SU(3) symmetry breaking in charmed baryon decays

Abstract

We explore the breaking effects of the SU(3) flavor symmetry in the singly Cabibbo-suppressed anti-triplet charmed baryon decays of $\mathbf{B}_c\rightarrow \mathbf{B}_n M$, with $\mathbf{B}_c=(\Xi _c^0,\Xi _c^+,\Lambda _c^+)$ and $\mathbf{B}_n(M)$ the baryon (pseudo-scalar) octets. We find that these breaking effects can be used to account for the experimental data on the decay branching ratios of ${\mathcal {B}}(\Lambda _c^+\rightarrow \Sigma ^{0} K^{+},\Lambda ^{0} K^{+})$ and $R'_{K/\pi }={\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- K^+)$/${\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- \pi ^+)$. In addition, we obtain that ${\mathcal {B}}(\Xi _{c}^{0} \rightarrow \Xi ^{-} K^{+},\Sigma ^{-} \pi ^{+})=(4.6 \pm 1.7,12.8 \pm 3.1)\times 10^{-4}$, ${\mathcal {B}}(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-)=(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}$ and ${\mathcal {B}}(\Xi _c^+\rightarrow \Sigma ^{0(+)} \pi ^{+(0)})=(10.3 \pm 1.7)\times 10^{-4}$, which all receive significant contributions from the breaking effects, and can be tested by the BESIII and LHCb experiments.

1 Introduction

It is known that the theoretical approach based on the factorization and quantum chromodynamics (QCD) barely explains the charmed hadron decays [1]. This is due to the fact that the mass of the charm quark, $m_c\simeq 1.5$ GeV, is not as heavy as that of the bottom one, $m_b\simeq 4.8$ GeV, resulting in an undetermined correction to the heavy quark expansion, such that the alternative models have to take place for this correction [2,3,4,5,6,7,8]. On the other hand, the SU(3) flavor ($SU(3)_f$) symmetry that works in the b-hadron decays [9,10,11,12,13] can be well applied to $D\rightarrow MM$ and $\mathbf{B}_c\rightarrow \mathbf{B}_n M$ [14,15,16,17,18,19,20,21,22,23,24,25], where $\mathbf{B}_c=(\Xi _c^{0},-\Xi _c^{+},\Lambda _c^+)$ are the lowest-lying anti-triplet charmed baryon states, while $\mathbf{B}_n$ and M represent baryon and pseudoscalar meson states, respectively. Particularly, the $SU(3)_f$ symmetry has been extended to investigate the singly charmed baryon sextet states as well as the doubly and triply charmed baryon ones [23, 24]. For $D\rightarrow MM$ decays, the measurements produce [26]

$$\begin{aligned} {\mathcal {R}}_{D^0(K/\pi )}\equiv & {} \frac{{\mathcal {B}}(D^0\rightarrow K^+K^-)}{{\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)} = 2.82\pm 0.07, \nonumber \\ {\mathcal {B}}_{D^0(2K_s^0)}\equiv & {} {\mathcal {B}}(D^0\rightarrow K_s^0K_s^0)=(1.70\pm 0.12)\times 10^{-4},\nonumber \\ \end{aligned}$$

(1)

in comparison with $({\mathcal {R}}_{D^0(K/\pi )},{\mathcal {B}}_{D^0(2K_s^0)})\simeq (1,0)$ given by the theoretical calculations based on the $SU(3)_f$ symmetry. The disagreements between the theory and experiment imply that the breaking effects of the $SU(3)_f$ symmetry cannot be ignored in the singly Cabibbo-suppressed (SCS) processes. We note that, in the literature, the $SU(3)_f$ breaking effects were used to relate ${\mathcal {R}}_{K/\pi }$ to the possible large difference of the CP violating asymmetries of $\Delta {\mathcal {A}}_{CP}\equiv {\mathcal {A}}_{CP}(D^0\rightarrow K^+K^-) -{\mathcal {A}}_{CP}(D^0\rightarrow \pi ^+\pi ^-)$ [15, 27, 28], which is recently measured to be $(-0.10\pm 0.08\pm 0.03)\%$ by LHCb [29].

For the two-body $\mathbf{B}_c\rightarrow \mathbf{B}_n M$ decays, both Cabibbo flavored (CF) and SCS decays are not well explained. In particular, the experimental measurements show that

$$\begin{aligned} {\mathcal {B}}_{p\pi ^0}\equiv & {} {\mathcal {B}} (\Lambda _c^+ \rightarrow p \pi ^0)\nonumber \\< & {} 3\times 10^{-4}\;(\text {90\% C.L.})\;{\text {[29, 30]}},\nonumber \\ {\mathcal {R}}'_{K/\pi }\equiv & {} \frac{{\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- K^+)}{{\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- \pi ^+)}\nonumber \\= & {} 0.028\pm 0.006\simeq (0.6\pm 0.2)s_c^2\;\text {[25]},\nonumber \\ {\mathcal {B}}(\Lambda _c^+ \rightarrow \Lambda ^0 K^+)= & {} (6.1\pm 1.2)\times 10^{-4}\;\text {[25]},\nonumber \\ {\mathcal {B}}(\Lambda _c^+ \rightarrow \Sigma ^0 K^+)= & {} (5.2\pm 0.8)\times 10^{-4}\;\text {[25]}, \end{aligned}$$

(2)

where $s_c\equiv \sin \theta _c=0.2248$ [26] with $\theta _c$ the well-known Cabbibo angle. However, theoretical evaluations based on the $SU(3)_f$ symmetry lead to ${\mathcal {B}}_{p\pi ^0}=(5.7\pm 1.5)\times 10^{-4}$ and ${\mathcal {R}}'_{K/\pi }\simeq 1.0 s_c^2$ [21], and those in the factorization approach give ${\mathcal {B}}_{p\pi ^0}=f_{\pi }^2/(2f_K^2) s_c^2\,{\mathcal {B}}(\Lambda _c^+\rightarrow p\bar{K}^0) =(5.5\pm 0.3)\times 10^{-4}$ and ${\mathcal {R}}'_{K/\pi }=(f_K/f_\pi )^2 s_c^2\simeq 1.4 s_c^2$, where we have used the data of ${\mathcal {B}}(\Lambda _c^+\rightarrow p\bar{K}^0)=(3.16\pm 0.16)\times 10^{-2}$ [26]. In addition, the fitted results of ${\mathcal {B}}(\Lambda _c^+ \rightarrow \Lambda ^0 K^+,\Sigma ^0 K^+) =(4.6\pm 0.9,4.0\pm 0.8)\times 10^{-4}$ [22] are $(1.3-1.6)\sigma$ away from the data in Eq. (2). In this study, we will consider the breaking effects of the $SU(3)_f$ symmetry due to the fact of $m_s\gg m_{u,d}$ in the $\mathbf{B}_c\rightarrow \mathbf{B}_n M$ decays, particularly, the SCS processes, in accordance with the $D\rightarrow MM$ ones. Our goal is to find out whether the data in Eq. (2) can be understood by introducing the breaking effects.

The paper is organized as follows. In Sect. 2, we provide the formalism, in which the amplitudes of the $\mathbf{B}_c\rightarrow \mathbf{B}_n M$ decays with and without the breaking effects of $SU(3)_f$ symmetry are presented. The numerical analysis is performed in Sect. 3. In Sect. 4, we discuss our results and give the conclusions.

2 Formalism

The two-body charmed baryon weak decays, such as $\Xi ^0_c \rightarrow \Xi ^- \pi ^+(\Xi ^- K^+)$ and $\Lambda _c^+ \rightarrow p \pi ^0$, proceed through the quark-level transitions of $c\rightarrow s u\bar{d}$, $c\rightarrow u d\bar{d}$ and $c\rightarrow u s\bar{s}$, with the effective Hamiltonian given by [32]

$$\begin{aligned}&{\mathcal {H}}_{eff}\nonumber \\&\quad =\sum _{i=+,-}\frac{G_F}{\sqrt{2}}c_i \left( V_{cs}V_{ud}O_i+V_{cd}V_{ud} O_i^d+V_{cs}V_{us}O_i^s\right) , \end{aligned}$$

(3)

where $G_F$ is the Fermi constant, $c_{\pm }$ are the scale-dependent Wilson coefficients, and the CKM matrix elements $V_{cs}V_{ud}\simeq 1$ and $V_{cs}V_{us}\simeq -V_{cd}V_{ud}\simeq s_c$ correspond to the Cabibbo-favored (CF) and singly Cabibbo-suppressed (SCS) charmed hadron decays, respectively, while $O^{(d,s)}_\pm$ are the four-quark operators, written as

$$\begin{aligned} O_\pm= & {} {1\over 2}\left[ (\bar{u} d)(\bar{s} c)\pm (\bar{s} d)(\bar{u} c)\right] ,\;\nonumber \\ O_\pm ^q= & {} {1\over 2}\left[ (\bar{u} q)(\bar{q} c)\pm (\bar{q} q)(\bar{u} c)\right] , \end{aligned}$$

(4)

where $q=(d,s)$ and $(\bar{q}_1 q_2)=\bar{q}_1\gamma _\mu (1-\gamma _5)q_2$. With $q_i=(u,d,s)$ as the triplet of 3, the operator of $(\bar{q}^i q_k \bar{q}^j)c$ can be decomposed as the irreducible forms, that is, $(\bar{3}\times 3\times \bar{3})c=(\bar{3}+\bar{3}'+6+\overline{15})c$. Accordingly, the operators $O_-^{(q)}$ and $O_+^{(q)}$ belong to 6 and $\overline{15}$, respectively, given by [17]

$$\begin{aligned} O_{\mp }\simeq & {} {\mathcal {O}}_{6(\overline{15})}=\frac{1}{2}(\bar{u} d\bar{s}\mp \bar{s} d\bar{u})c,\nonumber \\ O_{\mp }^q\simeq & {} {\mathcal {O}}_{6(\overline{15})}^q=\frac{1}{2}(\bar{u} q\bar{q}\mp \bar{q} q\bar{u})c, \end{aligned}$$

(5)

such that the effective Hamiltonian in Eq. (3) can be transformed into the tensor form of

$$\begin{aligned} {\mathcal {H}}_{eff}= & {} \frac{G_F}{\sqrt{2}}\left[ c_- \frac{\epsilon ^{ijl}}{2}H(6)_{lk}+c_+H(\overline{15})_k^{ij}\right] , \end{aligned}$$

(6)

with the non-zero entries:

$$\begin{aligned} H_{22}(6)= & {} 2,H_2^{13}(\overline{15})=H_2^{31}(\overline{15})=1,\nonumber \\ H_{23}(6)= & {} H_{32}(6)=-2s_c, H^2_{12}(\overline{15})=H^2_{21}(\overline{15})=s_c, \end{aligned}$$

(7)

where the notations of (i, j, k) are quark indices, to be connected to the initial and final states in the amplitudes. Note that $H_{23}(6)$ and $H_{32}(6)$ are derived from ${\mathcal {O}}^{s}_6$ and ${\mathcal {O}}^{d}_6$, respectively. The lowest-lying charmed baryon states $\mathbf{B}_c$ are an anti-triplet of $\bar{3}$ to consist of $(ds-sd)c$, $(us-su)c$ and $(ud-du)c$, presented as

$$\begin{aligned} \mathbf{B}_{c}= & {} (\Xi _c^0,-\Xi _c^+,\Lambda _c^+), \end{aligned}$$

(8)

together with the baryon and meson octets, given by

$$\begin{aligned} \mathbf{B}_n= & {} \left( \begin{array}{ccc} \frac{1}{\sqrt{6}}\Lambda ^0+\frac{1}{\sqrt{2}}\Sigma ^0 &{} \Sigma ^+ &{} p\\ \Sigma ^- &{}\frac{1}{\sqrt{6}}\Lambda ^0 -\frac{1}{\sqrt{2}}\Sigma ^0 &{} n\\ \Xi ^- &{} \Xi ^0 &{}-\sqrt{\frac{2}{3}}\Lambda ^0 \end{array}\right) ,\nonumber \\ M= & {} \left( \begin{array}{ccc} \frac{1}{\sqrt{2}}\pi ^0&{} \pi ^- &{} K^-\\ \pi ^+ &{}-\frac{1}{\sqrt{2}}\pi ^0&{} \bar{K}^0\\ K^+ &{} K^0&{} 0 \end{array}\right) , \end{aligned}$$

(9)

where we have removed the octet $\eta _8$ and singlet $\eta _1$ meson states to simplify our discussions. Subsequently, the amplitudes of $\mathbf{B}_c\rightarrow \mathbf{B}_n M$ can be derived as

$$\begin{aligned} {\mathcal {A}}(\mathbf{B}_c\rightarrow \mathbf{B}_n M)= & {} \langle \mathbf{B}_n M|{\mathcal {H}}_{eff}|\mathbf{B}_c\rangle \nonumber \\= & {} \frac{G_F}{\sqrt{2}}T(\mathbf{B}_{c}\rightarrow \mathbf{B}_nM), \end{aligned}$$

(10)

with $T(\mathbf{B}_{c}\rightarrow \mathbf{B}_nM)=T({\mathcal {H}}_6)+T({\mathcal {H}}_{\overline{15}})$, where $T({\mathcal {H}}_{6,\overline{15}})$ are decomposed as [21,22,23]

$$\begin{aligned} T({\mathcal {H}}_6)= & {} a_1 H_{ij}(6)T^{ik}(\mathbf{B}_n)_k^l (M)_l^j\nonumber \\&+\,a_2 H_{ij}(6)T^{ik}(M)_k^l (\mathbf{B}_n)_l^j\nonumber \\&+\, a_3 H_{ij}(6)(\mathbf{B}_n)_k^i (M)_l^j T^{kl} , \nonumber \\ T({\mathcal {H}}_{\overline{15}})= & {} a_4H^{k}_{li}(\overline{15})(\mathbf{B}_{c})^j (M)^i_j (\mathbf{B}_n)^l_k\nonumber \\&+\,a_5(\mathbf{B}_n)^i_j (M)^l_i H(\overline{15})^{jk}_l (\mathbf{B}_{c})_k\nonumber \\&+\, a_6(\mathbf{B}_n)^k_l (M)^i_j H(\overline{15})^{jl}_i (\mathbf{B}_{c})_k \nonumber \\&+\,a_7(\mathbf{B}_n)^l_i (M)^i_j H(\overline{15})^{jk}_l (\mathbf{B}_{c})_k, \end{aligned}$$

(11)

with $T^{ij} \equiv (\mathbf{B}_c)_k\epsilon ^{ijk}$. In Eq. (11), $a_{1,2,3}$ and $a_{4,5,6,7}$ are the SU(3) parameters from H(6) and $H(\overline{15})$, respectively, in which $c_{\mp }$ have been absorbed. We hence obtain [22]

$$\begin{aligned} T(\Lambda _c^+\rightarrow p \pi ^{0})= & {} -\sqrt{2}\left( a_{2}+a_3-\frac{a_{6} - a_{7}}{2}\right) s_c,\nonumber \\ T(\Xi _c^0\rightarrow \Xi ^{-} \pi ^{+})= & {} 2\left( a_{1}+\frac{a_{5} + a_{6}}{2}\right) ,\nonumber \\ T(\Xi _c^0\rightarrow \Xi ^{-} K^{+})= & {} -2\left( a_{1}+\frac{a_{5} + a_{6}}{2}\right) s_c, \end{aligned}$$

(12)

based on the exact $SU(3)_f$ symmetry.

According to Refs. [21, 22], the numerical analysis with the minimum $\chi ^2$ fit has well explained the ten observed $\mathbf{B}_c\rightarrow \mathbf{B}_nM$ decays by neglecting the terms associated with $a_{4,5,6,7}$ to reduce the parameters [20,21,22,23, 25]. This reduction is due to the fact that the contributions to the branching rates from $H(\overline{15})$ and H(6) lead to a small ratio of ${\mathcal {R}}(\overline{15}/6)=c_+^2/c_-^2\simeq 17\%$ with $(c_+,c_-)=(0.76,1.78)$ calculated at the scale $\mu =1$ GeV in the NDR scheme [33, 34]. There remain two measurements to be explained. In Eq. (2), the prediction for ${\mathcal {B}}_{p\pi ^0}$ has the 2$\sigma$ gap to reach the edge of the experimental upper bound. However, with ${\mathcal {R}}(\overline{15}/6)$ to be small, it is nearly impossible that, by restoring $a_{4,5,6,7}$ that have been ignored in the literature [20,21,22,23, 25], one can accommodate the data of ${\mathcal {B}}_{p\pi ^0}$ but without having impacts on the other decay modes, which are correlated with the same sets of parameters. Moreover, as seen from Eq. (12), there is no room for ${\mathcal {R}}'_{K/\pi }$ as it is fixed to be $(1.0)s_c^2$. On the other hand, the results for $D\rightarrow MM$ decays in Eq. (1) suggest some possible corrections from the breaking effects of the $SU(3)_f$ symmetry in the SCS processes. In the charm baryon decays, we consider the similar effects. Due to $m_s\gg m_{u,d}$, we present the matrix of $M_s=\epsilon (\lambda _s)^i_j$ [14] to break $SU(3)_f$, where $\epsilon \sim 0.2-0.3$ and $\lambda _s$ is given by [14, 15, 18]

$$\begin{aligned} \lambda _s= & {} \left( \begin{array}{c@{\qquad }c@{\qquad }c} 1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}-2 \end{array}\right) , \end{aligned}$$

(13)

which transforms as an octet of 8, such that its coupling to H(6) is in the form of $8\times 6=\bar{3}+6+\overline{15}+24$, and $\bar{3}$ is for the simplest break effects to be confined in the SCS processes [18, 35]. Note that from $\frac{1}{8}\Big (\delta ^i_l(\lambda _s)^n_jH(6)_{kn}-\delta ^i_l(\lambda _s)^n_kH(6)_{jn}+\delta ^i_k(\lambda _s)^n_jH(6)_{ln}-\delta ^i_k(\lambda _s)^n_lH(6)_{jn}\Big )$ and the nonzero entry of $H(\bar{3})^1=s_c$ from the coupling of $H(6)_{23}$ and $H(6)_{32}$ [14], one can trace back to the break effect between SCS $c\rightarrow us\bar{s}$ and $c\rightarrow ud\bar{d}$ transitions. As a result, the $SU(3)_f$ symmetry breaking gives rise to the new T-amplitudes, given by

$$\begin{aligned} T({\mathcal {H}}_3)= & {} v_1(\mathbf{B}_c)_iH(\overline{3})^i(\mathbf{B}_n)^j_k(M)^k_j\nonumber \\&+\,v_2(\mathbf{B}_c)_i H(\overline{3})^j(\mathbf{B}_n)^i_k(M)^k_j\nonumber \\&+\, v_3(\mathbf{B}_c)_i H(\overline{3})^j(\mathbf{B}_n)^k_j(M)^i_k, \end{aligned}$$

(14)

where $v_{1,2,3}$ are the parameters related to the $SU(3)_f$ breaking. It is interesting to note that the $v_1$ terms associated with $(\mathbf{B}_c)_iH(\overline{3})^i$ in Eq. (14) occur in some of the $\Xi _c^{0,+}$ decays, but disappear in all $\Lambda _c^+$ modes. By adding $T({\mathcal {H}}_3)$ into $T(\mathbf{B}_c\rightarrow \mathbf{B}_n M)$ in Eq. (10), the full expansions of $T(\mathbf{B}_c\rightarrow \mathbf{B}_n M)$ are given in Table 1, to be used to calculate the decay widths, given by [26]

$$\begin{aligned} \Gamma (\mathbf{B}_c\rightarrow \mathbf{B}_n M)= \frac{|\vec {p}_{cm}|}{8\pi m_{\mathbf{B}_c}^2}|{\mathcal {A}}(\mathbf{B}_c\rightarrow \mathbf{B}_n M)|^2, \end{aligned}$$

(15)

where $|\vec {p}_{cm}|= \sqrt{[(m_{\mathbf{B}_c}^2-(m_{\mathbf{B}_n}+m_M)^2] [(m_{\mathbf{B}_c}^2-(m_{\mathbf{B}_n}-m_M)^2]}/(2 m_{\mathbf{B}_c})$.

Table 1 Amplitudes of $T(\mathbf{B}_{c}\rightarrow \mathbf{B}_n M)$, where T-amps refers to $T(\mathbf{B}_{c}\rightarrow \mathbf{B}_n M)$ and CF (SCS) represents Cabbibo favored (singly Cabbibo-suppressed)

3 Numerical analysis

In the numerical analysis, we examine ${\mathcal {B}}(\Lambda _c^+ \rightarrow \Lambda ^0 K^+,\Sigma ^0 K^+,p\pi ^0)$ and ${\mathcal {R}}'_{K/\pi }$ by including the breaking effects of the $SU(3)_f$ symmetry to see if one can explain their data in Eq. (2). The theoretical inputs for the CKM matrix elements are given by [26]

$$\begin{aligned} (V_{cs},V_{ud},V_{us},V_{cd})=(1-\lambda ^2/2,1-\lambda ^2/2, \lambda ,-\lambda ), \end{aligned}$$

(16)

with $\lambda =0.2248$ in the Wolfenstein parameterization. We perform the minimum $\chi ^2$ fit, in terms of the equation of [22]

$$\begin{aligned} \chi ^2= \sum _{i} \bigg (\frac{{\mathcal {B}}^i_{th}-{\mathcal {B}}^i_{ex}}{\sigma _{ex}^i}\bigg )^2+ \sum _{j}\bigg (\frac{{\mathcal {R}}^j_{th}-{\mathcal {R}}^j_{ex}}{\sigma _{ex}^j}\bigg )^2, \end{aligned}$$

(17)

with ${\mathcal {B}}={\mathcal {B}}(\Lambda _c^+\rightarrow \mathbf{B}_nM)$ and ${\mathcal {R}}={\mathcal {B}}(\Xi _c^0\rightarrow \mathbf{B}_nM)/{\mathcal {B}}(\Xi _c^0\rightarrow \Xi ^-\pi ^+)$, where the subscripts th and ex are denoted as the theoretical inputs from the amplitudes in Table 1 and the experimental data points in Table 2, respectively, while $\sigma _{i,j}$ correspond to the 1$\sigma$ errors. By following Refs. [21,22,23], we extract the parameters, which are in fact complex numbers, given by

$$\begin{aligned}&a_1, a_2e^{i\delta _{a_2}},a_3e^{i\delta _{a_3}}, v_1e^{i\delta _{v_1}}, v_2e^{i\delta _{v_2}},v_3e^{i\delta _{v_3}}, \end{aligned}$$

(18)

where $a_{4,5, \ldots ,7}$ have been ignored as discussed in Sect. 3. Since only the relative phases contribute to the branching ratios, $a_1$ is set to be real without losing generality. However, we take $v_i$ to be real numbers in order to fit 8 parameters with the 9 data points in Table 2. In the calculation, $\delta _{a_i}$ (i=2 or 3) from $a_i e^{\delta _{a_i}}$ is a fitting parameter, which can absorb the phase of $\delta _{v_i}$ from the interference in the data fitting. Note that $\delta _{a_i}\ (i=2,3)$ have been fitted with the imaginary parts [22]. As a result, we may set $\delta _{v_{2,3}}$ along with the overall phase of $\delta _{v_1}$ to be zero for the estimations of the decay branching ratios due to the SU(3) breaking effects. We will follow Ref. [25] to test our assumption, where a similar global fit in the approach of the $SU(3)_f$ symmetry has been done to extract $a_2 e^{\delta _{a_2}}$ by freely rotating the angle of $\delta _{a_2}$ from $-180^\circ$ to $180^\circ$ to estimate the uncertainties of the branching ratios. Subsequently, the fit with the breaking effects in the $SU(3)_f$ symmetry yields

$$\begin{aligned} (a_1,a_2,a_3)= & {} (0.252\pm 0.005,0.127\pm 0.009,0.091\nonumber \\&\pm \, 0.015)\,\text {GeV}^3,\nonumber \\ (\delta _{a_2},\delta _{a_3})= & {} (73.0\pm 27.3, 40.2\pm 4.7)^\circ ,\nonumber \\ (v_1,v_2,v_3)= & {} (0.090\pm 0.032,-0.037\pm 0.013,0.025\nonumber \\&\pm \, 0.012)\,\text {GeV}^3, \end{aligned}$$

(19)

with $\chi ^2/d.o.f=3.0/1$, where d.o.f represents the degree of freedom. Note that $a_{1,2,3}$ and their phases are nearly the same as those without the breaking of $SU(3)_f$ [22]. With the parameters in Eq. (19), we obtain the branching ratios of the CF and SCS $\mathbf{B}_c\rightarrow \mathbf{B}_nM$ decays, shown in Table 3.

Table 2 The data of the $\mathbf{B}_c\rightarrow \mathbf{B}_n M$ decays, together with the reproduction with the exact (broken) $SU(3)_f$ symmetry in the 3rd (4th) column

Table 3 The branching ratios of the $\mathbf{B}_{c}\rightarrow \mathbf{B}_n M$ decays, where the numbers with the dagger ($\dagger$) correspond to the reproductions of the experimental data input, instead of the predictions

4 Discussions and conclusions

As seen from Tables 2 and 3, the breaking effects associated with $v_2$ and $v_3$ on the branching ratios of the SCS $\Lambda _c^+\rightarrow \mathbf{B}_nM$ decays are at most around $30\%$, which is close to the naive estimation of $(f_K/f_\pi )^2\simeq 40\%$. In particular, we get ${\mathcal {B}}(\Lambda _c^+ \rightarrow \Lambda ^0 K^+,\Sigma ^0 K^+) =(6.1\pm 0.9,5.2\pm 0.7)\times 10^{-4}$, which explain the data in Eq. (2) well and alleviate the $(1.3-1.6)\sigma$ deviations by the fit with the exact $SU(3)_f$ symmetry [22]. Meanwhile, the branching ratios for the CF modes are fitted to be the same as those without the breaking except ${\mathcal {B}}(\Lambda _c^+ \rightarrow \Xi ^0 K^+)$, which is slightly different in order to account for the recent observational value given by BESIII [36].

Moreover, the fitted value of ${\mathcal {R}}'_{K/\pi }=(0.6\pm 0.2)s_c^2=0.03\,\pm \, 0.01$ explains the data very well for the first time. This leads to the prediction of ${\mathcal {B}}(\Xi _{c}^{0} \rightarrow \Xi ^{-} K^{+})=(4.6 \pm 1.7)\times 10^{-4}$, with $v_1+v_2$ as the destructive contribution to reduce the value of $(7.6 \pm 0.4)\times 10^{-4}$ under the exact $SU(3)_f$ symmetry, whereas ${\mathcal {B}}( \Xi _{c}^{0}\rightarrow \Sigma ^{-} \pi ^{+}) =(12.8 \pm 3.1)\times 10^{-4}$ receives the constructive contribution from $v_1+v_2$, with $T(\Xi _c^0\rightarrow \Xi ^{-} K^{+},\Sigma ^-\pi ^+)=\mp [a_1\mp (v_1+v_2)]s_c$. Since there are other similar interferences between $a_i$ and $v_i$, which come from $T(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-)=\mp [a_2\mp (v_1+v_3)]s_c$ and $T(\Xi _c^+\rightarrow \Sigma ^0 \pi ^+,\Sigma ^+\pi ^0)=\mp \sqrt{2} [(a_1-a_2)+(v_2-v_3)]s_c$, it is predicted that ${\mathcal {B}}(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-)=(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}$ and ${\mathcal {B}}(\Xi _c^+\rightarrow \Sigma ^0 \pi ^+(\Sigma ^+\pi ^0))=( 10.3 \pm 1.7)\times 10^{-4}$. It is interesting to note that the important roles of the terms associated with $v_1$ in the T amplitudes are also projected in the $\Xi _c^{0,+}$ modes, particularly, $\Xi _c^0\rightarrow \Xi ^{-} K^{+}$ and $\Xi _{c}^{0}\rightarrow \Sigma ^{0} \pi ^{0}$. Clearly, these SCS $\Xi _c$ decays all contain sizable $SU(3)_f$ breaking effects, and can be treated as golden modes to test the $SU(3)_f$ symmetry.

In our calculation, we treat $v_3$ as the norm in $T(\Lambda _c^+\rightarrow \Sigma ^{0} K^{+})\simeq \sqrt{2}(a_1-a_3+v_3)s_c$ of Table I, such that $\delta _{v_3}$ is allowed to rotate from $-90^\circ$ to $50^\circ$ without letting ${\mathcal {B}}( \Lambda _{c}^{+} \rightarrow \Sigma ^{0} K^{+})$ exceed the data. Since the allowed range for $\delta _{v_{3}}$ is large, it is clear that its value is insensitive to the data. On the other hand, in order to explain the experimental data of ${\mathcal {R}}'_{K/\pi }$ with the smallest corrections from $v_i e^{\delta _{v_i}}$, we assume maximumly destructive interferences between $a_i e^{\delta _{a_i}}$ and $v_i e^{\delta _{v_i}}$. Explicitly, in $T(\Xi _c^0\rightarrow \Xi ^{-} K^{+})\simeq -2(a_1-v_1-v_2)s_c$ for ${\mathcal {R}}'_{K/\pi }$, we can take $\delta _{v_1}=\delta _{v_2}=\delta _{a_1}=0$ as an overall phase in $T(\Xi _c^0\rightarrow \Xi ^{-} K^{+})$. Consequently, we are able to assume real values for $v_i$ (i = 1,2,3) without loss of generality. Finally, we remark that, even with the breaking effects, we are still unable to fit the data of $\Lambda _c^+\rightarrow p\pi ^0$ in Eq. (1) as our result for its branching ratio of $(5.4\pm 1.0)\times 10^{-4}$, which is close to $(5.5\pm 0.3)\times 10^{-4}$ from the factorization approach [22], is lower than the current experimental upper bound of $3\times 10^{-4}$ [30, 31]. However, it is possible that $H(\overline{15})$ would be non-negligible in $\Lambda _c^+\rightarrow p\pi ^0$. For example, with $T(\Lambda _c^+\rightarrow p\pi ^0) =\sqrt{2}(a_{2}+a_3+v_2-(a_{6} - a_{7})/2)s_c$ in Table 1, the contribution from $(a_6-a_7)/2$ of $H(\overline{15})$ might be comparable with that from $a_2+a_3+v_2$ of H(6), while $a_{2,3}$ and $v_2$ of Eq. (19) are taken to be small. In particular, with $(a_6-a_7)/2$ to be around $25\%$ of $a_{2}+a_3+v_2$, ${\mathcal {B}}(\Lambda _c^+\rightarrow p\pi ^0)$ can be reduced to be within the experimental upper bound due to the destructive interference. In this case, there is a corresponding constructive interference in $\Lambda _c^+\rightarrow n\pi ^+$, leading to ${\mathcal {B}}(\Lambda _c^+\rightarrow n\pi ^+)\sim 17\times 10^{-4}$, which breaks the relation of ${\mathcal {A}}(\Lambda _c^+\rightarrow n\pi ^+)= \sqrt{2} {\mathcal {A}}(\Lambda _c^+\rightarrow p\pi ^0)$ [8]. Clearly, in order to confirm the importance of $H(\overline{15})$, both experimental observations of $\Lambda _c^+\rightarrow p\pi ^0$ and $\Lambda _c^+\rightarrow n\pi ^+$ are needed.

In sum, we have studied the singly Cabibbo-suppressed charmed baryon decays. We have shown that the breaking effect of the $SU(3)_f$ symmetry can be used to understand the experimental data of ${\mathcal {B}}(\Lambda _c^+\rightarrow \Sigma ^{0} K^{+},\Lambda ^{0} K^{+})$ and $R'_{K/\pi }={\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- K^+)$/${\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- \pi ^+)$. With these effects, we have obtained that ${\mathcal {B}}(\Xi _{c}^{0} \rightarrow \Xi ^{-} K^{+},\Sigma ^{-} \pi ^{+}) =(4.6 \pm 1.7,12.8 \pm 3.1)\times 10^{-4}$, ${\mathcal {B}}(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-) =(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}$ and ${\mathcal {B}}(\Xi _c^+\rightarrow \Sigma ^0 \pi ^+(\Sigma ^+\pi ^0))=(10.3 \pm 1.7)\times 10^{-4}$, which are quite different from those predicted by the approach with the exact $SU(3)_f$ symmetry. However, even with the breaking effects, our result for the branching ratio of $\Lambda _c^+\rightarrow p\pi ^0$ is still higher than the current experimental upper bound, which clearly requires a close examination by a future dedicated experiment.