A diagrammatic analysis of two-body charmed baryon decays with flavor symmetry

In terms of the topological diagram approach with the SU(3) flavor symmetry, we study the two-body anti-triplet charmed baryon decays, whereas the symmetry properties of the baryon wave functions are not taken into account. Since each (non-)factorizable topological amplitude can be extracted with the data, we find that only one W-exchange decaying process can dominantly contribute to $\Lambda_c^+\to \Xi^0 K^+$. Besides, it is found that the non-factorizable contributions can cause the destructive interference, such that our result of ${\cal B}(\Lambda_c^+\to p\pi^0)=(0.8\pm 0.7)\times 10^{-4}$ agrees with the experimental upper bound. We also predict that ${\cal B}(\Lambda_c^+ \to \Sigma^+ \eta^\prime)=(0.5\pm 0.1)\times 10^{-2}$, ${\cal B}(\Lambda_c^+ \to p \eta^\prime)=(1.0\pm 0.3)\times 10^{-4}$, ${\cal B}(\Xi_c^0 \to \Xi^-K^+)=(5.23\pm 0.04)\times 10^{-4}$ and ${\cal B}(\Xi_c^0 \to \Lambda^0\bar K^0,\Xi^-\pi^+)=(0.80\pm 0.20,1.91\pm0.17)\times 10^{-2}$, to be compared with the future BESIII and LHCb experiments.

The inconsistencies indicate the possibility that, although being often neglected in the bhadron decays [7][8][9], the non-factorizable effects are able to give rise to sizeable contributions in the two-body charmed baryon decays. For example, the Λ + c → Ξ 0 K + and Λ + c → Ξ * 0 K + decays only proceed through the two non-factorizable W -exchange processes, where Ξ * 0 ≡ Ξ(1530) 0 . Indeed, the recent measurements of their absolute branching fractions by BESIII present that [10] where the more accurate results reconfirm the previous measurements relative to Λ + c → pK − π + [1,11,12]. The approach of the SU(3) flavor (SU(3) f ) symmetry is commonly used in the heavy hadron decays [4,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], where the initial and final states are related to the irreducible SU(3) f representation of the effective Hamiltonian, to derive the SU(3) f amplitudes. By merging all possible factorizable and non-factorizable effects, the SU(3) f amplitudes are enabled to explain B(Λ + c → Ξ 0 K + ) [4,[22][23][24]. However, one cannot distinguish which of the two W -exchange processes gives more contribution, while the similar bc → pK − , and Ξ + cc → Σ ++ c (2520)K − have been studied in [28][29][30][31]. For B(Λ + c → pπ 0 ) estimated to be higher than the experimental upper bound [24], which corresponds to a 2 ∼ O(1.0) from the observation of B(Λ + c → pK 0 ), the sizeable destructive interferences from the non-factorizable effects are proposed to reduce the estimation [5]. Therefore, the investigation of the all kinds of non-factorizble effects in B c → BM is necessary, which come from five different processes. In Refs. [5,32], since the topological diagrams have been drawn to identify different decaying processes, with the corresponding parameters extracted from the data, we can systematically analyze the different non-factorizable effects. This should be like the studies with the topological amplitudes in the D decays [33][34][35][36]. In this report, by including all the existing data, we will determine the topological amplitudes in the two-body charmed baryon decays, such that we will be able to explain the inconsistent extractions of a 2 from Λ + c → pK 0 and Λ + c → pπ 0 , together with the relation of . Moreover, some of the branching ratios can be predicted.

II. DIAGRAMMATIC APPROACH
For the two-body charmed baryon decays via the tree-level c → sud, c → udd and c → uss transitions, the effective Hamiltonian is given by [37] where G F is the Fermi constant, c 1,2 are the Wilson coefficients, and V ij the CKM matrix elements. The four-quark operators O (q) 1,2 are written as where q = (d, s) and (q 1 q 2 ) =q 1 γ µ (1 − γ 5 )q 2 . The decays with V cs V ud ≃ 1 and V cs V us ≃ −V cd V ud ≃ s c are classified as the Cabibbo-favored (CF) and singly Cabibbo-suppressed (SCS) processes, respectively.
In terms of the effective Hamiltonian in Eq. (2), the amplitudes of B c → BM can be depicted as the quark diagrams [5,32], where the quark lines should be in accordance with the operators in Eq. (3). For example, the color-allowed and color-suppressed amplitudes with the external and internal W emissions are drawn in Fig. 1a and b, respectively, being parameterized as T and C, instead of being calculated with the QCD-inspired models. Apart which can add a pre-factor of ± 1/2 to the parameters of (T, C (′) , E B(M ) , E ′ ). Likewise, the (η, η ′ ) meson states mix with η q = 1/2(uū + dd) and η s = ss, whose mixing matrix is with the mixing angle φ = (39.3 ± 1.0) • . Note that, unlike the topological quark-diagram approach in Ref. [32], the symmetry properties of the baryon wave functions are not taken into account, such that the topological amplitudes are presented in the simple forms. Subsequently, we obtain the amplitudes of B c → BM in Table I for the observed ones, of which the decay widths depend on the integration of the phase space for the two-body decays,

III. NUMERICAL RESULTS
In our numerical analysis, the topological amplitudes to be extracted with the data are in fact complex, presented as which are counted to be 11 parameters, with T set to be relatively real. As the theoretical inputs in the amplitudes of B c → BM, the CKM matrix elements in the Wolfenstein parameterization are given by [1] ( with λ = s c = 0.22453 ± 0.00044. We perform the numerical analysis with the minimum χ 2 -fit method, of which the equation is written as [24] where B (R) denotes (the ratios of) the branching ratios. Besides, the subscripts th and ex stand for the theoretical inputs from the amplitudes in Table I and the experimental data   points in Table II with d.o.f representing the degree of freedom. By taking the parameters in Eq. (9) as the inputs, we theoretically reproduce (the ratios of) the branching ratios, together with the predictions for B(Λ + c → Σ + η ′ , pη ′ ) and B(Ξ 0 c → Ξ − K + , Λ 0K 0 , Ξ − π + ), given in Table II.

IV. DISCUSSIONS AND CONCLUSIONS
The value of χ 2 /d.o.f ≃ 0.4 presents a reasonable fit, which demonstrates that the topological amplitudes based on the diagrammatic approach can explain the data of the 0.027 ± 0.002 0.028 ± 0.006 0.42 ± 0.11 0.5 ± 0.1 0.42 ± 0.06 a In the revision of the numerical analysis, the values have been fixed as 1.3 ± 0.7 [40].
two-body B c → BM decays. The contributions from the factorizable and non-factorizable decaying processes can be specifically quantified in Eq. (9). Moreover, the fit of |T | ≃ |E M | ≃ 0.4 and |C| ≃ |C ′ | ≃ |E B | ≃ 0.2 shows that the non-factorizable effects are extracted to be as large as the factorizable ones, except for E ′ ≃ 0, which can be the hint for the QCDinspired model calculations. Indeed, the (non-)factorizable effects have been explored in the Λ b decays, which relies on the soft-collinear effective theory [28].
, it is clear to see that, instead of E ′ , E B as one of the W -exchange processes in Fig. 1 dominantly contribute to the branching ratio, which is pointed out for the first time.
In the factorization approach, one tends to believe that there exist the universal effective Wilson coefficients a 1,2 for the color-allowed and color-suppressed decay modes, respectively, which is based on the assumption that the non-factorizable effects can give similar contributions to the different decays, and received by a 1,2 . This leads to the relations of with a 2 (ā 2 ) for Λ + c → pπ 0 (pK 0 ) and a 1 (ā 1 ) for Ξ 0 c → Ξ − K + (Ξ − π + ), where f M is the decay constant, presenting the meson production in the factorizable amplitude of A(B c → B n M) ∝ M|(q 1 q 2 )|0 B|(q 3 c)|B c . By means of a 2 =ā 2 and the data input of B(Λ + c → pK 0 ) in Table II, one estimates that B(Λ + c → pπ 0 ) = (5.5 ± 0.3) × 10 −4 [24] , apparently disagreeing with the experimental upper bound of 2.7 × 10 −4 or (0.8 ± 1.4) × 10 −4 [2,39]. While a large destructive interference between the factorizable and non-factorizable amplitudes for Λ + c → pπ 0 is proposed as the solution [5], we show that A(Λ + c → pK 0 ) ∝ C + E M and Table I, where E B − C ′ can give rise to the sizeable destructive interference, leading to B(Λ + c → pπ 0 ) = (0.8 ± 0.7) × 10 −4 in Table II. With the second relation in Eq. (10), a 1 =ā 1 causes R 1 (Ξ 0 c ) = 1.4s 2 c to be far away from the data of (0.56 ± 0.12)s 2 c . The exact SU(3) f symmetry also leads to the deviated , in agreement with the data. We hence conclude that, when the non-factorizable effects can be significant in B c → BM, the effective Wilson coefficients a 1,2 cannot have universal values.
Both by the SU(3) f symmetry, the topological diagrams and SU(3) f amplitudes can be used to explain the data well. Moreover, they are demonstrated to be the equivalent model-indepent approaches [41], while the diagrammatic one explicitly describes the different decaying processes, and the SU(3) f amplitudes merge all possible (non-) factorizable contributions. Since the absolute branching fractions of Ξ 0 c → BM have not been measured, we predict that B(Ξ 0 c → Ξ − K + ) = (5.23 ± 0.04) × 10 −4 and B(Ξ 0 c → Λ 0K 0 , Ξ − π + ) = (0.80 ± 0.20, 1.91 ± 0.17) × 10 −2 , together with B(Λ + c → Σ + η ′ ) = (0.5 ± 0.1) × 10 −2 and B(Λ + c → pη ′ ) = (1.0 ± 0.3) × 10 −4 , to be tested by the future measurements. In sum, we have globally analyzed all the measured B c → BM decays, which is in terms of the diagrammatic approach with the SU(3) f symmetry. We have determined the factorizable and non-factorizble amplitudes, being parameterized to correspond to the topological diagrams. Accordingly, we have been able to distinguish one W -exchange decaying process from the other, which dominantly contributes to Λ + c → Ξ 0 K + . The destructive interference between the factorizable and non-factorizable decaying processes have been given to contribute to B(Λ + c → pπ 0 ), such that the overestimation of B(Λ + c → pπ 0 ) in the factorization could be reduced to agree with the experimental upper bound. For B(Ξ 0 c → Ξ − K + )/B(Ξ 0 c → Ξ − π + ) = (0.56 ± 0.12)s 2 c measured to disapprove the predicted value of 1.4s 2 c in the factorization, it has been regarded to be due to the ignoring of one specific non-factorizble effect for Ξ 0 c → Ξ − π + . We have predicted that B(Λ + c → Σ + η ′ ) = (0.5 ± 0.1) × 10