Anti-triplet charmed baryon decays with SU(3) Flavor Symmetry

We study the decays of the anti-triplet charmed baryon state $(\Xi_c^0,\Xi_c^+,\Lambda_c^+)$ based on the $SU(3)$ flavor symmetry. In particular, after predicting ${\cal B}(\Xi_c^0\to \Xi^-\pi^+)=(15.7 \pm 0.7)\times 10^{-3}$ and ${\cal B}(\Xi_c^+\to\Xi^-\pi^+\pi^+)=(14.7\pm 8.4)\times 10^{-3}$, we extract that ${\cal B}(\Xi_c^0\to\Lambda K^-\pi^+,\Lambda K^+K^-,\Xi^- e^+\nu_e)=(16.8\pm 2.3,0.45\pm 0.11,48.7\pm 17.4)\times 10^{-3}$ and ${\cal B}(\Xi_c^+\to p K_s^0 K_s^0,\Sigma^+ K^-\pi^+,\Xi^0\pi^+\pi^0,\Xi^0 e^+\nu_e)=(1.3\pm 0.8,13.8\pm 8.0,33.8\pm 21.9,33.8^{+21.9}_{-22.6})\times 10^{-3}$. We also find that ${\cal B}(\Xi_c^0\to \Xi^{0} \eta,\Xi^{0} \eta')=(1.7 ^{+ 1.0 }_{- 1.7 },8.6 ^{+ 11.0 }_{-\;\,6.3 })\times 10^{-3}$, ${\cal B}(\Xi_c^0\to \Lambda^{0} \eta,\Lambda^{0} \eta')=(1.6 ^{+ 1.2 }_{- 0.8 },9.4 ^{+ 11.6 }_{-\;\,6.8 })\times 10^{-4}$ and ${\cal B}(\Xi_c^+\to \Sigma^{+} \eta,\Sigma^{+} \eta')=(28.4 ^{+ 8.2 }_{- 6.9 },13.2 ^{+ 24.0 }_{- 11.9 })\times 10^{-4}$. These $\Xi_c$ decays with the branching ratios of $O(10^{-4}-10^{-3})$ are clearly promising to be observed by the BESIII and LHCb experiments.


I. INTRODUCTION
In terms of the SU(3) flavor (SU(3) f ) symmetry, the Ξ c decays should be in association with the Λ + c ones as Ξ 0 c , Ξ + c and Λ + c are united as the lowest-lying anti-triplet of the charmed baryon states (B c ). Nonetheless, in accordance with f Ξ + c +f Ξ 0 c +f Ω 0 c ≃ 0.136 f Λ + c estimated in Refs. [1,2], where f Bc,Ω 0 c stand for the fragmentation fractions for the rates of the charmed baryon productions, the measurements of the Ξ c decays are not easy tasks compared to the Λ + c ones. For example, the two-body Λ + c → B n M decays with B n (M) the baryon (pseudoscalar-meson ) have been extensively studied by experiments. Interestingly, six decay Λ + c decay modes have been recently reexamined or measured by BESIII [3,4]. In addition, LHCb has just observed the three-body Λ + c → pMM decays [5], together with their CP violating asymmetries [6]. However, no much progress has been made in the Ξ c decays. In particular, none of the absolute branching fractions in the Ξ c decays has been given yet.

II. FORMALISM
For the two-body anti-triplet of the lowest-lying charmed baryon decays of B c → B n M, and B n (M) are the baryon (pseudoscalar) octet states, the effective Hamiltonian responsible for the tree-level c → sud, c → uqq and c → dus transitions are given by [39] with qq = dd or ss, G F the Fermi constant, V ij the CKM matrix elements, and c ± the scaledependent Wilson coefficients to take into account the sub-leading-order QCD corrections.
which correspond to the tensor notations of 1/2ǫ ijl H(6) lk and H(15) ij k , respectively, with (i, j, k) representing the quark indices and the non-zero entries being H 22 (6) = 2 and where the individual non-zero entries of H(6) lk and H(15 can be presented as the matrix forms: Correspondingly, the B c anti-triplet and B n octet states are written as The adding of the singlet η 1 to the octet (π, K, η 8 ) leads to the nonet of the pseudoscalar meson, given by [30] where (η, η ′ ) are the mixtures of (η 1 , η 8 ), with the mixing angle φ = (39.3 ± 1.0) • [40] for The amplitudes of the B c → B n M decays via the effective Hamiltonian in Eq. (1) appear forms, the amplitudes of B c → B n M can be further derived as with T (B c → B n M) given by [28] T where T ij ≡ (B c ) k ǫ ijk , and (c − , c + ) have been absorbed into the SU(3) parameters of (a 1 , a 2 , a 3 , h) and (a 4 , a 5 , a 6 , a 7 , h ′ ), respectively, and the h (′) terms correspond to the contributions from the singlet η 1 . With the T -amps expanded in

III. NUMERICAL RESULTS AND DISCUSSIONS
For the numerical analysis, we note that the contributions of the SU(3) parameters (a 4 , a 5 , a 6 , a 7 , h ′ ) from H(15) would be neglected based on the following reasons. First, the contributions to the decay branching rates from H(15) and H(6) lead to a small ratio of R(15/6) = c 2 + /c 2 − ≃ 17% in terms of (c + , c − ) = (0.76, 1.78) from the QCD calculation at the scale µ = 1 GeV in the naive dimensional regularization (NDR) scheme [41,42].
Second, it is pointed out in Ref. [19] that O ( †,′) + belong to H(15) in the group structure and behave as symmetric operators in color indices, whereas the baryon wave functions are totally antisymmetric, such that the mismatch causes the disappearance of c + O ( †,′) + in the calculation of the non-facotrizable effects, which are regarded to be significant in the charmed baryon decays. Note that even though the single ignoring of H(15) is viable, a possible interference between the amplitudes with H(6) and H(15) may be sizable to fail this assumption, which will be tested in the fit. Hence, being from H(6) the parameters (a 1 , a 2 , a 3 , h) in Eq. (9) are kept for the fit, which are in fact complex. Since an overall phase can be removed without losing generality, we set a 1 to be real, such that there are seven real independent parameters to be determined, given by We use the minimum χ 2 fit for the determination, given by where B i th and R j th stand for the separated decay branching ratios and the ratios of the two-decay branching fractions from the SU(3) amplitudes, while B i ex and R j ex are the corresponding experimental data, along with σ i ex and σ j ex the 1σ uncertainties, respectively. With the ten experimental data in Table 2, the global fit results in (a 1 , a 2 , a 3 , h) = (0.244 ± 0.006, 0.115 ± 0.014, 0.088 ± 0.019, 0.105 ± 0.073) GeV 3 , (δ a 2 , δ a 3 , δ h ) = (78.1 ± 7.1, 35.1 ± 8.7, 10.2 ± 29.6) • , where d.o.f represents the degree of freedom. The numerical values for the parameters in Eq. (14) are the theoretical inputs, which are used to predict the two-body B c → B decays in Table 3.

ACKNOWLEDGMENTS
This work was supported in part by National Center for Theoretical Sciences, MoST