Asymmetries of anti-triplet charmed baryon decays

TL;DR

This paper systematically studies anti-triplet charmed baryon decays using SU(3)_F symmetry, revealing significant differences from previous models in decay asymmetries and aligning with recent experimental data.

Contribution

It provides the first comprehensive SU(3)_F analysis of decay rates and asymmetries, including ${ m O}(ar{15})$ contributions, improving theoretical predictions.

Findings

Up-down asymmetry for $ o o o$ is predicted to be 0.94, matching BESIII data.

Significant differences in asymmetry predictions compared to previous literature.

Analysis of $K_S^0-K_L^0$ asymmetries in various decay channels.

Abstract

We analyze the decay processes of ${\bf B}_c \to {\bf B}_n M$ with the $SU(3)_F$ flavor symmetry and spin-dependent amplitudes, where ${\bf B}_c({\bf B}_n)$ and $M$ are the anti-triplet charmed (octet) baryon and nonet meson states, respectively. In the $SU(3)_F$ approach, it is the first time that the decay rates and up-down asymmetries are fully and systematically studied without neglecting the ${\cal O}(\overline{15})$ contributions of the color anti-symmetric parts in the effective Hamiltonian. Our results of the up-down asymmetries based on $SU(3)_F$ are quite different from the previous theoretical values in the literature. In particular, we find that the up-down symmetry of $ \alpha(\Lambda_c^+\to \Xi^0 K^+)_{SU(3)} = 0.94^{+0.06}_{-0.11}$, which is consistent with the recent experimental data of $0.77\pm0.78$ by the BESIII Collaboration, but predicted to be zero in the literature. We also examine the $K_S^0-K_L^0$ asymmetries between the decays of ${\bf B}_c \to {\bf B}_n K_S^0$ and ${\bf B}_c \to {\bf B}_n K_L^0$ with both Cabibbo-allowed and doubly Cabibbo-suppressed transitions.

Tables

Table 1. Table 1: Comparisons of the decay branching ratios and asymmetries between the experimental data  pdg ; Ablikim:2018bir ; Exp:eta' ; Exp:absXic0 ; YKC and theoretical reproductions with S ​ U ​ ( 3 ) F 𝑆 𝑈 subscript 3 𝐹 SU(3)_{F} .

Channel	${ℬ}_{exp}$	${\alpha }_{exp}$	${ℬ}_{SU{\left(3\right)}_F}$	${\alpha }_{SU{\left(3\right)}_F}$
${\Lambda }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$\left(13.0\pm 0.7\right)\times {10}^{-3}$	$-0.91\pm 0.15$	$\left(13.0\pm 0.7\right)\times {10}^{-3}$	$-0.87\pm 0.10$
${\Lambda }_c^{+}\to p{K}_S^0$	$\left(15.8\pm 0.8\right)\times {10}^{-3}$		$\left(15.7\pm 0.8\right)\times {10}^{-3}$	$-{0.89}_{-0.11}^{+0.26}$
${\Lambda }_c^{+}\to {\Sigma }^0{\pi }^{+}$	$\left(12.9\pm 0.7\right)\times {10}^{-3}$		$\left(12.7\pm 0.6\right)\times {10}^{-3}$	$-0.35\pm 0.27$
${\Lambda }_c^{+}\to {\Sigma }^{+}{\pi }^0$	$\left(12.4\pm 1.0\right)\times {10}^{-3}$	$-0.45\pm 0.32$	$\left(12.7\pm 0.6\right)\times {10}^{-3}$	$-0.35\pm 0.27$
${\Lambda }_c^{+}\to {\Sigma }^{+}\eta$	$\left(4.1\pm 2.0\right)\times {10}^{-3}$		$\left(3.2\pm 1.3\right)\times {10}^{-3}$	$-0.40\pm 0.47$
${\Lambda }_c^{+}\to {\Sigma }^{+}{\eta }^{\prime }$	$\left(13.4\pm 5.7\right)\times {10}^{-3}$		$\left(14.4\pm 5.6\right)\times {10}^{-3}$	${1.00}_{-0.17}^{+0.00}$
${\Lambda }_c^{+}\to {\Xi }^0{K}^{+}$	$\left(5.9\pm 1.0\right)\times {10}^{-3}$	${}_{}{}^{\ast }0.77\pm 0.78$	$\left(5.6\pm 0.9\right)\times {10}^{-3}$	${0.94}_{-0.11}^{+0.06}$
${\Lambda }_c^{+}\to p{\pi }^0$	$\left(0.8\pm 1.3\right)\times {10}^{-4}$YKC		$\left(1.2\pm 1.2\right)\times {10}^{-4}$	$-0.05\pm 0.72$
${\Lambda }_c^{+}\to p\eta$	$\left(12.4\pm 3.0\right)\times {10}^{-4}$		$\left(11.5\pm 2.7\right)\times {10}^{-4}$	$-{0.96}_{-0.04}^{+0.30}$
${\Lambda }_c^{+}\to {\Lambda }^0{K}^{+}$	$\left(6.1\pm 1.2\right)\times {10}^{-4}$		$\left(6.5\pm 1.0\right)\times {10}^{-4}$	$0.32\pm 0.30$
${\Lambda }_c^{+}\to {\Sigma }^0{K}^{+}$	$\left(5.2\pm 0.8\right)\times {10}^{-4}$		$\left(5.4\pm 0.7\right)\times {10}^{-4}$	$-{1.00}_{-0.00}^{+0.06}$
${\Xi }_c^0\to {\Xi }^{-}{\pi }^{+}$	$\left(1.80\pm 0.52\right)\times {10}^{-2}$	$-0.6\pm 0.4$	$\left(2.21\pm 0.14\right)\times {10}^{-2}$	$-{0.98}_{-0.02}^{+0.07}$
${\Xi }_c^0\to {\Lambda }^0{K}_S^0$			$\left(5.0\pm 0.3\right)\times {10}^{-3}$	$-0.70\pm 0.28$
${}_{}{}^{\ast \ast }ℛ_{{\Xi }_c^0}^{}$	$0.210\pm 0.028$

Table 2. Table 2: Predictions of the branching ratios and up-down asymmetries for the Cabibbo-allowed decays, where we have also listed the values of A and B in the unit of 10 − 1 ​ G F ​ GeV 2 superscript 10 1 subscript 𝐺 𝐹 superscript GeV 2 10^{-1}G_{F}\text{GeV}^{2} .

channel	$A$	$B$	${10}^3ℬ$	$\alpha$
${\Lambda }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$-0.33\pm 0.06$	$1.62\pm 0.12$	$13.0\pm 0.7$	$-0.87\pm 0.10$
${\Lambda }_c^{+}\to p{\overline{K}}^0$	$-0.89\pm 0.15$	$1.44\pm 0.62$	$31.2\pm 1.6$	$-{0.90}_{-0.10}^{+0.22}$
${\Lambda }_c^{+}\to {\Sigma }^0{\pi }^{+}$	$-0.63\pm 0.02$	$0.37\pm 0.29$	$12.7\pm 0.6$	$-0.35\pm 0.27$
${\Lambda }_c^{+}\to {\Sigma }^{+}{\pi }^0$	$0.63\pm 0.02$	$-0.37\pm 0.29$	$12.7\pm 0.6$	$-0.35\pm 0.27$
${\Lambda }_c^{+}\to {\Sigma }^{+}\eta$	$-0.34\pm 0.07$	$0.26\pm 0.44$	$3.2\pm 1.3$	$-0.40\pm 0.47$
${\Lambda }_c^{+}\to {\Sigma }^{+}{\eta }^{\prime }$	$-0.69\pm 0.26$	$-4.80\pm 1.54$	$14.4\pm 5.6$	${1.00}_{-0.17}^{+0.00}$
${\Lambda }_c^{+}\to {\Xi }^0{K}^{+}$	$0.27\pm 0.06$	$1.61\pm 0.24$	$5.6\pm 0.9$	${0.94}_{-0.11}^{+0.06}$
${\Xi }_c^{+}\to {\Sigma }^{+}{\overline{K}}^0$	$0.28\pm 0.12$	$0.69\pm 0.52$	${8.6}_{-7.8}^{+9.4}$	${0.98}_{-0.16}^{+0.02}$
${\Xi }_c^{+}\to {\Xi }^0{\pi }^{+}$	$-0.22\pm 0.06$	$0.12\pm 0.23$	$3.8\pm 2.0$	$-0.32\pm 0.52$
${\Xi }_c^0\to {\Xi }^{-}{\pi }^{+}$	$0.84\pm 0.08$	$-2.25\pm 0.3$	$22.1\pm 1.4$	$-{0.98}_{-0.02}^{+0.07}$
${\Xi }_c^0\to {\Lambda }^0{\overline{K}}^0$	$-0.73\pm 0.07$	$0.80\pm 0.39$	$10.5\pm 0.6$	$-0.68\pm 0.28$
${\Xi }_c^0\to {\Sigma }^0{\overline{K}}^0$	$-0.01\pm 0.10$	$0.65\pm 0.33$	$0.8\pm 0.8$	$-0.07\pm 0.90$
${\Xi }_c^0\to {\Sigma }^{+}{K}^{-}$	$-0.27\pm 0.06$	$-1.61\pm 0.24$	$5.9\pm 1.1$	$0.81\pm 0.16$
${\Xi }_c^0\to {\Xi }^0{\pi }^0$	$-0.44\pm 0.06$	$1.50\pm 0.23$	$7.6\pm 1.0$	$-{1.00}_{-0.00}^{+0.07}$
${\Xi }_c^0\to {\Xi }^0\eta$	$0.65\pm 0.08$	$1.64\pm 0.55$	$10.3\pm 2.0$	${0.93}_{-0.19}^{+0.07}$
${\Xi }_c^0\to {\Xi }^0{\eta }^{\prime }$	$0.61\pm 0.25$	$4.27\pm 1.51$	$9.1\pm 4.1$	${0.98}_{-0.27}^{+0.02}$

Table 3. Table 3: Legend is the same as Table  2 but for the singly Cabibbo-suppressed decays with an overall factor of sin ⁡ θ c subscript 𝜃 𝑐 \sin\theta_{c} for A 𝐴 A and B 𝐵 B omitted.

channel	$A$	$B$	${10}^4ℬ$	$\alpha$
${\Lambda }_c^{+}\to p{\pi }^0$	$0.01\pm 0.10$	$-0.65\pm 0.33$	$1.2\pm 1.2$	$-0.05\pm 0.72$
${\Lambda }_c^{+}\to p\eta$	$-0.75\pm 0.18$	$1.44\pm 0.77$	$12.4\pm 3.5$	$-{0.94}_{-0.06}^{+0.26}$
${\Lambda }_c^{+}\to p{\eta }^{\prime }$	$0.84\pm 0.27$	$4.33\pm 1.91$	$24.5\pm 14.6$	${0.91}_{-0.21}^{+0.09}$
${\Lambda }_c^{+}\to n{\pi }^{+}$	$-0.04\pm 0.07$	$-1.73\pm 0.20$	$8.5\pm 2.0$	$0.12\pm 0.19$
${\Lambda }_c^{+}\to {\Lambda }^0{K}^{+}$	$0.65\pm 0.06$	$0.35\pm 0.35$	$6.5\pm 1.0$	$0.32\pm 0.32$
${\Lambda }_c^{+}\to {\Sigma }^0{K}^{+}$	$0.44\pm 0.06$	$-1.50\pm 0.23$	$5.4\pm 0.7$	$-{1.00}_{-0.00}^{+0.06}$
${\Lambda }_c^{+}\to {\Sigma }^{+}{K}^0$	$0.62\pm 0.08$	$-2.12\pm 0.33$	$10.9\pm 1.5$	$-{1.0}_{-0.00}^{+0.06}$
${\Xi }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$0.05\pm 0.07$	$-1.47\pm 0.24$	$12.3\pm 4.1$	$-0.19\pm 0.24$
${\Xi }_c^{+}\to p{\overline{K}}^0$	$0.62\pm 0.08$	$-2.12\pm 0.33$	$43.3\pm 7.8$	$-{0.93}_{-0.07}^{+0.09}$
${\Xi }_c^{+}\to {\Sigma }^0{\pi }^{+}$	$0.78\pm 0.04$	$-0.45\pm 0.34$	$25.5\pm 2.6$	$-0.38\pm 0.27$
${\Xi }_c^{+}\to {\Sigma }^{+}{\pi }^0$	$-0.82\pm 0.09$	$-0.12\pm 0.53$	$26.9\pm 6.5$	$0.10\pm 0.43$
${\Xi }_c^{+}\to {\Sigma }^{+}\eta$	$0.63\pm 0.16$	$0.56\pm 0.85$	$15.5\pm 10.3$	${0.58}_{-0.59}^{+0.42}$
${\Xi }_c^{+}\to {\Sigma }^{+}{\eta }^{\prime }$	$0.46\pm 0.29$	$4.57\pm 1.84$	$34.6\pm 21.9$	${0.72}_{-0.41}^{+0.28}$
${\Xi }_c^{+}\to {\Xi }^0{K}^{+}$	$-0.04\pm 0.07$	$-1.73\pm 0.20$	$8.2\pm 1.9$	$0.17\pm 0.28$
${\Xi }_c^0\to {\Lambda }^0{\pi }^0$	$-0.02\pm 0.06$	$1.28\pm 0.21$	$2.3\pm 0.8$	$-0.09\pm 0.23$
${\Xi }_c^0\to {\Lambda }^0\eta$	$-0.10\pm 0.15$	$2.95\pm 0.65$	$6.4\pm 2.3$	$-0.42\pm 0.27$
${\Xi }_c^0\to {\Lambda }^0{\eta }^{\prime }$	$0.81\pm 0.33$	$4.97\pm 2.28$	$16.4\pm 10.6$	${0.87}_{-0.28}^{+0.13}$
${\Xi }_c^0\to p{K}^{-}$	$0.27\pm 0.06$	$1.61\pm 0.24$	$5.0\pm 1.1$	$0.67\pm 0.17$
${\Xi }_c^0\to n{\overline{K}}^0$	$0.88\pm 0.03$	$-0.52\pm 0.42$	$7.5\pm 0.5$	$-0.47\pm 0.34$
${\Xi }_c^0\to {\Sigma }^0{\pi }^0$	$0.31\pm 0.09$	$-1.52\pm 0.28$	$3.8\pm 0.7$	$-{0.88}_{-0.12}^{+0.19}$
${\Xi }_c^0\to {\Sigma }^0\eta$	$-0.44\pm 0.11$	$-0.39\pm 0.60$	$1.4\pm 0.8$	$0.09\pm 0.77$
${\Xi }_c^0\to {\Sigma }^0{\eta }^{\prime }$	$-0.33\pm 0.20$	$-3.23\pm 1.30$	$3.3\pm 2.2$	${0.70}_{-0.43}^{+0.30}$
${\Xi }_c^0\to {\Sigma }^{+}{\pi }^{-}$	$-0.27\pm 0.06$	$-1.61\pm 0.24$	$3.9\pm 0.8$	$0.78\pm 0.17$
${\Xi }_c^0\to {\Sigma }^{-}{\pi }^{+}$	$0.84\pm 0.08$	$-2.24\pm 0.30$	$13.3\pm 0.9$	$-{1.00}_{-0.00}^{+0.02}$
${\Xi }_c^0\to {\Xi }^0{K}^0$	$-0.88\pm 0.03$	$0.52\pm 0.42$	$7.2\pm 0.4$	$-0.32\pm 0.25$
${\Xi }_c^0\to {\Xi }^{-}{K}^{+}$	$-0.84\pm 0.08$	$2.24\pm 0.30$	$9.8\pm 0.6$	$-{0.95}_{-0.05}^{+0.06}$

Table 4. Table 4: Legend is the same as Table  2 but for the doubly Cabibbo-suppressed decays with an overall factor of sin 2 ⁡ θ c superscript 2 subscript 𝜃 𝑐 \sin^{2}\theta_{c} for A 𝐴 A and B 𝐵 B omitted.

channel	$A$	$B$	${10}^5ℬ$	$\alpha$
${\Lambda }_c^{+}\to p{K}^0$	$0.28\pm 0.12$	$0.69\pm 0.52$	${1.2}_{-1.2}^{+1.4}$	${1.00}_{-0.09}^{+0}$
${\Lambda }_c^{+}\to n{K}^{+}$	$-0.22\pm 0.06$	$0.12\pm 0.23$	$0.4\pm 0.2$	$-{0.41}_{-0.59}^{+0.62}$
${\Xi }_c^{+}\to {\Lambda }^0{K}^{+}$	$-0.38\pm 0.03$	$-0.49\pm 0.24$	$3.3\pm 0.8$	$0.76\pm 0.24$
${\Xi }_c^{+}\to p{\pi }^0$	$0.19\pm 0.05$	$1.14\pm 0.17$	$6.0\pm 1.4$	$0.65\pm 0.17$
${\Xi }_c^{+}\to p\eta$	$0.43\pm 0.10$	$-2.25\pm 0.53$	$20.4\pm 8.4$	$-0.75\pm 0.15$
${\Xi }_c^{+}\to p{\eta }^{\prime }$	$-0.75\pm 0.27$	$-4.10\pm 1.87$	$40.1\pm 27.7$	${0.80}_{-0.30}^{+0.20}$
${\Xi }_c^{+}\to n{\pi }^{+}$	$0.27\pm 0.06$	$1.61\pm 0.24$	$12.1\pm 2.8$	$0.65\pm 0.17$
${\Xi }_c^{+}\to {\Sigma }^0{K}^{+}$	$-0.60\pm 0.06$	$1.59\pm 0.21$	$11.9\pm 0.7$	$-{0.99}_{-0.01}^{+0.03}$
${\Xi }_c^{+}\to {\Sigma }^{+}{K}^0$	$-0.89\pm 0.15$	$1.44\pm 0.62$	$19.5\pm 1.7$	$-0.82{\pm }_{-0.18}^{+0.28}$
${\Xi }_c^0\to {\Lambda }^0{K}^0$	$-0.36\pm 0.05$	$-0.16\pm 0.26$	$0.6\pm 0.2$	$0.32\pm 0.45$
${\Xi }_c^0\to p{\pi }^{-}$	$0.27\pm 0.06$	$1.61\pm 0.24$	$3.1\pm 0.7$	$0.65\pm 0.17$
${\Xi }_c^0\to n{\pi }^0$	$-0.19\pm 0.05$	$-1.14\pm 0.17$	$1.5\pm 0.4$	$0.65\pm 0.17$
${\Xi }_c^0\to n\eta$	$0.43\pm 0.10$	$-2.25\pm 0.53$	$5.2\pm 2.1$	$-0.75\pm 0.15$
${\Xi }_c^0\to n{\eta }^{\prime }$	$-0.75\pm 0.27$	$-4.10\pm 1.87$	$10.2\pm 7.1$	${0.80}_{-0.30}^{+0.20}$
${\Xi }_c^0\to {\Sigma }^0{K}^0$	$0.63\pm 0.10$	$-1.01\pm 0.44$	$2.5\pm 0.2$	$-{0.82}_{-0.18}^{+0.28}$
${\Xi }_c^0\to {\Sigma }^{-}{K}^{+}$	$-0.84\pm 0.08$	$2.24\pm 0.30$	$6.1\pm 0.4$	$-{0.99}_{-0.01}^{+0.03}$

Table 5. Table 5: Summary of our results with S ​ U ​ ( 3 ) F 𝑆 𝑈 subscript 3 𝐹 SU(3)_{F} and those in the literature for the up-down asymmetries of the Cabibno-allowed charmed baryon decays, where the data, KK, XK, CT, UVK, Zen, Iva, SV1, and SV2 are from the PDG  pdg , Korner and Kramer  Korner92 , Xu and Kamal  Xu92 , Cheng and Tseng  Cheng:allow , Uppal, Verma and Khanna  Uppal:1994pt , Zenczykowski  xialpha2 , Ivanov e ​ l ​ a ​ l . 𝑒 𝑙 𝑎 𝑙 el~{}al.   xialpha1 , Sharma and Verma  Verma98 , and Sharma and Verma   Sharma:1996sc , respectively.

channel	our result	data	KK	XK	CT	UVK	Zen	Iva	SV1	SV2
					(CT′)	(UVK′)				(SV2′)
${\Lambda }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$-0.87\pm 0.10$	$-0.91\pm 0.15$	$-0.70$	$-0.67$	$-0.99$	$-0.87$	$-0.99$	$-0.95$	$-0.99$	input
					$\left(-0.95\right)$	$\left(-0.85\right)$
${\Lambda }_c^{+}\to p{\overline{K}}^0$	$-{0.90}_{-0.10}^{+0.22}$		$-1.0$	$0.51$	$-0.90$	$-0.99$	$-0.66$	$-0.97$	$-0.99$	$-0.99\pm 0.39$
					$\left(-0.49\right)$	$\left(-0.99\right)$
${\Lambda }_c^{+}\to {\Sigma }^0{\pi }^{+}$	$-0.35\pm 0.27$		$0.70$	$0.92$	$-0.49$	$-0.32$	$0.39$	$0.43$	$-0.31$	$-0.45\pm 0.32$
					$\left(0.78\right)$	$\left(-0.32\right)$
${\Lambda }_c^{+}\to {\Sigma }^{+}{\pi }^0$	$-0.35\pm 0.27$	$-0.45\pm 0.32$	$0.70$	$0.92$	$-0.49$	$-0.32$	$0.39$	$0.43$	$-0.31$	input
					$\left(0.78\right)$	$\left(-0.32\right)$
${\Lambda }_c^{+}\to {\Sigma }^{+}\eta$	$-0.40\pm 0.47$		$0.33$			$-0.94$	0	0.55	$-0.99$	$0.92\pm 0.47$
						$\left(-0.99\right)$				$\left(0.96\pm 0.34\right)$
${\Lambda }_c^{+}\to {\Sigma }^{+}{\eta }^{\prime }$	${1.00}_{-0.17}^{+0.00}$		$-0.45$			$0.68$	$-0.91$	$-0.05$	$0.44$	$-0.75\pm 0.38$
						$\left(0.68\right)$			$0.44$	$\left(-0.91\pm 0.40\right)$
${\Lambda }_c^{+}\to {\Xi }^0{K}^{+}$	${0.94}_{-0.11}^{+0.06}$	$0.77\pm 0.78$	0	0		0	0	0	0	0
${\Xi }_c^{+}\to {\Sigma }^{+}{\overline{K}}^0$	${0.98}_{-0.16}^{+0.02}$		$-1.0$	$0.24$	$0.43$		$1.0$	$-0.99$	$-0.38$	$0.03\pm 0.31$
					$\left(-0.09\right)$					$\left(-0.23\pm 0.22\right)$
${\Xi }_c^{+}\to {\Xi }^0{\pi }^{+}$	$-0.32\pm 0.52$		$-0.78$	$-0.81$	$-0.77$		$1.0$	$-1.0$	$-0.74$	$0.03\pm 0.29$
					$\left(-0.77\right)$					$\left(-0.24\pm 0.23\right)$
${\Xi }_c^0\to {\Xi }^{-}{\pi }^{+}$	$-{0.98}_{-0.02}^{+0.07}$	$-0.6\pm 0.4$	$-0.38$	$-0.38$	$-0.47$		$-0.79$	$-0.84$	$-0.99$	$-0.96\pm 0.38$
					$\left(-0.99\right)$
${\Xi }_c^0\to {\Lambda }^0{\overline{K}}^0$	$-0.68\pm 0.28$		$-0.76$	$1.0$	$-0.88$		$-0.29$	$-0.75$	$-0.85$	$-0.85\pm 0.36$
					$\left(-0.73\right)$
${\Xi }_c^0\to {\Sigma }^0{\overline{K}}^0$	$-0.07\pm 0.90$		$-0.96$	$-0.99$	$0.85$		$-0.50$	$-0.55$	$-0.15$	$0.07\pm 0.67$
					$\left(-0.59\right)$
${\Xi }_c^0\to {\Sigma }^{+}{K}^{-}$	$0.81\pm 0.16$		0	0			0	0	0	0
${\Xi }_c^0\to {\Xi }^0{\pi }^0$	$-{1.00}_{-0.00}^{+0.07}$		$0.92$	$0.92$	$-0.78$		$0.21$	$0.94$	$-0.80$	$-0.99\pm 0.37$
					$\left(-0.54\right)$
${\Xi }_c^0\to {\Xi }^0\eta$	${0.93}_{-0.19}^{+0.07}$		$-0.92$				$-0.04$	$-1.0$	$-0.45$	$-0.96\pm 0.38$
										$\left(0.14\pm 0.34\right)$
${\Xi }_c^0\to {\Xi }^0{\eta }^{\prime }$	${0.98}_{-0.27}^{+0.02}$		$-0.38$				$-1.0$	$-0.32$	$0.65$	$-0.63\pm 0.40$
										$\left(-0.99\pm 0.42\right)$

Table 6. Table 6: Summary of our results with S ​ U ​ ( 3 ) F 𝑆 𝑈 subscript 3 𝐹 SU(3)_{F} and those in the literature for the up-down asymmetries of the singly Cabibbo-suppressed charmed baryon decays, where UVK, SV2 and CKX are from Refs.  Uppal:1994pt , Sharma:1996sc and Cheng:sup , respectively.

channel	our result	UVK(′)	SV2(′)	CKX
${\Lambda }_c^{+}\to p{\pi }^0$	$-0.05\pm 0.72$	$0.82\left(0.85\right)$	$0.05\left(0.05\right)$	$-0.95$
${\Lambda }_c^{+}\to p\eta$	$-{0.94}_{-0.06}^{+0.26}$	$-1.00\left(-0.79\right)$	$-0.74\left(-0.45\right)$	$-0.56$
${\Lambda }_c^{+}\to p{\eta }^{\prime }$	${0.91}_{-0.21}^{+0.09}$	$0.87\left(0.87\right)$	$-0.97\left(-0.99\right)$
${\Lambda }_c^{+}\to n{\pi }^{+}$	$0.12\pm 0.19$	$-0.13\left(0.68\right)$	$0.05\left(0.05\right)$	$-0.90$
${\Lambda }_c^{+}\to {\Lambda }^0{K}^{+}$	$0.32\pm 0.32$	$-0.99\left(-0.99\right)$	$-0.54\left(0.97\right)$	$-0.96$
${\Lambda }_c^{+}\to {\Sigma }^0{K}^{+}$	$-{1.00}_{-0.00}^{+0.06}$	$-0.80\left(-0.80\right)$	$0.68\left(-0.98\right)$	$-0.73$
${\Lambda }_c^{+}\to {\Sigma }^{+}{K}^0$	$-{1.00}_{-0.00}^{+0.06}$	$-0.80\left(-0.80\right)$	$0.68\left(-0.98\right)$	$-0.74$

Table 7. Table 7: Irreducible amplitudes, decay branching ratios and up-down and K S 0 − K L 0 superscript subscript 𝐾 𝑆 0 superscript subscript 𝐾 𝐿 0 K_{S}^{0}-K_{L}^{0} asymmetries of 𝐁 c → 𝐁 n ​ K L 0 / K S 0 → subscript 𝐁 𝑐 subscript 𝐁 𝑛 superscript subscript 𝐾 𝐿 0 superscript subscript 𝐾 𝑆 0 {\bf B}_{c}\to{\bf B}_{n}K_{L}^{0}/K_{S}^{0} with both Cabibbo-allowed and doubly Cabibble-suppressed transitions, where the B 𝐵 B amplitudes can be obtained directly by substituting a i subscript 𝑎 𝑖 a_{i} with b i subscript 𝑏 𝑖 b_{i} .

channel	Irreducible amplitude for A	${10}^3{ℬ}_{SU{\left(3\right)}_F}$	${\alpha }_{SU{\left(3\right)}_F}$	${𝐑}_{K_{S,L}^0}$
${\Lambda }_c^{+}\to p{K}_S^0$	$\sqrt{2}\left(\left(a_1-\frac{a_6}{2}\right)+\left(a_3-\frac{a_6}{2}\right)s_c^2\right)$	$15.7\pm 0.8$	$-{0.89}_{-0.11}^{+0.26}$	$0.009\pm 0.011$
${\Lambda }_c^{+}\to p{K}_L^0$	$-\sqrt{2}\left(\left(a_1-\frac{a_6}{2}\right)-\left(a_3-\frac{a_6}{2}\right)s_c^2\right)$	$15.5\pm 0.8$	$-{0.92}_{-0.08}^{+0.21}$
${\Xi }_c^{+}\to {\Sigma }^{+}{K}_S^0$	$-\sqrt{2}\left(\left(a_3-\frac{a_6}{2}\right)+\left(a_1-\frac{a_6}{2}\right)s_c^2\right)$	${4.9}_{-4.2}^{+5.9}$	${0.89}_{-0.46}^{+0.11}$	$0.118\pm 0.078$
${\Xi }_c^{+}\to {\Sigma }^{+}{K}_L^0$	$\sqrt{2}\left(\left(a_3-\frac{a_6}{2}\right)-\left(a_1-\frac{a_6}{2}\right)s_c^2\right)$	${3.9}_{-3.5}^{+5.1}$	${1.00}_{-0.18}^{+0.00}$
${\Xi }_c^0\to {\Sigma }^0{K}_S^0$	$\left(a_2+a_3-\frac{a_6}{2}\right)+\left(a_1-\frac{a_6}{2}\right)s_c^2$	$0.5\pm 0.4$	$-{0.34}_{-0.66}^{+0.95}$	$0.170\pm 0.146$
${\Xi }_c^0\to {\Sigma }^0{K}_L^0$	$-\left(a_2+a_3-\frac{a_6}{2}\right)+\left(a_1-\frac{a_6}{2}\right)s_c^2$	${0.3}_{-0.3}^{+0.5}$	$0.28\pm 0.71$
${\Xi }_c^0\to {\Lambda }^0{K}_S^0$	$\frac{1}{\sqrt{3}}((2a_1-a_2-a_3-\frac{a_6}{2})$	$5.0\pm 0.3$	$-0.70\pm 0.28$	$-0.043\pm 0.003$
	$-(a_1-2a_2-2a_3+\frac{a_6}{2})s_c^2)$
${\Xi }_c^0\to {\Lambda }^0{K}_L^0$	$-\frac{1}{\sqrt{3}}((2a_1-a_2-a_3-\frac{a_6}{2})$	$5.5\pm 0.3$	$-0.66\pm 0.28$
	$+(a_1-2a_2-2a_3+\frac{a_6}{2})s_c^2)$

Table 8. Table 8: Cabibbo-allowed amplitudes for A 𝐁 c → 𝐁 n ​ M subscript 𝐴 → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑀 A_{{\bf B}_{c}\to{\bf B}_{n}M}

Channel	$A$
${\Lambda }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$\frac{\sqrt{6}}{3}\left(-a_1-a_2-a_3-a_6\right)$
$\to p{\overline{K}}^0$	$-2a_1+a_6$
$\to {\Sigma }^0{\pi }^{+}$	$\sqrt{2}\left(-a_1+a_2+a_3\right)$
$\to {\Sigma }^{+}{\pi }^0$	$\sqrt{2}\left(a_1-a_2-a_3\right)$
$\to {\Sigma }^{+}\eta$	$\frac{\sqrt{2}}{3}{c}_{\varphi }\left(-a_1-a_2+a_3-6\tilde{a}\right)+\frac{2}{3}{s}_{\varphi }\left(-a_1-a_2+a_3+3\tilde{a}\right)$
$\to {\Sigma }^{+}{\eta }^{\prime }$	$\frac{2}{3}{c}_{\varphi }\left(a_1+a_2-a_3-3\tilde{a}\right)+\frac{\sqrt{2}}{3}{s}_{\varphi }\left(-a_1-a_2+a_3-3\tilde{a}\right)$
$\to {\Xi }^0{K}^{+}$	$-2a_2$
${\Xi }_c^{+}\to {\Sigma }^{+}{\overline{K}}^0$	$2a_3-a_6$
$\to {\Xi }^0{\pi }^{+}$	$-2a_3-a_6$
${\Xi }_c^0\to {\Lambda }^0{\overline{K}}^0$	$\frac{\sqrt{6}}{3}\left(-2a_1+a_2+a_3+\frac{a_6}{2}\right)$
$\to {\Sigma }^0{\overline{K}}^0$	$\sqrt{2}\left(-a_2-a_3+\frac{a_6}{2}\right)$
$\to {\Sigma }^{+}{K}^{-}$	$2a_2$
$\to {\Xi }^0{\pi }^0$	$\sqrt{2}\left(-a_1+a_3\right)$
$\to {\Xi }^0\eta$	$\frac{\sqrt{2}}{3}{c}_{\varphi }\left(a_1-2a_2-a_3+6\tilde{a}\right)+\frac{2}{3}{s}_{\varphi }\left(a_1-2a_2-a_3-3\tilde{a}\right)$
$\to {\Xi }^0{\eta }^{\prime }$	$\frac{2}{3}{c}_{\varphi }\left(-a_1+2a_2+a_3+3\tilde{a}\right)+\frac{\sqrt{2}}{3}{s}_{\varphi }\left(a_1-2a_2-a_3+6\tilde{a}\right)$
$\to {\Xi }^{-}{\pi }^{+}$	$2a_1+a_6$

Table 9. Table 9: Singly Cabibbo-suppressed amplitudes for A 𝐁 c → 𝐁 n ​ M subscript 𝐴 → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑀 A_{{\bf B}_{c}\to{\bf B}_{n}M}

Channel	${\sin }^{-1}{\theta }_{c}A$
${\Lambda }_c^{+}\to {\Lambda }^0{K}^{+}$	$\frac{\sqrt{6}}{3}\left(a_1-2a_2+a_3+a_6\right)$
$\to p{\pi }^0$	$\sqrt{2}\left(a_2+a_3-\frac{a_6}{2}\right)$
$\to p\eta$	$\sqrt{2}{c}_{\varphi }\left(-\frac{2a_1}{3}+\frac{a_2}{3}+\frac{a_3}{3}+\frac{a_6}{2}+2\tilde{a}\right)+\frac{1}{3}{s}_{\varphi }\left(-4a_1+2a_2+2a_3+3a_6-6\tilde{a}\right)$
$\to p{\eta }^{\prime }$	$\frac{1}{3}{c}_{\varphi }\left(4a_1-2a_2-2a_3-3a_6+6\tilde{a}\right)+\sqrt{2}{s}_{\varphi }\left(-\frac{2a_1}{3}+\frac{a_2}{3}+\frac{a_3}{3}+\frac{a_6}{2}+2\tilde{a}\right)$
$\to n{\pi }^{+}$	$2a_2+2a_3+a_6$
$\to {\Sigma }^0{K}^{+}$	$\sqrt{2}\left(a_1-a_3\right)$
$\to {\Sigma }^{+}{K}_S$	$\sqrt{2}\left(a_1-a_3\right)$
$\to {\Sigma }^{+}{K}_L$	$\sqrt{2}\left(a_1-a_3\right)$
${\Xi }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$\frac{\sqrt{6}}{3}\left(a_1+a_2-2a_3-\frac{a_6}{2}\right)$
$\to p{K}_S$	$\sqrt{2}\left(-a_1+a_3\right)$
$\to p{K}_L$	$\sqrt{2}\left(a_1-a_3\right)$
$\to {\Sigma }^0{\pi }^{+}$	$\sqrt{2}\left(a_1-a_2+\frac{a_6}{2}\right)$
$\to {\Sigma }^{+}{\pi }^0$	$\sqrt{2}\left(-a_1+a_2+\frac{a_6}{2}\right)$
$\to {\Sigma }^{+}\eta$	$\sqrt{2}{c}_{\varphi }\left(\frac{a_1}{3}+\frac{a_2}{3}+\frac{2a_3}{3}-\frac{a_6}{2}+2\tilde{a}\right)+\frac{1}{3}{s}_{\varphi }\left(2a_1+2a_2+4a_3-3a_6-6\tilde{a}\right)$
$\to {\Sigma }^{+}{\eta }^{\prime }$	$\frac{1}{3}{c}_{\varphi }\left(-2a_1-2a_2-4a_3+3a_6+6\tilde{a}\right)+\sqrt{2}{s}_{\varphi }\left(\frac{a_1}{3}+\frac{a_2}{3}+\frac{2a_3}{3}-\frac{a_6}{2}+2\tilde{a}\right)$
$\to {\Xi }^0{K}^{+}$	$2a_2+2a_3+a_6$
${\Xi }_c^0\to {\Lambda }^0{\pi }^0$	$\frac{\sqrt{3}}{3}\left(-a_1-a_2+2a_3-\frac{a_6}{2}\right)$
$\to {\Lambda }^0\eta$	$\frac{\sqrt{3}}{3}{c}_{\varphi }\left(-a_1-a_2+\frac{a_6}{2}+6\tilde{a}\right)+\frac{\sqrt{6}}{3}{s}_{\varphi }\left(-a_1-a_2+\frac{a_6}{2}-3\tilde{a}\right)$
$\to {\Lambda }^0{\eta }^{\prime }$	$\frac{\sqrt{6}}{3}{c}_{\varphi }\left(a_1+a_2-\frac{a_6}{2}+3\tilde{a}\right)+\frac{\sqrt{3}}{3}{s}_{\varphi }\left(-a_1-a_2+\frac{a_6}{2}+6\tilde{a}\right)$
$\to p{K}^{-}$	$-2a_2$
$\to n{K}_S$	$\sqrt{2}\left(-a_1+a_2+a_3\right)$
$\to n{K}_L$	$\sqrt{2}\left(a_1-a_2-a_3\right)$
$\to {\Sigma }^0{\pi }^0$	$a_1+a_2-\frac{a_6}{2}$
$\to {\Sigma }^0\eta$	$c_{\varphi }\left(-\frac{a_1}{3}-\frac{a_2}{3}-\frac{2a_3}{3}+\frac{a_6}{2}-2\tilde{a}\right)+\sqrt{2}{s}_{\varphi }\left(-\frac{a_1}{3}-\frac{a_2}{3}-\frac{2a_3}{3}+\frac{a_6}{2}+\tilde{a}\right)$
$\to {\Sigma }^0{\eta }^{\prime }$	$\sqrt{2}{c}_{\varphi }\left(\frac{a_1}{3}+\frac{a_2}{3}+\frac{2a_3}{3}-\frac{a_6}{2}-\tilde{a}\right)+s_{\varphi }\left(-\frac{a_1}{3}-\frac{a_2}{3}-\frac{2a_3}{3}+\frac{a_6}{2}-2\tilde{a}\right)$
$\to {\Sigma }^{+}{\pi }^{-}$	$2a_2$
$\to {\Sigma }^{-}{\pi }^{+}$	$2a_1+a_6$
$\to {\Xi }^0{K}_S$	$\sqrt{2}\left(-a_1+a_2+a_3\right)$
$\to {\Xi }^0{K}_L$	$\sqrt{2}\left(-a_1+a_2+a_3\right)$
$\to {\Xi }^{-}{K}^{+}$	$-2a_1-a_6$

Table 10. Table 10: Doubly Cabibbo-suppressed amplitudes for A 𝐁 c → 𝐁 n ​ M subscript 𝐴 → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑀 A_{{\bf B}_{c}\to{\bf B}_{n}M}

Channel	${\sin }^{-2}{\theta }_{c}A$
${\Lambda }_c^{+}\to p{K}^0$	$2a_3-a_6$
$\to n{K}^{+}$	$-2a_3-a_6$
${\Xi }_c^{+}\to {\Lambda }^0{K}^{+}$	$\frac{\sqrt{6}}{3}\left(-a_1+2a_2+2a_3+\frac{a_6}{2}\right)$
$\to p{\pi }^0$	$-\sqrt{2}{a}_2$
$\to p\eta$	$\frac{\sqrt{2}}{3}{c}_{\varphi }\left(2a_1-a_2-2a_3-6\tilde{a}\right)+\frac{2}{3}{s}_{\varphi }\left(2a_1-a_2-2a_3+3\tilde{a}\right)$
$\to p{\eta }^{\prime }$	$\frac{2}{3}{c}_{\varphi }\left(-2a_1+a_2+2a_3-3\tilde{a}\right)+\frac{\sqrt{2}}{3}{s}_{\varphi }\left(2a_1-a_2-2a_3-6\tilde{a}\right)$
$\to n{\pi }^{+}$	$-2a_2$
$\to {\Sigma }^0{K}^{+}$	$\sqrt{2}\left(-a_1-\frac{a_6}{2}\right)$
$\to {\Sigma }^{+}{K}^0$	$-2a_1+a_6$
${\Xi }_c^0\to {\Lambda }^0{K}^0$	$\frac{\sqrt{6}}{3}\left(-a_1+2a_2+2a_3-\frac{a_6}{2}\right)$
$\to p{\pi }^{-}$	$-2a_2$
$\to n{\pi }^0$	$\sqrt{2}{a}_2$
$\to n\eta$	$\frac{\sqrt{2}}{3}{c}_{\varphi }\left(-6\tilde{a}+2a_1-a_2-2a_3\right)+\frac{2}{3}{s}_{\varphi }\left(3\tilde{a}+2a_1-a_2-2a_3\right)$
$\to n{\eta }^{\prime }$	$\frac{2}{3}{c}_{\varphi }\left(-3\tilde{a}-2a_1+a_2+2a_3\right)+\frac{\sqrt{2}}{3}{s}_{\varphi }\left(-6\tilde{a}+2a_1-a_2-2a_3\right)$
$\to {\Sigma }^0{K}^0$	$\sqrt{2}\left(a_1-\frac{a_6}{2}\right)$
$\to {\Sigma }^{-}{K}^{+}$	$-2a_1-a_6$

Equations

$$\alpha ({\Lambda }_c^{+}\to {\Xi }^0{K}^{+}{)}_{exp}=0.77\pm 0.78,$$

$${\mathbf{B}}_c=({\Xi }_c^0,-{\Xi }_c^{+},{\Lambda }_c^{+}),$$

$$\displaystyle{\bf B}_{n}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{6}}\Lambda+\frac{1}{\sqrt{2}}\Sigma^{0}&\Sigma^{+}&p\\ \Sigma^{-}&\frac{1}{\sqrt{6}}\Lambda-\frac{1}{\sqrt{2}}\Sigma^{0}&n\\ \Xi^{-}&\Xi^{0}&-\sqrt{\frac{2}{3}}\Lambda\end{array}\right)\,,$$

$$\displaystyle M=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}(\pi^{0}+c_{\phi}\eta+s_{\phi}\eta^{\prime})&\pi^{+}&K^{+}\\ \pi^{-}&\frac{1}{\sqrt{2}}(-\pi^{0}+c_{\phi}\eta+s_{\phi}\eta^{\prime})&K^{0}\\ K^{-}&\bar{K}^{0}&-s_{\phi}\eta+c_{\phi}\eta^{\prime}\end{array}\right)\,,$$

$${\mathcal{H}}_{eff}$$

$$O_{\pm }^{q_2{q}_1}$$

$$H(\overline{15}{)}_k^{ij}$$

$$H(6{)}_{ij}$$

$$\mathcal{M}({\mathbf{B}}_c\to {\mathbf{B}}_{n}M)=i{\overline{u}}_{{\mathbf{B}}_n}\left(A-B{\gamma }_5\right)u_{{\mathbf{B}}_c},$$

$$A_{({\mathbf{B}}_c\to {\mathbf{B}}_{n}M)}=$$

$$\frac{G_F}{\sqrt{2}}{V}_{cq}{V}_{u{q}_m}{\chi }_{\pm }⟨{\mathbf{B}}_n|({\bar{q}}_{i}c{)}_{V-A}|{\mathbf{B}}_c⟩⟨M|({\bar{q}}_l{q}_m{)}_{V-A}|0⟩,$$

$$\frac{d\Gamma }{d\theta }$$

$$\alpha$$

$$\Gamma$$

$${\mathbf{R}}_{K_{S,L}^0}({\mathbf{B}}_c\to {\mathbf{B}}_n)=\frac{\Gamma ({\mathbf{B}}_c\to {\mathbf{B}}_n{K}_S^0)-\Gamma ({\mathbf{B}}_c\to {\mathbf{B}}_n{K}_L^0)}{\Gamma ({\mathbf{B}}_c\to {\mathbf{B}}_n{K}_S^0)+\Gamma ({\mathbf{B}}_c\to {\mathbf{B}}_n{K}_L^0)}.$$

$$(a_1,a_2,a_3,a_6,\tilde{a},b_1,b_2,b_3,b_6,\tilde{b}),$$

$$\tilde{a}\equiv a_0+\frac{1}{3}\left(a_1+a_2-a_3\right),\tilde{b}\equiv b_0+\frac{1}{3}\left(b_1+b_2-b_3\right)$$

$$(a_1,a_2,a_3,a_6,\tilde{a})=(4.34\pm 0.50,-1.33\pm 0.32,1.25\pm 0.36,-0.26\pm 0.64,1.77\pm 0.83)10^{-2}{G}_F{\text{GeV}}^2,$$

$$(b_1,b_2,b_3,b_6,\tilde{b})=(-9.20\pm 2.09,-8.03\pm 1.19,1.42\pm 1.61,-4.05\pm 2.48,13.15\pm 5.56)10^{-2}{G}_F{\text{GeV}}^2.$$

$$R$$

$$⟨0|j_5^{\mu }|M⟩=i{f}_M{q}^{\mu },$$

$$q^{\mu }⟨B_n|\bar{q}{\gamma }_{\mu }{\gamma }_{5}c|B_c⟩\gg q^{\mu }⟨B_n|\bar{q}{\gamma }_{\mu }c|B_c⟩=i⟨B_n|{\partial }^{\mu }(\bar{q}{\gamma }_{\mu }c)|B_c⟩,$$

$$A({\Lambda }_c^{+}\to p{\pi }^0)=\sqrt{2}\left(a_2+a_3-\frac{a_6}{2}\right),$$

$$\alpha ({\Lambda }_c^{+}\to {\Xi }^0{K}^{+}{)}_{SU(3)}=0.94_{-0.11}^{+0.06}$$

$$A({\Lambda }_c^{+}\to {\Sigma }^0{K}^{+})$$

$$B({\Lambda }_c^{+}\to {\Sigma }^0{K}^{+})$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Sigma }^0{K}^{+},{\Sigma }^{+}{K}_S^0,{\Sigma }^{+}{K}_L^0)=(5.4\pm 0.7)\times 10^{-4},$$

$$\alpha ({\Lambda }_c^{+}\to {\Sigma }^0{K}^{+},{\Sigma }^{+}{K}_S^0,{\Sigma }^{+}{K}_L^0)=-1.00_{-0.00}^{+0.06},$$

$$\alpha$$

Full text

I Introduction

Recently, the Belle collaboration has measured the absolute branching ratio of $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ with high precision Belle , resulting in the world average value of ${\cal B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})=(6.23\pm 0.33)\%$, given by the Particle Data Group (PDG) pdg . This decay mode is the so-called golden channel as most of $\Lambda_{c}^{+}$ decay branching fractions are presented relative to it. Subsequently, this golden mode, along with many other $\Lambda_{c}^{+}$ ones, has also been observed by the BESIII Collaboration Ablikim:2015flg ; Ablikim:2015prg ; Ablikim:2016tze ; Ablikim:2016mcr ; Ablikim:2017ors ; Ablikim:2016vqd ; Ablikim:2017iqd ; Ablikim:2018jfs ; Ablikim:2018bir ; Exp:eta' with $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs, produced by $e^{+}e^{-}$ collisions at a center-of-mass energy of $\sqrt{s}=4.6$ GeV, having a uniquely clean background to study the anti-triplet charmed baryon state of $\Lambda_{c}^{+}$. In particular, the decay of $\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ has been seen for the first time with $\eta^{\prime}$ in the final states for the charmed baryon decays Exp:eta' . In addition, the absolute decay branching fraction of $\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$, which involves the anti-triplet charmed baryon state of $\Xi_{c}^{0}$, has also been measured by the Belle collaboration Exp:absXic0 . Clearly, a new experimental physics era for charmed baryons has started.

On the other hand, the theoretical study of the charmed baryon decays has faced several difficulties. The most serious one is that the factorization approach in the non-leptonic decays of charmed baryons is not working. For example, the Cabibbo-allowed decays of $\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ do not receive any factorizable contributions, whereas the experimental data show that their branching fractions are all close to $O(10^{-2})$ pdg , indicating the failure of the factorization method. In addition, the complication of the charmed baryon structure makes us impossible to directly evaluate the decay amplitude in a model-independent way. It is known that the most reliable and simple way to examine the charmed baryon processes is to use the flavor symmetry of $SU(3)_{F}$ Sharma:1996sc ; Savage:1989qr ; Savage:1991wu ; Lu:2016ogy ; first ; zero ; second ; third ; fourth ; fifth ; Wang:2017gxe ; Geng:2019bfz ; Zhao:2018mov . Indeed, it has been recently demonstrated that the results for the charmed baryon decays based on the $SU(3)_{F}$ approach Lu:2016ogy ; first ; zero ; second ; third ; fourth ; fifth ; Wang:2017gxe ; Geng:2019bfz ; Zhao:2018mov are consistent with the current experimental data. Nevertheless, the charmed baryon decays have been extensively studied in various dynamical models Korner92 ; Xu92 ; Uppal:1994pt ; Cheng:general ; Cheng:allow ; Chen:quark ; Cheng:sup ; Chen:2002jr ; xialpha1 ; Verma98 ; xialpha2 , particularly, the recent dynamical calculations of the singly Cabibbo-suppressed $\Lambda_{c}^{+}$ decays by Cheng, Kang and Xu (CKX) Cheng:sup based on current algebra.

For the two-body charmed baryon decay of ${\bf B}_{c}\to{\bf B}_{n}M$, with ${\bf B}_{c}({\bf B}_{n})$ and $M$ the anti-triplet charmed (octet) baryon and nonet meson states, respectively, beside its decay branching fraction, there exits another interesting physical observable, the up-down asymmetry $\alpha$, which is related to the longitudinal polarization of ${\bf B}_{n}$. Currently, there are three experimental measurements of the up-down asymmetries in the charmed baryon decays pdg , along with the recent one by BESIII Ablikim:2018bir , given by

[TABLE]

which has been suggested to be approximately zero in the previous theoretical studies with the dynamical models Korner92 ; Xu92 ; xialpha1 ; xialpha2 ; Cheng:allow ; Cheng:general ; Uppal:1994pt ; Verma98 as well as the $SU(3)_{F}$ approach Sharma:1996sc . However, the up-down asymmetries in ${\bf B}_{c}\to{\bf B}_{n}M$ were not discussed in the previous studies with $SU(3)_{F}$ in Refs. Lu:2016ogy ; first ; zero ; second ; third ; fourth ; fifth ; Wang:2017gxe ; Geng:2019bfz ; Zhao:2018mov .

In addition, it has been noticed that the physical Cabibble-allowed dominated decay processes of ${\bf B}_{c}\to{\bf B}_{n}K_{L}^{0}$ and ${\bf B}_{c}\to{\bf B}_{n}K_{S}^{0}$ are the same when the doubly Cabibbo-suppressed contributions are taking to be zero Wang:2017gxe . However, in some of these processes, the doubly Cabibbo-suppressed transitions are not negligible, which can be examined by defining the $K_{S}^{0}-K_{L}^{0}$ asymmetries between the $K_{S}^{0}$ and $K_{L}^{0}$ modes Wang:2017gxe to track the interferences.

In this work, we will systematically analyze the decay processes of ${\bf B}_{c}\to{\bf B}_{n}M$ with the $SU(3)_{F}$ symmetry with all operators under $SU(3)_{F}$. We will also include the effect of the $\eta-\eta^{\prime}$ mixing. There are two different ways to link the amplitudes among the processes by $SU(3)_{F}$. The first one is a purely mathematical consideration. By imposing the $SU(3)$ group, we are able to write down the amplitude by tensor contractions. The second one is the diagrammatic approach, in which one draws down all the possible diagrams for the decay process with ascertaining that the amplitude from each diagram shall be the same by interchanging up, down and strange quarks. Both ways have their own advantages. The tensor method is easier to cooperate with the other symmetry and it allows us to estimate the order of the contribution from the amplitude with the Wilson coefficients. Explicitly, it could cooperate with the $SU(3)$ color symmetry and take account of the strange quark mass as the source of the $SU(3)_{F}$ symmetry breaking Savage:1991wu ; third . On the other hand, the diagrammatic approach can distinguish the factorizable and non-factorizable amplitudes Chen:quark . The close relations between the two methods have been examined in Ref. He:2018joe . In Ref. fifth , it has been proved to be useful if one combines both methods.

This paper is organized as follows. In Sec. II, we give the formalism for the two-body charmed baryon decays of ${\bf B}_{c}\to{\bf B}_{n}M$, in which we first write the decay amplitudes in terms of parity conserved and violated parts under the $SU(3)_{F}$ flavor symmetry, and then display the decay rates and asymmetries. In Sec. III, we show our numerical results and present discussions. We conclude in Sec. IV. In Appendix A, we list the all decay amplitudes of the anti triplet baryon states in terms of the $SU(3)_{F}$ parameters. We give the definitions of the up-down and longitudinal polarization asymmetries in Appendix B.

II Formalism

To study the two-body decays of the anti-triplet charmed baryon (${\bf B}_{c}$) to octet baryon (${\bf B}_{n}$) and nonet pseudoscalar meson ($M$) states, we write the hadronic state representations under the $SU(3)_{F}$ flavor symmetry to be

[TABLE]

respectively, where $(c_{\phi},s_{\phi})=(\cos\phi,\sin\phi)$ and $\phi=39.3^{\circ}$ FKS to describe the mixing between $\eta_{8}$ and $\eta_{0}$ of the octet and nonet sates for $\eta$.

From $c\to u\bar{d}s$, $c\to u$ and $c\to u\bar{s}d$ transitions at tree level, the effective Hamiltonian is given by Buras:1998raa

[TABLE]

with

[TABLE]

where $(|V_{cs}V_{ud}|,|V_{cd}V_{ud}|,|V_{cd}V_{us}|)\simeq(1,s_{c},s_{c}^{2})$ with $s_{c}\equiv\sin\theta_{c}\approx 0.225$ pdg and $\theta_{c}$ the Cabibbo angle, $c_{i}$ (i=+,-) represent the Wilson coefficients, $G_{F}$ is the Fermi constant, $O_{\pm}^{q_{2}q_{1}}$ and $O_{\pm}^{qq}\equiv O_{\pm}^{dd}-O_{\pm}^{ss}$ are the four-quark operators, and $(\bar{q}_{1}q_{2})\equiv\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}$. In Eq. (10), the decays associated with $O_{\pm}^{ds}$, $O_{\pm}^{qq}$ and $O_{\pm}^{sd}$ are the so-called Cabibbo-allowed (favored), singly Cabibbo-suppressed and doubly Cabibbo-suppressed processes, respectively.

Note that $O_{+(-)}$, corresponding to the ${\cal O}(\overline{15}(6))$ representation, is (anti)symmetric in flavor and color indices. The tensor forms of $H(\overline{15})$ and $H(6)$ under $SU(3)_{F}$ are given by

[TABLE]

respectively, where we have used the conversion of $V_{cd}=-V_{us}=s_{c}$. In general, we write the spin-dependent amplitude of ${\bf B}_{c}\to{\bf B}_{n}M$ as

[TABLE]

where $A$ and $B$ are the $s$-wave and $p$-wave amplitudes, corresponding to the parity violating and conserving ones, and $u_{{\bf B}_{n,c}}$ are the baryon Dirac spinors, respectively. From Eqs. (10) and (27), we can decompose $A$ in terms of the tensor forms under $SU(3)_{F}$ as

[TABLE]

where $({\bf B}_{c}^{\prime})^{ij}\equiv\epsilon^{ijk}({\bf B}_{c})_{k}$. Here, we have assumed that the mass dependence of $A$ and $B$ are negligible, while the Wilson coefficients of $c_{i}$ have been absorbed into the $SU(3)_{F}$ parameters $a_{i}^{(\prime)}$ and $b_{i}^{(\prime)}$. Note that we treat the $SU(3)_{F}$ flavor symmetry to be exact. To obtain more precise results, one has to include the $SU(3)_{F}$ breaking terms in the amplitudes as shown in Refs. Savage:1991wu ; third . Note that the analysis with $SU(3)_{F}$ breaking effect can be done when more experimental data are available in the future. The expansions of $A_{({\bf B}_{c}\to{\bf B}_{n}M)}$ are listed in Appendix A, while those of $B_{({\bf B}_{c}\to{\bf B}_{n}M)}$ can be derived by replacing $a_{i}$ in $A_{({\bf B}_{c}\to{\bf B}_{n}M)}$ with $b_{i}$.

Since the operator ${\cal O}(\overline{15})\sim(\bar{u}q_{1})(\bar{q}_{2}c)+(\bar{q}_{2}q_{1})(\bar{u}c)$ is symmetric in color index, whereas the baryon states are antisymmetric, the contributions of ${\cal O}(\overline{15})$ from the nonfactorizable part to the amplitude vanish, so that we only need to consider the factorizable amplitude from ${\cal O}(\overline{15})$ fifth .

The factorizable diagram is shown in Fig. 1 with the bubble representing the four-quark interaction, which corresponds to the factorized amplitude, given by Korner92

[TABLE]

where $q=q_{i}(q_{l})$ and $\chi_{\pm}$ are related to the effective Wilson coefficients for the charged (neutral) meson in the final states.

From the topological diagram in Fig. 1, one concludes that only $a_{6}$ and $b_{6}$ terms in Eq. (28) contain the factorizable contributions in ${\cal O}(\overline{15})$, in which the octet meson state $M$ is directly given by the weak interaction alone as demonstrated in Ref. fifth . As a result, in our calculations we will neglect the terms associated with $a_{0}^{\prime}$, $a_{4}$, $a_{5}$ and $a_{7}$ and $b_{0}^{\prime}$, $b_{4}$, $b_{5}$ and $b_{7}$ in Eq. (27).

The decay angular distribution of the direction $\hat{p}_{{\bf B}_{n}}=\vec{p}_{{\bf B}_{n}}/{p}_{{\bf B}_{n}}$ ($p_{{\bf B}_{n}}\equiv|\vec{p}_{{\bf B}_{n}}|$) of ${\bf B}_{n}$ in the rest frame of ${\bf B}_{c}$ is found to be

[TABLE]

where $\vec{P}_{{\bf B}_{n}}$ is the polarization vector of ${\bf B}_{n}$ with the longitudinal component being $P_{{\bf B}_{n}}=\alpha$, $\theta$ is the angle between $\vec{P}_{{\bf B}_{n}}$ and $\hat{p}_{{\bf B}_{n}}$ and $\alpha$ is the so-called up-down asymmetry parameter, given by

[TABLE]

with $E_{{\bf B}_{n}}$ and $\vec{p}_{{\bf B}_{n}}$ the energy and three momentum of ${\bf B}_{n}$. The definitions of the up-down and longitudinal asymmetries can be found in Appendix B. Consequently, we obtain the decay rate as

[TABLE]

To extract the doubly Cabibbo-suppressed contributions in the Cabibbo-allowed dominating decays of ${\bf B}_{c}\to{\bf B}_{n}K_{L}^{0}/K_{S}^{0}$, we also define the $K_{S}^{0}-K_{L}^{0}$ asymmetry parameter as Wang:2017gxe

[TABLE]

III Numerical Results and Discussions

We now determine the $SU3)_{F}$ parameters through the experimental data pdg ; Ablikim:2018bir ; Exp:eta' ; Exp:absXic0 ; YKC , listed in Table LABEL:Table1, where we have also shown the reproduced values for the observables. In the following analysis, we take the amplitudes of $A$ and $B$ as real by using the fact that CP is mainly conserved in charmed decays and assuming the final state interaction is negligible Pakvasa:1990if 111We note that $A$ and $B$ are relative real if $CP$ is conserved and the final state interactions are negligible. This statement has been given in many textbooks, such as those in Refs. Weinberg:1995mt ; Sozzi:2008zza .. Note that in our fit, we have used the original data point of ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})=(0.8\pm 1.3)\times 10^{-4}$ from the BESIII Collaboration YKC , but the result of $\alpha(\Lambda_{c}^{+}\to\Xi^{0}K^{+})=0.77\pm 0.78$ Ablikim:2018bir is not included.

Consequently, there are 16 experimental data inputs to fit with 10 $SU(3)_{F}$ parameters in Eq. (28), given by

[TABLE]

resulting in the degree of freedom (d.o.f) to be 6. In order to separate the amplitudes from $\eta_{0}$ and octet meson states, we define $\tilde{a}$ and $\tilde{b}$ by

[TABLE]

respectively. As a result, the $\eta_{0}$ amplitude depends only on $\tilde{a}$ and $\tilde{b}$. By performing the minimal $\chi^{2}$ fitting as shown in Ref. second , we obtain

[TABLE]

The correlation coefficients between i-th and j-th $SU(3)_{F}$ parameters in Eq. (34) are given by

[TABLE]

In our fit, we find that $\chi^{2}/d.o.f=0.5$, which indicates that our results with the $SU(3)_{F}$ symmetry can well explain all current existing experimental data for the decay branching ratios and up-down asymmetries. Indeed, as seen in Table LABEL:Table1, our reproductions based on $SU(3)_{F}$ are all consistent with the corresponding experimental measurements. However, it is important to pointed out that our values of ${\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})=(2.21\pm 0.14)\times 10^{-2}$ and $|\alpha(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})|=0.98^{+0.02}_{-0.07}$ are consistent with, but higher than, the corresponding data of $(1.80\pm 0.52)\times 10^{-2}$ Exp:absXic0 and $0.6\pm 0.4$ pdg , respectively.

It is worth to take a closer look on the parameters in Eq. (III). As mentioned early, $H(\overline{15})$ only contributes to the factorization amplitudes, which can be parametrized only in terms of $a_{6}$ and $b_{6}$ terms, corresponding to the vector and axial-vector currents in the baryonic matrix elements, respectively. Our result of $b_{6}\gg a_{6}$ in Eq. (III) suggests that the axial-vector part of the factorization contribution is much larger that the vector one. This can be understood as follows. In the decay of ${\bf B}_{c}\to{\bf B}_{n}M$, the pseudoscalar meson part of the factorization approach is given by

[TABLE]

where $f_{M}$ is the meson decay constant, while $q^{\mu}$ is the four-momentum of $M$, which is also equal to the four-momentum difference between the initial and final baryons of ${\bf B}_{c}$ and ${\bf B}_{n}$. Consequently, we get that

[TABLE]

where $q$ stands for the light quarks. In the case of the $SU(4)$ flavor symmetry, in which the charm quark is also treated as $q$, Eq. (49) is automatically satisfied as the right-handed part is zero. It is clear that the inequality in Eq. (49) depends on the parameters $a_{6}$ and $b_{6}$, which are not quite determined yet, particularly $a_{6}$. In fact, from Table 9 in Appendix A, we have that

[TABLE]

in which $a_{2}$ and $a_{3}$ get almost canceled out each other, resulting in that it could be dominated by the $a_{6}$ terms. In this case, the experimental search for the up-down asymmetry as well as the future measurement on the branching ration of $\Lambda_{c}^{+}\to p\pi^{0}$ will be helpful to obtain the precise value of $a_{6}$.

In Tables 2, 3 and 4, we list our predictions of the branching ratios and up-down asymmetries for the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed decays, respectively. In the tables, we have also presented the values of A and B, which are useful to understand the up-down asymmetries as well as the comparisons with those given by specific theoretical models. We note that some of our results for the up-down asymmetries have been discussed for the first time in the literature, while the decay branching ratios are almost the same as those in Refs. Lu:2016ogy ; first ; zero ; second ; third ; fifth ; Wang:2017gxe . In particular, we find that ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})=(1.2\pm 1.2)\times 10^{-4}$, which is consistent with our previous value of $(1.3\pm 0.7)\times 10^{-4}$ in Ref. fifth and $0.8\times 10^{-4}$ calculated by the pole model with current algebra in Ref. Cheng:sup as well as the current experimental upper limit of $2.7\times 10^{-4}$ pdg . In addition, the decay branching ratio for the related Cabibbo-suppressed mode of $\Lambda_{c}^{+}\to n\pi^{+}$ is predicted to be $(8.5\pm 1.9)\times 10^{-4}$, in comparison with $(6.1\pm 2.0)\times 10^{-4}$ in Ref. fifth and $2.7\times 10^{-4}$ in Ref. Cheng:sup . We remark that most of the branching ratios in the present work with the spin-dependent amplitudes have small uncertainties comparing to those of our previous study with $SU(3)_{F}$ in Ref. fifth except the decay of $\Lambda_{c}^{+}\to p\pi^{0}$ due to the cancellation effect as well as the correlations in Eq. (47). Explicitly, as shown Table III, the sign in $A_{(\Lambda_{c}^{+}\to p\pi^{0})}=(-0.01\pm 0.10)\text{sin}\theta_{c}\times 10^{-1}G_{F}\text{GeV}^{2}$ is not well determined, resulting in a large error in $\alpha(\Lambda_{c}^{+}\to p\pi^{0})_{SU(3)}=-0.05\pm 0.72$. To determine the asymmetry precisely, the experiment with a smaller uncertainty is clearly needed.

To compare our predictions of the up-down asymmetries with those in the literature, we summarize the values of $\alpha$ for the Cabibbo-allowed and singly Cabibbo-suppressed decays of ${\bf B}_{c}\to{\bf B}_{n}M$ in Tables 5 and 6, respectively.

In the tables, the data are taken from the experimental values in Ref. pdg , KK and Iva correspond to the calculations with the covariant quark models by Korner and Kramer (KK) Korner92 and Ivanov $el~{}al.$ (Iva) xialpha1 , XK, CT and Zen are based on the pole models by Xu and Kamal (XK) Xu92 , Cheng and Tseng (CT) Cheng:allow and Zenczykowski (Zen) xialpha2 , SV1, CT*′, UVK(′)* and CKX are related to the considerations of current algebra by Sharma and Verma (SV1) Verma98 , Cheng and Tseng (CT) Cheng:allow , Uppal, Verma and Khanna (UVK) without (with) the baryon wave function scale variation Uppal:1994pt and Cheng, Kang and Xu (CKX) Cheng:sup , and SV2*(′)* represent the results with $SU(3)_{F}$ by Sharma and Verma with two different signs of $B(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ Sharma:1996sc , respectively. As seen in Table 5, our results of the up-down asymmetries are quite different from those in the literature Korner92 ; Xu92 ; xialpha1 ; xialpha2 ; Cheng:allow ; Cheng:general ; Uppal:1994pt ; Verma98 ; Sharma:1996sc . In particular, it is interesting to see that we predict that

[TABLE]

which is consistent with the current experimental data of $0.77\pm 0.78$ in Eq. (1) Ablikim:2018bir , but different from all theoretical predictions in the literature. For example, it has been suggested that this asymmetry is approximately zero in dynamical models Korner92 ; Xu92 ; xialpha1 ; xialpha2 ; Cheng:allow ; Cheng:general ; Uppal:1994pt ; Verma98 , while the authors in Ref. Sharma:1996sc have also taken it to be zero as a data input when the $SU(3)_{F}$ symmetry is imposed. In our fit, the value in Eq. (1) has not been included as an input in order to see its value based on the $SU(3)_{F}$ approach. Since the error of our predicted result in Eq. (51) is small, we are confident that $\alpha(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ should be much lager than zero and close to one. Moreover, our result of $\alpha(\Lambda_{c}^{+}\to\Lambda_{0}K^{+})_{SU(3)}=0.32\pm 0.32$ is different from the CKX one of $\alpha(\Lambda_{c}^{+}\to\Lambda_{0}K^{+})_{CKX}=-0.96$ in Ref. Cheng:sup . The reason for the difference is due to the signs in the parity violated amplitudes of $A_{(\Lambda_{c}^{+}\to\Lambda_{0}K^{+})_{SU(3)}}=(1.5\pm 0.1)\times 10^{-2}G_{F}\text{GeV}^{2}$ in our calculation and $A_{(\Lambda_{c}^{+}\to\Lambda_{0}K^{+})_{CKX}}=-1.57\times 10^{-2}G_{F}\text{GeV}^{2}$ in Ref. Cheng:sup . To clarify these issues, further precision measurements on these asymmetries are highly recommended.

In addition, due to the vanishing contributions to the decays from the $a_{4}$, $a_{5}$, $a_{7}$ and $a^{\prime}_{0}$ terms of ${\cal O}(\overline{15})$, we get

[TABLE]

leading to the fitted values of

[TABLE]

as given in Table 3. Note that the decay branching ratio of $\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$ has been measured to be $(5.2\pm 0.8)\times 10^{-4}$ pdg , which agrees with with that in Eq. (III). Future measurements on $\Lambda_{c}^{+}\to\Sigma^{+}K^{0}_{S}$ and $\Lambda_{c}^{+}\to\Sigma^{+}K^{0}_{L}$ are important as they can tell us if Eqs. (III) and (III), which can also be derived through the isospin symmetry, are right or wrong.

We now concentrate on the decay processes of ${\bf B}_{c}\to{\bf B}_{n}K_{L}^{0}$ and ${\bf B}_{c}\to{\bf B}_{n}K_{S}^{0}$, which involve both Cabibbo-allowed and doubly suppressed transitions, as shown in Table 7.

If we ignore the later contributions associated with $\sin^{2}\theta_{c}$, ${\cal B}({\bf B}_{c}\to{\bf B}_{n}K_{S}^{0})={\cal B}({\bf B}_{c}\to{\bf B}_{n}K_{L}^{0})$. Clearly, the $K_{S}^{0}-K_{L}^{0}$ asymmetry depends on the doubly Cabibbo-suppressed parts of the decays. As shown in Table 7, the central values for the first three asymmetries are predicted to be around $10\%$ or more, which are consistent with those in Ref. Wang:2017gxe . For $\Xi_{c}^{0}\to\Sigma^{0}K_{S}^{0}/K_{L}^{0}$, the up-down asymmetry of ${\bf R}_{K^{0}_{S,L}}(\Xi_{c}^{0}\to\Sigma^{0})$ has different sign, indicating that the effect of the doublyCabibbo-suppressed transition is not ignorable in these decay processes. Explicitly, we find out that ${\cal B}(\Xi_{c}^{0}\to\Lambda^{0}K_{L}^{0})$ can be a little larger than ${\cal B}(\Xi_{c}^{0}\to\Lambda^{0}K_{S}^{0})$, in which the $K_{S}^{0}-K_{L}^{0}$ asymmetry is predicted to be $-(4.3\pm 0.3)\%$ with a tiny uncertainty, which agrees well with $-(3.7\pm 0.4)\%$ in Ref. Wang:2017gxe .

IV Conclusions

We have studied the two-body decays of ${\bf B}_{c}\to{\bf B}_{n}M$ with the $SU(3)_{F}$ flavor symmetry based on the spin-dependent $s$ and $p$-wave amplitudes of $A$ and $B$, respectively. These amplitudes, which have been decomposed in terms of the $SU(3)_{F}$ parameters $a_{i}^{(\prime)}$ and $b_{i}^{(\prime)}$, allow us to examine the longitudinal polarization of $P_{{\bf B}_{n}}$, which is related to the up-down asymmetry of $\alpha$. We have obtained a good $\chi$ fit for the ten $SU(3)_{F}$ parameters in Eq. (III) from the all possible contributions of ${\cal O}(6)$ and ${\cal O}(\overline{15})$ with 16 data points in Table LABEL:Table1 in the $SU(3)_{F}$ approach, in which all experimental data for the decay branching ratios and up-down asymmetries can be explained. Consequently, we have systematically predicted all decay branching ratios and up-down asymmetries of the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed charmed baryon decays. In particular, our results of ${\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})=(2.21\pm 0.14)\times 10^{-2}$ and $\alpha(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})=-0.98^{+0.07}_{-0.02}$ are consistent with the data of $(1.80\pm 0.52)\times 10^{-2}$ Exp:absXic0 and $-0.6\pm 0.4$ pdg , respectively. We have also found that ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})=(1.2\pm 1.2)\times 10^{-4}$, which is consistent with the current experimental upper limit of $2.7\times 10^{-4}$ pdg . In addition, we have gotten that ${\cal B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+},\Sigma^{+}K^{0}_{S},\Sigma^{+}K^{0}_{L})=(5.4\pm 0.7)\times 10^{-4}$ and $\alpha(\Lambda_{c}^{+}\to\Sigma^{0}K^{+},\Sigma^{+}K^{0}_{S},\Sigma^{+}K^{0}_{L})=-1.00^{+0.06}_{-0.00}$, which are also guaranteed by the isospin symmetry.

We have shown in Table 5 that our predictions of the up-down asymmetries are quite different from the theoretical values in the literature for most of the decay modes. In particular, we have found that $\alpha(\Lambda_{c}^{+}\to\Xi^{0}K^{+})_{SU(3)}=0.94^{+0.06}_{-0.11}$ in Eq. (51), which is consistent with the current experimental data of $0.77\pm 0.78$ in Eq. (1) Ablikim:2018bir , but much larger than zero predicted in the literature. A future precision measurement on this asymmetry is clearly very important as our prediction based on $SU(3)_{F}$ is close to one with a small uncertainty, which can be viewed as a benchmark for the $SU(3)_{F}$ approach.

We have also explored the $K_{S}^{0}-K_{L}^{0}$asymmetries in the decays of ${\bf B}_{c}\to{\bf B}_{n}K_{L}^{0}/K_{S}^{0}$ with both Cabibbo-allowed and doubly Cabibbo-suppressed transitions. The asymmetries depend strongly on the contributions from the doubly Cabibbo-suppressed contributions. Clearly, the measurements of these asymmetries are good tests for the doubly Cabibbo-suppressed transitions.

In conclusion, we give a systematic consideration of the up-down asymmetries in the two-body charmed baryon decays of ${\bf B}_{c}\to{\bf B}_{n}M$ as well as the $K_{S}^{0}-K_{L}^{0}$asymmetries in the decays of ${\bf B}_{c}\to{\bf B}_{n}K_{L}^{0}/K_{S}^{0}$ in the $SU(3)_{F}$ approach. Some of our predictions based on $SU(3)_{F}$ are different from those in the dynamical models, can be tested by the experiments at BESIII and Belle.

Appendix A Irreducible Amplitudes

In this Appendix, we provide the irreducible amplitudes $A_{{\bf B}_{c}\to{\bf B}_{n}M}$ from Eq. (28) based on the flavor $SU(3)_{F}$ symmetry, while those of $B_{{\bf B}_{c}\to{\bf B}_{n}M}$ can be obtained by substituting $b_{i}$ with $a_{i}$ in $A_{{\bf B}_{c}\to{\bf B}_{n}M}$ . Note that in the limits of $\eta=\eta_{8}$ and $\eta^{\prime}=\eta_{0}$, one has that $s_{\phi}=\sqrt{2}c_{\phi}$, resulting in the $\eta^{\prime}=\eta_{0}$ modes only contain $\tilde{a}$. In Tables 8, 9 and 10, we show the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed amplitudes of $A_{{\bf B}_{c}\to{\bf B}_{n}M}$, respectively. Here, we have only considered the factorizable amplitudes from ${\cal O}(\overline{15})$, so that the terms associated with $a_{0,4,5,7}$ and $b_{0,4,5,7}$ are set to be zero.

Appendix B Up-down and Longitudinal Polarization Asymmetries

From Eq. (30), the up-down is defined by

[TABLE]

which is equal to the longitudinal polarization asymmetry, $i.e.$ $P_{{\bf B}_{n}}=\alpha$.

ACKNOWLEDGMENTS

This work was supported in part by National Center for Theoretical Sciences and MoST (MoST-104-2112-M-007-003-MY3 and MoST-107-2119-M-007-013-MY3).