Charmed Baryon Weak Decays with Decuplet Baryon and SU(3) Flavor Symmetry

We study the branching ratios and up-down asymmetries in the charmed baryon weak decays of ${\bf B}_c\to {\bf B}_DM$ with ${\bf B}_{c(D)}$ anti-triplet charmed (decuplet) baryon and $M$ pseudo-scalar meson states based on the flavor symmetry of $SU(3)_F$. We propose equal and physical-mass schemes for the hadronic states to deal with the large variations of the decuplet baryon momenta in the decays in order to fit with the current experimental data. We find that our fitting results of ${\cal B}({\bf B}_c\to {\bf B}_DM) $ are consistent with the current experimental data in both schemes, while the up-down asymmetries in all decays are found to be sizable, consistent with the current experimental data, but different from zero predicted in the literature. We also examine the processes of $\Xi_c^0 \to \Sigma^{\prime 0}K_S/K_L$ and derive the asymmetry between the $K_L/K_S$ modes being a constant.

However, most of the recent experimental and theoretical activities have been concentrated on the charmed baryon decays with the octet baryon in the final states, whereas there has been a little studies for the decuplet modes. Note that most of the charmed baryon experiments with the decuplet baryon were all done before the millennium. In this work, we will examine the two-body weak decays of B c → B D M, where B c(D) and M represent the anti-triplet charmed (decuplet) baryon and octet pseudo-scalar meson states based on SU(3) F . There are two important features for the decays of B c → B D M. The first one is that all factorizable amplitudes vanish, resulting in theoretically clean predictions for the non-factorizable contributions of the decays. The other one is that the decays involve only a few SU(3) F parameters, which are able to be determined by the current experimental data.
On the other hand, the up-down asymmetries of α in Λ + c → Ξ 0 K + and Λ + c → Ξ ′0 K + have also been given recently by the BESIII Collaboration with the results of α(Λ + c → Ξ 0 K + , Ξ ′0 K + ) = (0.77 ± 0.78, −1.00 ± 0.34) [11], respectively, where Ξ ′0 belongs to the decuplet baryon state with spin-3/2. Although the former experimental result is still consistent with zero, the later one is clearly sizable. This non-vanishing large asymmetry is different from the prediction in the most theoretical calculations in the literature [15][16][17][18][19][20][21][22][23]34]. Recently, based on the flavor symmetry of SU(3) F , we show that α(Λ + c → Ξ 0 K + ) = (0.94 +0.06 −0.11 ) [38], which is consistent with the data, but much larger than zero. In this work, we will particular check the up-down asymmetry in Λ + c → Ξ ′0 K + to see if it agrees with the experimental non-zero result in the SU(3) F approach. This paper is organized as the follow. In Sec. II, we present the formalism. We show how the decay amplitudes are related based on SU(3) F . In Sec. III, we provide the numerical results of the decay branching ratios and up-down asymmetries in B c → B D M. We conclude our study in Sec. IV.

II. FORMALISM
In order to investigate the two-body decays of the anti-triplet charmed baryon (B c ) to decuplet baryon (B D ) and octet pseudoscalar meson (M) states, we write their representations under the flavor symmetry of SU(3) F as Here, we have assumed that the physical meson η is solely made of the octet state [2]. The effective Hamiltonian associated with c → uds, c → uqq (q = d, s) and c → usd transitions is given by [27] where (|V * cs V ud |, |V * cd V ud |, |V * cd V us |) ≃ (1, s c , s 2 c ) with s c ≡ sin θ c ≈ 0.225 [2], c i (i=+,-) correspond to the Wilson coefficients, G F is the Fermi constant, and O q 2 q 1 ± with (q 1 q 2 ) ≡ q 1 γ µ (1 − γ 5 )q 2 represent the four-quark operators.
As O ± belong to 15 and 6 representations under SU(3) F , respectively, which are symmetric and antisymmetric in flavor and color indices, we can decompose the effective Hamiltonian in the tensor forms of H (15) and H(6), given by respectively, where we have used the convention of V cd = −V us = s c . 1 The most significant feature for B c → B D M is that the decay amplitude is essentially nonfactorizable due to the vanishing matrix element of the baryonic transition, i.e.
The reason is that the light quark pair in the anti-triplet charmed baryon state is anti-symmetric, whereas that in the decuplet one is totally symmetric. As a result, we can safely neglect H (15), which only contributes to the factorizable processes [15,[39][40][41][42].
In general, the decay amplitude of B c → B D M is given by where q µ is the four-momentum of the meson, w µ B D is the Rarita-Schwinger spinor vector for the spin-3/2 particle of B D , P (D) corresponds to the P (D)-wave amplitude, and u Bc is the spin-1/2 Dirac spinor of B c . By assuming CP invariance, P and D can be taken to be real.
Under SU(3) F , the amplitudes associated with P and D are related by respectively, where P 0 (D 0 ) is the real parameter to be determined and f BcB D M is the SU(3) F overlapping factor, defined by The value of f BcB D M in Eq. (6) depends on the specific mode in B c → B D M, for example, We will list the values of f BcB D M in the next section. We note that the factors of f BcB D M with M being a singlet under SU(3) F vanish so that the corresponding decays with a physical meson η ′ are suppressed. The reason is that 3 ⊗ 6 ⊗ 10 ⊗ 1 cannot form a singlet to be invariant under SU(3) F , where 3, 6, 10 and 1 are the SU(3) F representations for the antitriplet charmed baryon, anti-symmetric part of the effective Hamiltonian, decuplet baryon and singlet meson states, respectively. In practice, since P and D share the same overlapping factor, one can alternatively combine these two real parameters into one complex parameter for convenience [26][27][28][29][30][31][32][33][35][36][37][38].
Consequently, the decay width (Γ) for B c → B D M is given by while the up-down asymmetry (α) has the form where | q| represents the absolute value of the three-momentum of the octet pesudoscalar meson M or the decuplet baryon B D in the CM frame, m Bc is the mass of the charmed baryon Under the exact flavor symmetry of SU(3) F , one can simply impose the equal-mass (em) conditions, given by

III. NUMERICAL RESULTS
In the em scheme, from Eq. (9) we see that there is only one combined parameter ofr ≡ ξr for α(B c → B D M). By using the experimental data of α(Λ + c → Ξ ′0 K + ) = −1.00 ± 0.34 in Ref. [11], we expect that the up-down asymmetry in every decay mode of B c → B D M should have the same value as where the lower uncertainty of "0" reflects that the physical value of α cannot be less than −1. From Eqs. (9) and (11), we obtain On the other hand, the decay branching ratios in Eq. (8) also depend on one unknown parameter, defined by which can be determined by only one experimental data point. However, there are four experimental branching ratios as shown in Table I. To obtain the most plausible value for P 0 under the current experimental data, we adopt the minimal χ 2 fitting, defined by where B em is the decay branching ratio generated by P 0 in the em scheme with the experimental measured lifetime in Ref. [2] and B ex (σ ex ) corresponds to the measured branching ratio (uncertainty) in the data. By performing χ 2 fit with the minimal value of χ 2 em , we obtain that where d.o.f. represents "degree of freedom." The small value of χ 2 em /d.o.f. for the fit in Eq. (15) indicates that the em scheme is good to explain the current experimental data.
a The data has not been included in the data fitting.
We now discuss in the pm scheme. From the data of α ex (Λ + c → Ξ ′0 K + ), we find that With the value in Eq. (16), our predictions for α pm (B c → B D M) are shown in Tables I-IV.
To valuate the decay branching ratios, we have to refit the data, found to be (P 0 , D 0 ) pm = (3.2 ± 0.4, −5.1 ± 2.5)10 −2 G F GeV , R P 0 D 0 = 0.70 , where R P 0 D 0 stands for the correlation between the two fitted parameters of P 0 and D 0 and the data point of α ex (Λ + c → Ξ ′0 K + ) has also been included in the fit. Our results of B(B c → B D M) are listed in Tables I-IV. We note that unlike the cases in the em scheme, ζ In Table I  show the experimental data [2,11,13,14] as well as the theoretical calculations in the literature [15,16,34]. In particular, Xu and Kamal in Ref. [15] consider the baryon pole term as the nonfactorizable amplitude, Korner and Kramer in Ref. [16] take account of the heavy quark symmetry and covariant quark model for the baryon wave function, and Sharma and Verma in Ref. [34] study the branching ratios with SU(3) F based on the old experimental data. Note that our result of B em (Λ + c → Ξ ′0 K + ) = (4.1 ± 0.3) × 10 −3 is smaller than, but still consistent with, the current experimental value of (5.02 ± 1.04) × 10 −3 . However, it fits well with the previous experimental result of B(Λ + c → Ξ ′0 K + ) = (4.0 ± 1.0) × 10 −3 as shown in Table 1 of Ref. [11]. However, our result of B pm (Λ + c → Ξ ′0 K + ) = (1.0 ± 0.2) × 10 −3 is inconsistent with the data. It is interesting to point out that the up-down asymmetries for all decays are expected to be zero by theoretical studies in Refs. [15,16,34] due to the vanishing D-wave amplitudes, which are different from our nonzero results and inconsistent with the current experimental result of α ex (Λ + c → Ξ ′0 K + ) = −1.00 ± 0.34 [11]. We recommend to measure α(Λ + c → ∆ ++ K − ) in the future experiment as this decay channel has the largest decay branching rate, which will be a clean justification of the SU(3) F   [34].
There are some common features between our results and those in Refs. [15,16,34].
The most important one is that the vanishing amplitudes in the Cabibbo allowed decays of  [2] are insufficient to rule out this feature yet. It is interesting to note that the decay branching ratios given in the various theoretical calculations may not obey the flavor symmetry of SU(3) F in general, but they all preserve the isospin symmetry. In particular, the isospin relations in the Cabibbo allowed decays can be summarized as follows: Similar relations in the singly and doubly-Cabibbo suppressed decays are also expected.
Finally, we explore the decay processes of Ξ 0 c → Σ ′0 K S /K L , which contain both Cabibbo allowed and doubly-suppressed contributions as shown in Table IV, resulting in an asymmetry due to the interference between the two contributions. Explicitly, the K S − K L asymmetry is found to be  It is also interesting to note that the vanishing rates for the Cabibbo allowed decays of Ξ + c → Σ ′+K 0 and Ξ + c → Ξ ′0 π + have not been supported by the experimental data yet. For the up-down asymmetries, we have found that they are sizable, which are different from the prediction of zero due to the vanishing D-wave contributions in the literature. In for all decay modes in the em scheme, while they range from −1 to −0.42 at 1σ level in the pm scheme, consistent with the current only available data of α ex (Λ + c → Ξ ′0 K + ) = −1.00 ± 0.34 [11] for the up-down asymmetry. To justify the SU(3) F approach, we have proposed to search for α(Λ + c → ∆ ++ K − ), which is predicted to be −0.86 +0.44 −0.14 , in the future experiments, as the the decay has the largest branching rate among B c → B D M.
In addition, we have examined the processes of Ξ 0 c → Σ ′0 K S /K L , which contain both Cabibbo allowed and doubly-suppressed contributions. We have predicted the K L − K S asymmetry of R(Ξ 0 c → Σ ′0 K S /K L ) is −0.106, which depends on neither model/scheme nor the data fitting. Clearly, this asymmetry is a clean result in the SU(3) F approach, which should be tested by the experiments.