Singly Cabibbo suppressed decays of $\Lambda_{c}^+$ with SU(3) flavor symmetry

TL;DR

This paper uses SU(3) flavor symmetry and topological quark diagrams to analyze and predict decay rates of $ ext{Lambda}_c^+$ baryons, including previously neglected contributions, aligning well with existing data and providing testable predictions.

Contribution

It introduces the first analysis of $ ext{Lambda}_c^+$ decays considering ${ m O}(ar{15})$ contributions within the SU(3) symmetry framework, enhancing the understanding of singly Cabibbo suppressed decays.

Findings

Predicted $ ext{Lambda}_c^+ o p \pi^0$ branching ratio of $(1.3 extpm0.7) imes 10^{-4}$.

Results are consistent with experimental data for Cabibbo allowed and suppressed decays.

Provides testable predictions for ongoing and future experiments at BESIII and Belle-II.

Abstract

We analyze the weak processes of anti-triplet charmed baryons decaying to octet baryons and mesons with the SU(3) flavor symmetry and topological quark diagram scheme. We study the decay branching ratios without neglecting the contributions from ${\cal O}(\overline{15})$ for the first time in the SU(3) flavor symmetry approach. The fitting results for the Cabibbo allowed and suppressed decays of $\Lambda_{c}^+$ are all consistent with the experimental data. We predict all singly Cabibbo suppressed decays. In particular, we find that ${\cal B}(\Lambda_c^+\to p \pi^0)=(1.3\pm0.7)\times 10^{-4}$, which is slightly below the current experimental upper limit of $2.7\times 10^{-4}$ and can be tested by the ongoing experiment at BESIII as well as the future one at Belle-II.

Tables

Table 1. Table 1 : Decay amplitudes related to the S ​ U ​ ( 3 ) F 𝑆 𝑈 subscript 3 𝐹 SU(3)_{F} parameters and the experimental data for the absolute branching fractions and ℛ Ξ c 0 subscript ℛ superscript subscript Ξ 𝑐 0 {\cal R}_{\Xi_{c}^{0}}   pdg ; Ablikim:2017ors ; Ablikim:2015flg ; YKC .

Channel	Amplitude	Data
${\Lambda }_c^{+}\to {\Sigma }^{+}{\pi }^0$	$\sqrt{2}\left(a_1-a_2-a_3\right)$	$\left(12.4\pm 1.0\right)\times {10}^{-3}$
${\Lambda }_c^{+}\to {\Sigma }^{+}\eta$	$\frac{\sqrt{6}}{3}\left(-a_1-a_2+a_3\right)$	$\left(7.0\pm 2.3\right)\times {10}^{-3}$
${\Lambda }_c^{+}\to {\Sigma }^0{\pi }^{+}$	$\sqrt{2}\left(-a_1+a_2+a_3\right)$	$\left(12.9\pm 0.7\right)\times {10}^{-3}$
${\Lambda }_c^{+}\to {\Xi }^0{K}^{+}$	$-2a_2$	$\left(5.9\pm 1.0\right)\times {10}^{-3}$
${\Lambda }_c^{+}\to p{\overline{K}}^0$	$-2a_1+a_6$	$\left(31.6\pm 1.6\right)\times {10}^{-3}$
${\Lambda }_c^{+}\to {\Lambda }^0{\pi }^{+}$	$\frac{\sqrt{6}}{3}\left(-a_1-a_2-a_3-a_6\right)$	$\left(13.0\pm 0.7\right)\times {10}^{-3}$
${\Lambda }_c^{+}\to {\Sigma }^0{K}^{+}$	$\sqrt{2}\left(a_1-a_3\right)$	$\left(5.2\pm 0.8\right)\times {10}^{-4}$
${\Lambda }_c^{+}\to {\Sigma }^{+}{K}^0$	$2\left(a_1-a_3\right)$	-
${\Lambda }_c^{+}\to p{\pi }^0$	$\sqrt{2}\left(a_2+a_3-\frac{a_6}{2}\right)$	$\left(0.8\pm 1.3\right)\times {10}^{-4}$
${\Lambda }_c^{+}\to n{\pi }^{+}$	$2\left(a_2+a_3+\frac{a_6}{2}\right)$	-
${\Lambda }_c^{+}\to p\eta$	$\frac{\sqrt{6}}{3}\left(-2a_1+a_2-a_3+\frac{3}{2}{a}_6\right)$	$\left(12.4\pm 3.0\right)\times {10}^{-4}$
${\Lambda }_c^{+}\to {\Lambda }^0{K}^{+}$	$\frac{\sqrt{6}}{3}\left(a_1-2a_2+a_3+a_6\right)$	$\left(6.1\pm 1.2\right)\times {10}^{-4}$
${\Xi }_c^0\to {\Xi }^{-}{\pi }^{+}$	$2a_1+a_6$	-
${\Xi }_c^0\to {\Lambda }^0{\overline{K}}^0$	$\frac{\sqrt{6}}{3}\left(-2a_1+a_2+a_3+\frac{a_6}{2}\right)$	-
${ℛ}_{{\Xi }_c^0}$		$0.420\pm 0.056$

Table 2. Table 2 : Branching ratios for the Cabibbo allowed and singly Cabibbo suppressed decays of Λ c + superscript subscript Λ 𝑐 \Lambda_{c}^{+} .

Decay branching ratio	This work	Data	$SU{\left(3\right)}_F$ second	CKX Cheng:2018hwl
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}{\pi }^0\right)$	$12.6\pm 2.1$	$12.4\pm 1.0$	$12.8\pm 2.3$	-
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}\eta \right)$	$5.4\pm 1.0$	$7.0\pm 2.3$	$7.1\pm 3.8$	-
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Sigma }^0{\pi }^{+}\right)$	$12.6\pm 2.1$	$12.9\pm 0.7$	$12.8\pm 2.3$	-
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Xi }^0{K}^{+}\right)$	$5.9\pm 1.0$	$5.9\pm 0.9$	$5.5\pm 1.4$	-
${10}^3ℬ\left({\Lambda }_c^{+}\to p{\overline{K}}^0\right)$	$31.3\pm 1.6$	$31.6\pm 1.6$	$32.7\pm 1.5$	-
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Lambda }^0{\pi }^{+}\right)$	$13.1\pm 1.6$	$13.0\pm 0.7$	$12.8\pm 1.7$	-
${10}^4ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}{K}^0\right)$	$11.4\pm 2.0$	-	$8.0\pm 1.6$	14.4
${10}^4ℬ\left({\Lambda }_c^{+}\to {\Sigma }^0{K}^{+}\right)$	$5.7\pm 1.0$	$5.2\pm 0.8$	$4.0\pm 0.8$	7.18
${10}^4ℬ\left({\Lambda }_c^{+}\to p{\pi }^0\right)$	$1.3\pm 0.7$		$5.7\pm 1.5$	$0.8$
${10}^4ℬ({\Lambda }_c^{+}\to p\eta$)	$13.0\pm 1.0$	$12.4\pm 3.0$	${12.5}_{-3.6}^{+3.8}$	12.8
${10}^4ℬ\left({\Lambda }_c^{+}\to n{\pi }^{+}\right)$	$6.1\pm 2.0$	-	$11.3\pm 2.9$	$2.7$
${10}^4ℬ\left({\Lambda }_c^{+}\to {\Lambda }^0{K}^{+}\right)$	$6.4\pm 0.9$	$6.1\pm 1.2$	$4.6\pm 0.9$	10.6

Equations

$${\mathbf{B}}_n$$

$$M$$

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

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

$$\left(H(6{)}_{ij}\right)$$

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

$$H_k^{ij}=\frac{1}{2}\left(H(\overline{15}{)}_k^{ij}+\frac{1}{2}{ϵ}^{ijl}H(6{)}_{kl}\right).$$

$$\mathcal{A}({\mathbf{B}}_c\to {\mathbf{B}}_{n}M)$$

$$T_{{\mathcal{O}}_6}$$

$$T_{{\mathcal{O}}_{\overline{15}}}$$

$$\mathcal{A}(\mathcal{O}(\overline{15}))$$

$${\mathcal{A}}_F({\mathbf{B}}_c\to {\mathbf{B}}_{n}M)=T({\mathbf{B}}_n^{\prime }{)}_{ijk}({\mathbf{B}}_c^{\prime }{)}^{jk}{H}_m^{li}{M}_l^m+C({\mathbf{B}}_n^{\prime }{)}_{ijk}({\mathbf{B}}_c^{\prime }{)}^{jk}{H}_m^{il}{M}_l^m$$

$${\mathcal{A}}_F({\mathbf{B}}_c\to {\mathbf{B}}_{n}M)$$

$$\mathcal{A}({\Lambda }_c^{+}\to p{\pi }^0)$$

$$\mathcal{A}({\Lambda }_c^{+}\to n{\pi }^{+})$$

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

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

$$(a_1,|a_2|,|a_3|,|a_6|)$$

$$({\delta }_{a_2},{\delta }_{a_3},{\delta }_{a_6})$$

$${\chi }^2/d.o.f$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Sigma }^{+}{K}_S^0)=(5.7\pm 1.0)\times 10^{-4},$$

Full text

Recently, the study of the charmed baryons has been receiving increasing attention both theoretically and experimentally. The main reason for this is the recent measurement of the absolute branching fraction of the golden channel $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ by the Belle Collaboration Belle . This mode and many other $\Lambda_{c}^{+}$ ones have 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 with using $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs produced by $e^{+}e^{-}$ collisions at a center-of-mass energy of $\sqrt{s}=4.6$ GeV, which provides a uniquely clean background to study charmed baryons. Consequently, the Particle Data Group (PDG) pdg has given a new average of ${\cal B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})=(6.23\pm 0.33)\%$. The precision measurement on this mode is very important as it can be used to determine the absolute branching fractions of other $\Lambda_{c}^{+}$ decays pdg as well as processes involving $\Lambda_{c}^{+}$, such as the extractions of the CKM element from $\Lambda_{b}\to\Lambda_{c}^{+}\mu^{-}\bar{\nu}_{\mu}$ Aaij:2015bfa ; Hsiao:2018zqd . It is clear that a new era of physics for charmed baryons has begun. For a review on the theoretical progress of charmed baryons, please see Ref. Cheng:2015iom .

On the other hand, the singly Cabibbo suppressed decays of $\Lambda_{c}^{+}\to p\eta$ and $\Lambda_{c}^{+}\to p\pi^{0}$ have been recently investigated by BESIII  Ablikim:2017ors . The branching fraction of the former mode has been measured for the first time with ${\cal B}(\Lambda_{c}^{+}\to p\eta)=(1.24\pm 0.28\pm 0.10)\times 10^{-3}$, whereas that of the later one has also been searched with no significant signal observed, resulting in an upper limit of ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})<2.7\times 10^{-4}$ at the 90% confidence level. These two decays have been extensively studied in the literature based on various dynamical models Uppal:1994pt ; Chen:2002jr ; Cheng:2018hwl as well as the flavor $SU(3)_{F}$ symmetry Sharma:1996sc ; Lu:2016ogy ; zero ; first ; second ; third . In particular, Cheng, Kang and Xu (CKX) Cheng:2018hwl have performed a dynamical calculation based on current algebra to examine the decay of $\Lambda_{c}^{+}\to p\pi^{0}$ and found that its branching fraction is $0.8\times 10^{-4}$, which is consistent with the current experimental upper limit. However, those with $SU(3)_{F}$ have given an inconsistent larger value, e.g., $(5.7\pm 1.5)\times 10^{-4}$ in Ref. second .

It is known that it is difficult to make reliable predictions on the charmed baryon decay rates due to the lack of theoretical understanding of underlined dynamics for the charmed baryon structure. Since the Cabbibo allowed decays of $\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ do not receive any factorizable contributions, the nonzero experimental observed values of their branching fractions imply that the factorization approach is not working in charmed baryon decays. Without the use of a dynamical model, it is clear that the most reliable way to analyze charmed baryon processes is to impose $SU(3)_{F}$  Sharma:1996sc ; Lu:2016ogy ; first ; zero ; second ; third ; fourth ; Savage:1989qr ; Savage:1991wu . In fact, it has been demonstrated first ; zero ; second ; third that all the existing data of the Cabbibo favored and suppressed charmed baryon decays except ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})$ can be fitted well. In these calculations under $SU(3)_{F}$, the contributions to the decays from the sextet ${\bf 6}$ are assumed to be the dominant ones, whereas those from ${\bf\overline{15}}$ are neglected, by taking into account of the enhancements of the QCD running Wilson coefficients associated with the sextet ${\bf 6}$ part old1 ; old2 ; Abbott:1979fw ; Zeppenfeld:1980ex and the vanishing baryonic transition matrix elements from the nonfactorizable contributions with ${\bf\overline{15}}$ Cheng:2018hwl . However, it is interesting to ask what the contributions to the decay rates from the factorizable parts of ${\bf\overline{15}}$ are. In this note, we will try to answer this question. Specifically, we examine all possible contributions to the charmed baryon decays under $SU(3)_{F}$ without neglecting those from ${\bf\overline{15}}$. We examine the singly Cabibbo suppressed $\Lambda_{c}^{+}$ decays to check if our results are consistent with the data, in particular, the $\Lambda_{c}^{+}\to p\pi^{0}$ channel.

There are two approaches to write down the irreducible decay amplitude through $SU(3)_{F}$. One is to generalize the Wigner Eckart theorem deSwart:1963pdg by writing the decay amplitude to be invariant and singlet under $SU(3)_{F}$. The other is to use topological quark diagrams, where the decay amplitude is represented by all possible diagrams connected by quark lines which satisfy $SU(3)_{F}$. Both two have their own advantages. For the former, one is able to compare the contributions from different representations of operators. In this case, it is also possible to include the $SU(3)_{F}$ breaking effect by introducing the strange quark mass third ; Savage:1991wu . On the other hand, the irreducible amplitude in the later approach is more intuitive and gives an insight on dynamics Chau:1995gk . In particular, it could shed light for us on distinguishing the nonfactorizable and factorizable contributions in the processes. It is expected that these two approaches should give the same results under $SU(3)_{F}$. The close connections between the two have been recently examined in Ref. He:2018joe .

To study the two-body anti-triplet of the lowest-lying charmed baryon decays of ${\bf B}_{c}\to{\bf B}_{n}M$, where ${\bf B}_{c}=(\Xi_{c}^{0},-\Xi_{c}^{+},\Lambda_{c}^{+})$ and ${\bf B}_{n}$ and $M$ are the baryon and pseudoscalar octet states, given by

[TABLE]

[TABLE]

Here, we have assumed that the physical state of $\eta$ is solely made of $\eta_{8}$ due to the small mixing between the weak eigenstates of $\eta_{0}$ and $\eta_{8}$ pdg to reduce our fitting parameters.

We start with the effective Hamiltonian responsible for the tree-level $c\to su\bar{d}$, $c\to uq\bar{q}$ and $c\to du\bar{s}$ transitions, given by Buras:1998raa

[TABLE]

with

[TABLE]

where $O_{\pm}^{q_{2}q_{1}}$ and $O_{\pm}^{qq}\equiv O_{\pm}^{dd}-O_{\pm}^{ss}$ are the four-quark operators, $(\bar{q}_{1}q_{2})\equiv\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}$, $G_{F}$ is the Fermi constant, $V_{ij}$ are the CKM matrix elements, and $(c_{+}\,,c_{-})=(0.76\,,1.78)$, corresponding to the scale-dependent Wilson coefficients with the QCD corrections. By using $(V_{cs}V_{ud},V_{cd}V_{ud},V_{cd}V_{us})\simeq(1,-s_{c},-s_{c}^{2})$ in Eq. (9) with $s_{c}\equiv\sin\theta_{c}=0.2248$ pdg representing the well-known Cabbibo angle $\theta_{c}$, the decays associated with $O_{\pm}^{ds}$, $O_{\pm}^{qq}$ and $O_{\pm}^{sd}$ are the so-called Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed processes, respectively.

Under $SU(3)_{F}$, the operators in Eq. (10) correspond to $(\bar{q}^{i}q_{k})(\bar{q}^{j}c)$ with $q_{i}=(u,d,s)$ as the triplet of ${\bf 3}$, which can be decomposed as the irreducible forms of ${(\bf\bar{3}\times 3\times\bar{3}})c=({\bf\bar{3}+\bar{3}^{\prime}+6+\overline{15}})c$ with $c$ as a flavor singlet. As a result, $(O_{-},O_{+})$ fall into the irreducible presentations of $({\cal O}_{6},{\cal O}_{\overline{15}})$ Savage:1989qr . In analogy to octet baryons and mesons, we can write down the operators related to ${\cal O}_{6}$ and ${\cal O}_{\overline{15}}$ in tensor forms, given by

[TABLE]

with $(i,j,k)$=1,2 and 3, where $H(\overline{15})$ is traceless and symmetric in upper indies, while $H(6)_{ij}$ is symmetric in lower indies. One can also write the matrix elements of $H(6)_{ij}$ and $H(\overline{15})^{ij}_{k}$ in Eq. (14) as a single one, given by

[TABLE]

Now, we can write down the SU(3) irreducible amplitude for ${\bf B}_{c}\to{\bf B}_{n}M$ as first ; Savage:1989qr

[TABLE]

where

[TABLE]

with $({\bf B}^{\prime}_{c})^{jk}\equiv({\bf B}_{c})_{i}\epsilon^{ijk}$. Here, the Wilson coefficients have been absorbed in the parameters $a_{i}$.

In order to reduce the fitting parameters for the processes based on the amplitudes in Eqs. (27) and ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry), as mentioned early, the contributions related to ${\cal O}(\overline{15})$ in Eq. (27) have been neglected due to the fact that $c_{-}/c_{+}\approx 2.5$ and the vanishing contributions of ${\cal O}(\overline{15})$ from the nonfactorizable part to the amplitude. To see the later reason, we write the amplitude of ${\bf B}_{c}\to{\bf B}_{n}M$ due to ${\cal O}(\overline{15})$ in terms of the matrix element

[TABLE]

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 indices, whereas the baryon states ${\bf B}_{i}$ are antisymmetric, one easily arrives that $\langle{\bf B}_{i}|{\cal O}(\overline{15})|{\bf B}_{j}\rangle=0$. From the calculations of the nonfactorizable (NF) contributions in terms of the baryon poles (${\bf B^{*}}$), one has that ${\cal A}({\cal O}_{NF}(\overline{15}))$ is related to the combination of $g_{{\bf B_{c}B^{*}}M}\langle{\bf B}^{*}|{\cal O}(\overline{15})|{\bf B}_{c}\rangle$ and $g_{{\bf B^{*}B_{n}}M}\langle{\bf B}_{n}|{\cal O}(\overline{15})|{\bf B}^{*}\rangle$ as illustrated in Fig. 1 Cheng:1991sn , indicating that ${\cal O}(\overline{15})$ does not contribute the nonfactorizable amplitude Cheng:2018hwl . As a result, the amplitude in Eq. (29) only contains the factorizable (F) contributions in the decays of ${\bf B}_{c}\to{\bf B}_{n}M$, and can be factorized as

[TABLE]

To evaluate ${\cal A}_{F}({\cal O}(\overline{15}))$, we need the help of topological quark diagrams. In other words, we have to find out the terms in $T({\cal O}_{\overline{15}})$ of Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry), which can be factorizable. In Figs. 2a and 2b, we illustrate the factorizable contributions for the color allowed and suppressed processes in the topological diagram approach,111It is clear that we have ignored the soft gluon interactions whenever the factorization problem is discussed. respectively.

Note that the quark indices represent the light quark lines of hadrons or operators with $q_{i}=(u,d,s)$. From Fig. 2, we obtain that

[TABLE]

where $({\bf B}^{\prime}_{n})_{ijk}\equiv({\bf B}_{n})^{n}_{i}\epsilon_{njk}$, $({\bf B}^{\prime}_{c})^{jk}\equiv({\bf B}_{c})_{m}\epsilon^{mjk}$, and $T(C)$ represents the color allowed (suppressed) amplitude. By using Eq. (26) and the tensor identity $\epsilon_{njk}\epsilon^{mjk}=2\delta^{m}_{n}$, we find that

[TABLE]

where ${\cal A}_{F}({\cal O}(6))$ corresponds to the factorizable amplitude from ${\cal O}(6)$. Here, we have used that $H(\overline{15})$ is symmetry in upper indices in the second line of Eq. (32). By comparing Eq. (32) with Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry), we immediately identify that only the $a_{6}$ term in Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry) contains the factorizable amplitude of ${\cal O}(\overline{15}).$222In general, the term associated with $a_{6}$ also contribute the non-factorizable part. Consequently, we can safely neglect the $a_{4}$, $a_{5}$ and $a_{7}$ terms in $T_{{\cal O}_{\overline{15}}}$ of Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry) as they do not have the factorizable contributions to the processes. We remark that in Eq. (32), if $T$ and $C$ both exist in ${\cal A}_{F}({\cal O}(\overline{15}))$, one of them should be canceled out by the corresponding term in ${\cal A}_{F}({\cal O}(6))$, resulting in the process to be either color allowed or color suppressed. This can be explicitly demonstrated by the recent work in Ref. He:2018joe on the connection between the topological and $SU(3)_{F}$ approaches.

To illustrate the effect of the only $a_{6}$ term from ${\cal O}(\overline{15})$, we show the decay amplitudes of $\Lambda_{c}^{+}\to p\pi^{0}$ and $\Lambda_{c}^{+}\to n\pi^{+}$, given by third

[TABLE]

It is clear that the relation of ${\cal A}(\Lambda_{c}^{+}\to n\pi^{+})=\sqrt{2}{\cal A}(\Lambda_{c}^{+}\to p\pi^{0})$ Lu:2016ogy is violated with the contributions from $a_{6}$. This violation has been explicitly pointed out in Ref. Cheng:2018hwl based on a dynamical model. On the other hand, some direct relations still exist in some modes. For example, one has that

[TABLE]

Future experimental searches for these decays will confirm if the discussions based on $SU(3)_{F}$ are right or not.

We are now ready to perform our numerical calculation. Since the $SU(3)_{F}$ flavor symmetry does not involve the dynamical details, we have to determine the parameters in the irreducible amplitude by the experimental data, which can be found in the PDG pdg along with the recent measurements by BESIII Ablikim:2017ors ; Ablikim:2015flg . Currently, there are 9 data points from the absolute branching fractions, along with the original data of ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})=(0.8\pm 1.3)\times 10^{-4}$ by BESIII YKC , which are summarized in Table LABEL:Table1. In addition, we include the relative branching ratio of ${\cal R}_{\Xi_{c}^{0}}\equiv{\cal B}(\Xi_{c}^{0}\to\Lambda^{0}\bar{K}^{0})/{\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})=0.420\pm 0.056$ in our fitting.

Altogether, there are seven $SU(3)_{F}$ parameters ($a_{1},|a_{2}|e^{i\delta_{a_{2}}},|a_{3}|e^{i\delta_{a_{3}}},|a_{6}|e^{i\delta_{a_{6}}}$) to fit with eleven data pointes in Table LABEL:Table1. Here, we have set $a_{1}$ to be a real parameter due to the removal of an overall phase. We use the minimum $\chi^{2}$ fit as shown in Ref. second . Explicitly, we obtain

[TABLE]

where $d.o.f$ represents the degree of freedom. The value of $\chi^{2}/d.o.f$ indicates that our fit is good. In the previous studies of the $\Lambda^{+}_{c}$ decays based on $SU(3)_{F}$ Lu:2016ogy ; zero ; first ; second ; third , the contributions of ${\cal O}(\overline{15})$ have been neglected. Our results in Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry) show that the absolute value of $a_{6}$ is about $1/6$ compared to that of the leading one $a_{1}$ in $T({\cal O}_{6})$, so that the ignorance of ${\cal O}(\overline{15})$ is indeed valid.

In Table LABEL:Table2, we list our fitting results for the branching ratios of the Cabibbo allowed and singly Cabibbo suppressed $\Lambda_{c}^{+}$ decays. In the table, we have also included the previous results based on $SU(3)_{F}$ second without ${\cal O}(\overline{15})$ along with the data as well as those from the dynamical model calculations by CKX Cheng:2018hwl .

As seen in Table LABEL:Table2, our results for the Cabibbo allowed $\Lambda_{c}^{+}$ decays with the consideration of ${\cal O}(\overline{15})$ are slightly better than those without ${\cal O}(\overline{15})$, but they all fit the data well. On the other hand, the decay branching ratios for singly Cabibbo suppressed modes of $\Lambda_{c}^{+}$ with and without ${\cal O}(\overline{15})$ are quite different. In particular, we predict that ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})=(1.3\pm 0.7)\times 10^{-4}$, which is consistent with the experiments upper limit of $2.7\times 10^{-4}$ as well as the result of $0.8\times 10^{-4}$ calculated by the pole model with current algebra in Ref. Cheng:2018hwl . It is clear that the inconsistent branching ratio of $(5.7\pm 1.5)\times 10^{-4}$ in the previous study with $SU(3)_{F}$ second results from the ignorance of ${\cal O}(\overline{15})$, in which a large destructive interference occurs between ${\cal O}(\overline{15})$ and ${\cal O}(6)$. It is also interesting to note that ${\cal B}(\Lambda_{c}^{+}\to n\pi^{+})$ is found to be $(6.1\pm 2.0)\times 10^{-4}$, which is reduced by almost a factor 2 in comparing with that in Ref. second . Although the signs for the contributions from $a_{6}$ to $\Lambda_{c}^{+}\to p\pi^{0}$ and $\Lambda_{c}^{+}\to n\pi^{+}$ in Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry) are opposite, the resulting values are both reduced due to the complex numbers of $a_{2,3}$ and $a_{6}$ in Eq. ( Singly Cabibbo suppressed decays of $\Lambda_{c}^{+}$ with SU(3) flavor symmetry).

In addition, from Table LABEL:Table2, we have that

[TABLE]

which agrees with the experimental value of ${\cal B}(\Lambda_{c}^{+}\to\Sigma^{0}K^{+})=(5.2\pm 0.8)\times 10^{-4}$ pdg . The future search for $\Lambda_{c}^{+}\to\Sigma^{+}K^{0}_{S}$ is a good test for $SU(3)_{F}$.

Finally, we remark that we are unable to discuss the $SU(3)_{F}$ breaking effects after including the contributions of ${\cal O}(\overline{15})$ in the fit due to the insufficient experimental data points. Once more experimental data are available in the future, the studies of these effects along with the $\eta^{\prime}$ channels would be possible.

In sum, we have studied the two-body decays of $\Lambda_{c}^{+}\to{\bf B}_{n}M$ based on the approach with the $SU(3)_{F}$ flavor symmetry, which is a powerful tool to examine charmed baryon physics and allows us to connect the physical quantities without knowing the underlined dynamics. We have successfully fitted all the existing experimental data from the Cabibbo allowed and suppressed decays of $\Lambda_{c}^{+}$. By considering the approach with the topological quark diagrams, for the first time, the contributions from ${\cal O}(\overline{15})$ have been included in the calculations with the $SU(3)_{F}$ method. As a result, we have predicted all singly Cabibbo suppressed decays. In particular, we have found that ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})=(1.3\pm 0.7)\times 10^{-4}$, which is slightly below the current experimental upper limit of $2.7\times 10^{-4}$. This result can be tested by the experiments at BESIII and Belle-II.

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).