decay width and the ratio ${\Gamma(\Sigma^{\star}_{c}\pi^{\pm})}/{\Gamma(\Sigma_{c}\pi^{\pm})}$, which agrees very well with the indication from Belle experiment that $\Lambda_{c}(2880)^{+}$ favors $J^{P}=\frac{5}{2}^{+}$ by the analysis of the angular distribution belle-2880 .

Unfortunately the experiment information about the $\Lambda_{c}(2940)^{+}$, $\Xi(2980,3077)^{+,0}$ is scarce at present. From their calculated decay widths, we can only exclude some assignments which are marked with crosses in Tables 7, 8 and 9. The decay width ratios of $\Lambda_{c}(2940)^{+}$, $\Xi(2980,3077)^{+,0}$ from the ${}^{3}P_{0}$ model will be useful in the identification of their quantum numbers in the future since the inherent uncertainty cancels largely.

We have also discussed the strong decays of $\Xi(2980,3077)^{+,0}$ assuming they are radial excitations. Unfortunately the numerical results in Fig. 8 depend quite strongly on the node of the spatial wave function which is related to the parameters of the harmonic oscillator wave functions as shown in Fig. 8. We are unable to make strong conclusions here.

Appendix

VI.1 The harmonic oscillator wave functions used in our

calculation

For the S-wave charmed baryon,

[TABLE]

For the P-wave charmed baryon,

[TABLE]

For the D-wave charmed baryon,

[TABLE]

Here $\mathcal{Y}_{l}^{m}(\mathbf{p})$ is the solid harmonic polynomial.

The ground state wave function of meson is

[TABLE]

The the wave function of the first radially excited charmed baryon $\psi(n_{\rho},n_{\lambda})$ reads as

[TABLE]

where $n_{\rho}$ and $n_{\lambda}$ denote the radial quantum number between the two light quarks and between heavy quark and the two light quarks respectively. Here ${\mathbf{p}}_{\rho}=\sqrt{\frac{1}{2}}({\mathbf{p}}_{1}-{\mathbf{p}}_{2})$ and $\mathbf{p}_{\lambda}=\sqrt{\frac{1}{6}}(\mathbf{p}_{1}+\mathbf{p}_{2}-2\mathbf{p}_{3})$ for the above expressions. All the above harmonic oscillator wave functions can be normalized as $\int d\mathbf{p}_{1}d\mathbf{p}_{2}d\mathbf{p}_{3}|\psi|^{2}=1$.

VI.2 The momentum space integration

The momentum space integration $\Pi(l_{\rho A},m_{\rho A},l_{\lambda A},m_{\lambda A},m)$ includes:

For the S-wave charmed baryon decay,

[TABLE]

For the P-wave charmed baryon decay,

[TABLE]

For the D-wave charmed baryon decay,

[TABLE]

For the strong decay of the radial excitation, the momentum space integrals denoted as $\Pi(n_{\rho},n_{\lambda})$ are:

[TABLE]

where

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the above expressions, $|\textbf{p}|$ reads as

[TABLE]

Acknowledgments

C.C. thanks W.J. Fu for the help in the numerical calculation and Y.R. Liu and B. Zhang for useful discussions. This project was supported by the National Natural Science Foundation of China under Grants 10421503 and 10625521, Chinese Ministry of Education and the China Postdoctoral Science foundation (No. 20060400376).