Two-body hadronic weak decays of antitriplet charmed baryons

We study Cabibbo-favored (CF) and singly Cabibbo-suppressed (SCS) two-body hadronic weak decays of the antitriplet charmed baryons Λc , Ξ 0 c and Ξ + c with more focus on the last two. Both factorizable and nonfactorizable contributions are considered in the topologic diagram approach. The estimation of nonfactorizable contributions from W -exchange and inner W -emission diagrams relies on the pole model and current algebra. The non-perturbative parameters in both factorizable and nonfactorizable parts are calculated in the MIT bag model. Branching fractions and up-down decay asymmetries for all the CF and SCS decays of antitriplet charmed baryons are presented. The prediction of B(Ξc → Ξπ) agrees well with the measurements inferred from Belle and CLEO, while the calculated B(Ξc → Ξ−π+) is too large compared to the recent Belle measurement. We conclude that these two Ξc → Ξπ modes cannot be simultaneously explained within the currentalgebra framework for S-wave amplitudes. This issue needs to be resolved in future study. The longstanding puzzle with the branching fraction and decay asymmetry of Λc → ΞK is resolved by noting that only type-IIW -exchange diagram will contribute to this mode. We find that not only the calculated rate agrees with experiment but also the predicted decay asymmetry is consistent with the SU(3)-flavor symmetry approach in sign and magnitude. Likewise, the CF mode Ξc → Σ+K− and the SCS decays Ξc → pK−,Σ+π− proceed only through type-II W -exchange. They are predicted to have large and positive decay asymmetries. These features can be tested in the near future.


I. INTRODUCTION
Recently, there has been a significant progress in the experimental study of charm physics. In the meson sector, LHCb measured the CP asymmetry difference between D 0 → K − K + and D 0 → π − π + , giving ∆ACP = (−15.4 ± 2.9) × 10 −4 [1], which is the first observation of CP violation in the charm sector. The progress in charmed baryon physics is also impressive. The long-quested doubly charmed baryon was first observed through the process Ξ ++ cc → Λ + c K − π + π + at LHCb in 2017 [2]. Later in 2018, the lifetime of Ξ ++ cc [3], its mass and the two-body weak decay channel Ξ ++ cc → Ξ + c π + [4] were measured by LHCb. Some breakthrough has also been made in singly charmed baryons as well, especially the lightest one Λ + c . Both Belle [5] and BESIII [6] have measured the absolute branching fraction of the decay Λ + c → pK − π + . A new average of (6.28 ± 0.32)% for this benchmark mode is quoted by the Particle Data Group (PDG) [7]. The measurement of Λ + c → pπ 0 , pη [8] indicated that singly Cabibbo-suppressed (SCS) decays are ready to access.
Inspired by latest experimental results of Ξc decays, there have been some efforts from theorists [12][13][14][15][16][17]. Indeed, the study of charmed baryon weak decays, including the charged and neutral Ξc baryons, is an old subject. To understand the underlying dynamical mechanism in hadronic weak decays, one may draw the topological diagrams according to the hadron's content [18]. In charmed baryon decays, nonfactorizable contributions from W -exchange or inner W -emission diagrams play an essential role and they cannot be neglected, in contrast with the negligible effects in heavy meson decays. The fact that all the decays of Ξ +,0 c receive nonfactorizable contributions, especially some decays such as Ξ 0 c → Σ + K − , Ξ 0 π 0 proceed only through purely nonfactorizable diagrams, will allow us to check the importance and necessity of nonfactorizable contributions. However, we still do not have a reliable phenomenological model to calculate charmed baryon hadronic decays so far. In the 1990s various techniques were developed, including relativistic quark model (RQM) [19,20], pole model [21][22][23] and current algebra [22,24], to estimate the nonfactorizable effects in Cabbibo-favored Ξ +,0 c decays. The predicted branching fractions and decay asymmetries in various early model calculations are summarized in Table I. 1 Now with more experimental data accumulated, there are some updated studies in theory [12][13][14][15][16]. In these works except [16], the experimental results are taken as input and global fitting analyses are carried out at the hadron level based on SU(3) flavor symmetry without resorting to the detailed dynamics. Apparently, a reconsideration of charmed baryon weak decays, revealing the dynamics at the quark level, is timely and necessary. Pole model is one of the choices. In the pole model, important low-lying 1/2 + and 1/2 − states are usually considered under the pole approximation. In the decay with a pseudoscalar in the final state, Bc → B ′ + P , the nonfactorizable S-and P -wave amplitudes are dominated by 1/2 − low-lying baryon resonances and 1/2 + ground state baryons, respectively. The S-wave amplitude can be further reduced to current algebra in the soft-pseudoscalar limit. That is, the evaluation of the S-wave amplitude does not require the information of the troublesome negative-parity baryon resonances which are not well understood in the quark model. The methodology was developed and applied in the earlier work [22]. It turns out if the S-wave amplitude is evaluated in the pole model or in the covariant quark model and its variant, the decay asymmetries for both Λ + c → Σ + π 0 and Σ 0 π + were always predicted to be positive, while it was measured to be −0.45±0.31±0.06 for Σ + π 0 by CLEO [26]. In contrast, current algebra always leads to a negative decay asymmetry for aforementioned two modes: −0.49 in [22], −0.31 in [24], −0.76 in [27] and −0.47 in [28]. The issue with the sign of α(Λ + c → Σ + π 0 ) was finally resolved by BESIII. The decay asymmetry parameters of Λ + c → Λπ + , Σ 0 π + , Σ + π 0 and pKS were recently measured by BESIII [29] (see Table III below), for example, α(Λ + c → Σ + π 0 ) = −0.57 ± 0.12 was obtained. Hence, the negative sign of α(Λ + c → Σ + π 0 ) measured by CLEO is nicely confirmed by BESIII. This is one of the strong reasons why we adapt current algebra to work out parity-violating amplitudes.
It is well known that there is a long-standing puzzle with the branching fraction and decay asymmetry of Λ + c → Ξ 0 K + . The calculated branching fraction turns out to be too small compared to experiment and the decay asymmetry is predicted to be zero owing to the vanishing S-wave amplitude. We shall examine this issue in this work and point out a solution to this puzzle. This has important implications to the Ξ 0 c sector where the CF mode Ξ 0 c → Σ + K − and the SCS decays Ξ 0 c → pK − , Σ + π − will encounter similar problems in naive calculations.
Recently, we have followed this approach to calculate singly Cabibbo-suppressed (SCS) decays of Λ + c [25], in which the predictions of Λ + c → pπ 0 , pη are in good agreement with the BESIII measurement. In this work, we shall continue working in the pole model together with current algebra to compute both CF and SCS two-body weak decays of Ξc baryons. This paper is organized as follows. In Sec. II we will set up the formalism for computing branching fractions and up-down decay asymmetries, including contributions from both factorizable and nonfactorizable terms. Numerical results are presented in Sec. III. A conclusion will be given in Sec. IV. In Appendix A, we write down the baryon wave functions to fix our convention and then examine their behavior under U -, V -, and I-spin in Appendix B. Appendix C is devoted to the form factors for Λ + c → B transitions evaluated in the MIT bag model. The expressions of baryon matrix elements and axial-vector form factor calculated in MIT bag model will be presented in Appendix D.

A. Kinematics
Without loss of generality, the amplitude for the two-body weak decay Bi → B f P can be parameterized as where P denotes a pseudoscalar meson. Based on the S-and P -wave amplitudes, A and B, the decay width and up-down spin asymmetry are given by and pc is the three-momentum in the rest frame of mother particle. To proceed, we need the lifetimes of the relevant charmed baryons which are quoted as the new world averages τ (Ξ + c ) = (4.56 ± 0.05) × 10 −13 s, τ (Ξ 0 c ) = (1.53 ± 0.02) × 10 −13 s, dominated by the most recent lifetime measurements by the LHCb [30]. Note that the measured Ξ 0 c lifetime by the LHCb is approximately 3.3 standard deviations larger than the old world average value [7].
The S-and P -wave amplitudes of the two-body decay generically receive both factorizable and nonfactorizable contributions We should keep in mind that the above formal decomposition is process dependent, not all the processes contain both contributions shown in Eq. (7). To identify the explicit components, one way is to resort to the topological diagram method. In the topological diagram approach, the external W -emission and internal W -emission from the external quark are usually classified as factorization contributions, while the nonfactorizable contributions arise from inner W -emission and W -exchange diagrams. Contrary to weak decays of Λ + c , decay modes proceeding only through factorizable contributions cannot be found in Ξ +,0 c decays.

B. Factorizable contribution
The description of the factorizable contribution of the charmed baryon decay Bc → BP is based on the effective Hamiltonian approach.

General expression of factorizable amplitudes
The effective Hamiltonian for CF process is where the four-quark operators are given by with (q1q2) ≡q1γµ(1 − γ5)q2. The Wilson coefficients to the leading order are given as c1 = 1.346 and c2 = −0.636 at µ = 1.25 GeV and Λ MS = 325 MeV [31]. Under naive factorization the amplitude can be written down as where the choice of ai can be referred to Eq. (10).
Likewise, the S-and P -wave amplitudes for SCS processes are given by where the flavor of the down-type quark q, d or s, depends on the process. If P = η8, both flavors contribute, for example, where the decay constants are defined by We shall follow [32] to use f q η = 107 MeV and f s η = −112 MeV. Notice that in the case of π 0 production in the final state, one should replace a2 by −a2/ √ 2 in the factorizable amplitude, where the extra factor of −1/ √ 2 comes from the wave function of the π 0 , π 0 = (uū − dd)/ √ 2.

The parameterization of form factors
There are two different non-perturbative parameters in factorizable amplitudes, the decay constant and the form factor (FF). There exist some efforts for estimating the FFs for Ξc → B transition [16,[33][34][35]. In this work we prefer to work out FFs for Ξc-B transition and baryonic matrix elements all within the MIT bag model [36]. Since the decay rates and decay asymmetries are sensitive to the relative sign between factorizable and non-factorizable amplitudes, it is also desired to have an estimation of FFs in a globally consistent convention.
In this work we follow [34] to write the q 2 dependence of FF as where mV = 2.01 GeV, mA = 2.42 GeV for the (cd) quark content, and mV = 2.11 GeV, mA = 2.54 GeV for (cs) quark content. In the zero recoil limit where q 2 max = (mi − m f ) 2 , FFs can be expressed within the MIT bag model as [22] where u(r) and v(r) are the large and small components, respectively, of the quark wave function in the bag model. FFs at different q 2 are related via are presented in the third/fifth column. With different involved quark content shown in the second column, the evolution coefficients are shown in fourth/sixth column. The physical FF f1(m 2 P ) can be obtained from the coefficient in fourth column and the model result in third column, likewise for g1(m 2 P ).
This allows us to obtain the physical FF at q 2 = m 2 P . It is obvious that the FF at q 2 max is determined only by the baryons in initial and final states. However, its evolution with q 2 is governed by both the final-state meson and relevant quark content. Such dependence is reflected in Table II, in which the quark contents are shown in the second column. In the zero recoil limit, the FFs at q 2 max calculated from Eq. (18) are presented in the third and fifth columns. And then in the fourth and sixth columns, the evolution of FFs from q 2 = q 2 max to q 2 = m 2 P are derived according to Eq. (20). The auxiliary quantities Y (s) 1,2 are defined in terms of The model parameters are adopted from [25] and references therein. Numerically, we have Y1 = 0.88, Y s 1 = 0.95, Y2 = 0.77, Y s 2 = 0.86, which are consistent with the corresponding numbers in [22].
In the second column of Table V in [16], we see that the sign of the FFs f is positive, which differs from our result shown in Table II. As mentioned before, such a sign difference will affect the decay rates and asymmetries for processes involving both factorizable and nonfactorizable terms.

C. Nonfactorizable contribution
We work in the framework of the pole model to estimate nonfactorizable contributions. It is known that the S-wave amplitude is dominated by the low-lying 1/2 − resonances, while the P -wave one governed by the ground-state 1/2 + pole. The general formulas for A (S-wave) and B (P -wave) terms in the pole model are given by [34] where aij, bij are the baryon matrix elements defined by In the soft-meson limit, the intermediate excited 1/2 − states in the S-wave amplitude can be summed up and reduced to a commutator term [25], 4 By applying the generalized Goldberger-Treiman relation the P -wave amplitude can be simplified to Therefore, the two master equations Eq. (24) and Eq. (27) for the nonfactorizable contributions in the pole model rely on the commutator terms and the axial-vector form factor g A B ′ B which will be calculated in the MIT bag model in this work.

S-wave amplitude
We have deduced that the S-wave amplitude is determined by the commutator terms of conserving charge Q a and the parity-conserving part of the Hamiltonian. In the following we list the expressions of A com according to Eq. (24): where we have introduced the isospin, U -spin and V -spin ladder operators with In Eq. (28), η8 is the octet component of the η and η ′ η = cos θη8 − sin θη0, 32]. For the decay constant fη 8 , we shall follow [32] to use fη 8 = f8 cos θ with f8 = 1.26fπ . Hypercharge Y , the conserving charge for processes involving η8 in the final state, is taken to be Y = B + S − C as argued in [25]. The baryon matrix elements of commutators in Eq. (28), after the action of the ladder operators on baryon wave functions shown in Appendix B, can be further reduced to pure matrix elements of effective Hamiltonian, denoted by a B ′ B ≡ B ′ |H PC eff |B . Then in terms of a B ′ B , nonfactorizable contributions to S-wave amplitudes for charmed baryon decays are calculable.
For the Cabibbo-favored processes, we have For singly Cabibbo-suppressed processes we have The nonfactorizable S-wave amplitudes for SCS decays of Λ + c can be found in [25]. The evaluation of the baryon matrix elements a B ′ B in the MIT bag model and results are presented in Appendix D 1.

P -wave amplitude
Through the generalized Goldberger-Treiman relation Eq. (26), the strong coupling of B ′ BM can be expressed in terms of the axial-vector form factor g A B ′ B . Based on Eq. (27), P -wave amplitudes are given as follows. For Cabibbo-favored processes we have The nonfactorizable P -wave amplitudes for SCS decays of Λ + c can be found in [25]. In addition to the baryon matrix element a BB ′ , another quantity in the nonfactorizable part of P -wave amplitude is the axial-vector form factor g A(P ) B ′ B . For consistency, the estimation of g A(P ) B ′ B is carried out in the MIT bag model and the results are shown in Sec. D 2. As seen in the next section, one of the W -exchange diagrams, the so-called type-III diagram in which the quark pair is produced between the two quark lines without W -exchange, does not contribute to the nonfactorizable S-and P -wave amplitudes. This will be discussed in detail there.

III. NUMERICAL RESULTS AND DISCUSSIONS
A. Λ + c decays Before proceeding to the Ξc sector, we first discuss Λ + c decays as the measurements of branching fractions and decay asymmetries are well established for many of the channels. The goal is to see what we can learn from the Λ + c physics. We show in Table III the results of calculations for CF and SCS Λ + c decays. For the form factors f1 and g1, we follow [37] to use 5 for Λc-p transition and rescale the form factors for Λc-Λ transition to fit the decay Λ + c → Λπ + so that f Λc Λ 1 (0) = 0.406 and g ΛcΛ 1 (0) = 0.370. 6 We see from Table III that the calculated branching fractions and decay asymmetries are in general consistent with experiment except for the decay asymmetry in the decay Λ + c → pK 0 . While all the predictions of α(Λ + c → pK 0 ) in the literature are all negative except [21], the measured asymmetry by BESIII turns out to be positive with a large uncertainty, 0.18 ± 0.45 [29]. This issue needs to be resolved in future study. We next turn to the mode Λ + c → Ξ 0 K + which deserves a special attention. It has been shown that its S-and P -wave amplitudes are very small due to strong cancellation between various contributions. More specifically (see e.g. [22]), Since the matrix elements a Σ + Λ + c and a Ξ 0 Ξ 0 c are identical in the SU(3) limit and since there is a large cancellation between the first and third terms in B ca (no contribution from the second term due to the vanishing g ; for details see [22]), the calculated branching fraction turns out to be too small compared to experiment and the decay asymmetry is predicted to be zero owing to the vanishing S-wave amplitude [19-21, 23, 24]. However, a recent BESIII measurement leads to α(Λ + c → Ξ 0 K + ) = 0.77 ± 0.78 [38], though it is still dominated by the statistic uncertainty. This is a long-standing puzzle.
To solve the above-mentioned puzzle, we notice that one of the W -exchange diagrams depicted in the left panel of Fig. 1(a) can be described by two distinct pole diagrams at the hadron level shown in the right panel of the diagram 1(a). These two pole diagrams are called type-III diagrams in [19] and (d1) and (d2) in [23]. As first pointed out by Körner and Krämer [19], type-III diagram contributes only to the P -wave amplitude. Moreover, they showed that this diagram is empirically observed to be strongly suppressed. It was argued byŻenczykowski [23] that contributions from diagrams (d1) and (d2) cancel due to the spin-flavor structure. Hence, its S-and P -wave amplitudes vanish. The smallness of type-III W -exchange diagram also can be numerically checked through Eq. (40). In other words, the conventional expression of parity-violating and -conserving amplitudes given in Eq. (40) is actually for the type-III W -exchange diagram in Fig. 1(a). As a result, non-vanishing nonfactorizable S-and P -wave amplitudes arise solely from the W -exchange diagram depicted in Fig. 1(b) (called type-II W -exchange diagram in [19] and (b)-type diagram in [23]). The nonfactorizable amplitudes induced from type-II W -exchange now read 5 The sign of the form factors is fixed by Eq. (19). 6 We have checked if the form factors for Λ + c -p and Λ + c -Λ transitions given in Appendix C are used, the resulting decay asymmetries will remain stable, but the calculated branching fractions are not as good as those shown in Table III but within a factor of 2.
The corresponding pole diagrams are also shown.
TABLE III. The predicted S-and P -wave amplitudes of Cabibbo-favored (upper entry) and singly Cabibbosuppressed (lower entry) Λ + c → B + P decays in units of 10 −2 GF GeV 2 . Branching fractions and the asymmetry parameter α are shown in the last four columns. Experimental results for decay asymmetries are taken from [29] except the modes Λπ + and Σ + π 0 where the world averages are obtained from [29] and [7]. Consequently, both partial wave amplitudes are not subject to large cancellations. Note that the pole diagram induced by type-II W -exchange is the same as the second pole diagram (i.e. a weak transition of Λ + c -Σ + followed by a strong emission of K + ) in Fig. 1(a), but it is no longer canceled by the first pole diagram. A vanishing S-wave amplitude was often claimed in the literature. We wish to stress again that the parity-violating amplitude can be induced from type-II W -exchange through current algebra. 7 Eq. (41) leads to B(Λ + c → Ξ 0 K + ) = 0.71%, which is consistent with the data of (0.55 ± 0.07)% [7]. Moreover, the predicted positive decay asymmetry of order 0.90 is consistent with the BESIII's measurement of 0.77 ± 0.78 [38]. It is interesting to notice that α is also predicted to be 0.94 +0.06 −0.11 in the SU(3)-flavor approach [14]. Therefore, the long-standing puzzle with the branching fraction and the decay asymmetry of Λ + c → Ξ 0 K + is resolved. In the Ξc sector, vanishing type-III W -exchange contributions also occur in the CF decay Ξ 0 c → Σ + K − and the SCS modes Ξ 0 c → pK − , Σ + π − . We will come to this point later. Comparing Table III with Table II of [25] for SCS Λ + c decays, we see some changes in the P -wave amplitudes of Λ + c → pπ 0 , pη, nπ + . This is because the first equation in (C2) of [25] should read Consequently, we find B(Λ + c → pπ 0 ) is modified from 0.75×10 −4 [25] to the current value of 1.26×10 −4 . As for Λ + c → nπ + , after correcting the error with the axial-vector form factor g A(π + ) Σ 0 c Λc we find large cancellation in both S-and P -wave amplitudes, resulting very small branching fraction of order 0.9 × 10 −4 .

B. Ξc decays
Branching fractions and up-down decay asymmetries of CF and SCS Ξ +,0 c weak decays are calculated according to Eqs. (4), (5) and (7), yielding the numerical results shown in Tables IV and V, respectively. One interesting point is that there does not exist any decay mode which proceeds only through the factorizable diagram. Among all the processes, the three modes Ξ 0 c → Σ + K − , Ξ 0 π 0 , Ξ 0 η8 in CF processes and the five SCS modes Ξ + c → pK 0 , Ξ 0 c → Ξ 0 K 0 , pK − , nK 0 , Σ + π − proceed only through the nonfactorizable diagrams, while all the other channels receive contributions from both factorizable and nonfactorizable terms. The relative sign between factorizable and nonfactorizable is crucial governing whether the interference term is destructive or constructive. For example, factorizable and nonfactorizable terms in both the S-and P -wave amplitudes of the decays Ξ + c → Σ + K 0 , Ξ 0 π + and Ξ 0 c → Σ 0 K 0 interfere destructively, leading to small branching fractions, especially for the last mode. On the contrary, interference in the channels Ξ 0 c → ΛK 0 , Ξ − π + is found to be constructive.
The CF decay Ξ 0 c → Σ + K − and the SCS modes Ξ 0 c → pK − , Σ + π − are of special interest among all the Ξc weak decays. Their naive S-wave amplitudes are given by From Eqs. (D2) and (D3) for baryon matrix elements, it is easily seen that they all vanish in the SU(3) limit. Likewise, their P -wave amplitudes are also subject to large cancellations. Just as the decay Λ + c → Ξ 0 K + discussed in Sec. III.A, we should neglect the contributions from type-III W -exchange diagrams and focus TABLE IV. The Cabibbo-favored decays Ξc → B f P in units of 10 −2 GF GeV 2 . Branching fractions (in percent) and the up-down decay asymmetry α in theory and experiment are shown in the last four columns. Experimental results are taken from [9][10][11] for branching fractions and [39] for decay asymmetry. on type-II W -exchange ones. The resulting amplitudes for these three modes are given by modes denoted by A com are the same in magnitude but opposite in sign. Consequently, the interference between A fact and A com is destructive in Ξ + c → Ξ 0 π + but constructive in Ξ 0 c → Ξ − π + (see also [21]). As a result, the predicted branching fraction of order 6.5% for the latter is too large. If we use the form factors f Ξc Ξ 1 (0) = −0.590 and g ΞcΞ 1 (0) = −0.582 [35] in conjunction with the q 2 dependence given by Eq. (18), the branching fraction will be reduced only slightly from 6.5% to 6.2%. Hence, we conclude that these two modes cannot be simultaneously explained within the framework of current algebra for S-wave amplitudes.
To circumvent the difficulty with Ξ 0 c → Ξ − π + , one possibility is to consider the correction to the currentalgebra calculation of the parity-violating amplitude by writing where the term (A − A CA ) can be regarded as an on-shell correction to the current-algebra ressult. It turns out that in the existing pole model calculations [21,22,34], the on-shell correction (A − A CA ) always has a sign opposite to that of A CA . Moreover, the on-shell correction is sometimes large enough to flip the sign of the parity-violating amplitudes. It is conceivable that on-shell corrections could be large for Ξ − π + but small for Ξ 0 π + . This issue needs to be clarified in the future. Nevertheless, we have learned from Table III that current algebra generally works well in Λ + c → B + P decays, For the up-down decay asymmetry, there is only one measurement thus far. In 2001, CLEO collaboration measured Ξ 0 c → Ξ − π + and found α(Ξ 0 c → Ξ − π + ) = −0.6 ± 0.4 [39]. Our prediction is consistent with the CLEO's value. Decay asymmetries are usually negative in most of the channels. However, besides the three modes Ξ 0 c → Σ + K − , pK − , Σ + π − as discussed before, the following channels Ξ 0 c → Ξ 0 η, Σ 0 η and Ξ + c → Σ + π 0 , Σ + η in the Ξc sector are also predicted to have positive decay asymmetries (see Tables IV  and V). We hope that these predictions could be tested in the near future by Belle/Belle II.  [14,46] for the branching fractions in units of 10 −2 for Cabibbofavored Λ + c decays (upper entry) and 10 −3 for singly Cabibbo-suppressed ones (lower entry). Decay asymmetries are shown in parentheses.

C. Comparison with the SU(3) approach
Besides dynamical model calculations, two-body nonleptonic decays of charmed baryons have been analyzed in terms of SU(3)-irreducible-representation amplitudes [40,41]. There are two distinct approaches to implement this idea. One is to write down the SU(3)-irreducible-representation amplitudes by decomposing the effective Hamiltonian through the Wigner-Eckart theorem. The other is to use the topological quark diagrams which are related in different decay channels via SU(3) flavor symmetry. Each approach has its own advantage. A general formulation of the quark-diagram scheme for charmed baryons is given in [42] (see also [43]). Analysis of Cabibbo-suppressed decays using SU(3) flavor symmetry was first carried out in [44]. This approach became very popular recently [12][13][14]45]. Although SU(3) flavor symmetry is approximate, it does provide very useful information. In Tables VI and VII we compare our results for Λ + c and Ξ +,0 c decays, respectively, with the SU (3)F approach in [14,46] in which the parameters for both Sand P -wave amplitudes are obtained by fitting to the data. 9 We see from Table VI that it appears the SU(3) approach gives a better description of the measured branching fractions because it fits to the data. However, it is worth of mentioning that in the beginning the SU(3) practitioners tended to make the assumption of the sextet 6 dominance over 15. Under this hypothesis, one will lead to B(Λ + c → pπ 0 ) ∼ 5 × 10 −4 [12,45], which exceeds the current experimental limit of 2.7 × 10 −4 [47]. Our dynamic calculation in [25] predicted B(Λ + c → pπ 0 ) ∼ 1 × 10 −4 . As far as the branching fraction is concerned, it is important to measure the mode Λ + c → nπ + to distinguish our prediction from the SU(3) approach. As for decay asymmetries, while we agree on the sign and magnitude of α(Ξ 0 K + ), we disagree on the signs of α in Λ + c → nπ + and ΛK + . Hopefully, these can be tested in the future.  [14,46] for the branching fractions in units of 10 −2 for Cabibbofavored Ξ +,0 c decays (upper entry) and 10 −3 for singly Cabibbo-suppressed ones (lower entry). Decay asymmetries are shown in parentheses. Experimental results are taken from [9][10][11] for branching fractions and [39] for decay asymmetry.

Modes
This work Geng et al. [14,46] Expt.  Table VII that except Ξ + c → Σ + K 0 , Ξ 0 π + , Ξ 0 K + and Ξ 0 c → Ξ − π + , Ξ − K + all the branching fractions of Ξ +,0 c decays in this work and in the SU(3) approach are consistent with each other within a factor of 2. Furthermore, we agree on the signs of decay asymmetries except Ξ + c → Σ + K 0 and Ξ + c → Ξ 0 K + . 10 Notice that both approaches lead to B(Ξ 0 c → Ξ − π + ) ≫ B(Ξ + c → Ξ 0 π + ), contrary to the current data. Hence, it is of great importance to measure the branching fractions of them more accurately in order to test their underlying mechanism.

IV. CONCLUSION
In this work we have systematically studied the branching fractions and up-down decay asymmetries of CF and SCS decays of antitriplet charmed baryons. To estimate the nonfactorizable contributions, we work in the pole model for the P -wave amplitudes and current algebra for S-wave ones. Throughout the whole calculations, all the non-perturbative parameters, including form factors, baryon matrix elements and axial-vector form factors are evaluated using the MIT bag model.
We draw some conclusions from our analysis: • The long-standing puzzle with the branching fraction and decay asymmetry of Λ + c → Ξ 0 K + is resolved by realizing that only type-II W -exchange diagram will contribute to this mode. We find that not only the predicted rate agrees with experiment but also the decay asymmetry is consistent in sign and magnitude with the SU(3) flavor approach and the recent BESIII measurement, though the latter is still dominated by the statistic uncertainty. Hence, it is most likely that α(Λ + c → Ξ 0 K + ) is large and positive.
• In analog to Λ + c → Ξ 0 K + , the CF mode Ξ 0 c → Σ + K − and the SCS decays Ξ 0 c → pK − , Σ + π − proceed only through type-II W -exchange. They are predicted to have large and positive decay asymmetries. This can be tested in the near future.
• The predicted B(Ξ + c → Ξ 0 π + ) agrees well with the measurement inferred from Belle and CLEO, while the calculated B(Ξ 0 c → Ξ − π + ) is too large compared to the recent Belle measurement. We find B(Ξ 0 c → Ξ − π + ) ≫ B(Ξ + c → Ξ 0 π + ) and conclude that these two modes cannot be simultaneously explained within the current-algebra framework for S-wave amplitudes. More accurate measurements of them are called for to set the issue.
• Owing to large cancellation between factorizable and nonfactorizable contributions, the rate of Λ + c → nπ + is found to be of the same order as that of Λ + c → pπ 0 . It is important to measure both SCS modes to understand their underlying mechanism.
• Although Ξ 0 c → Σ 0 K 0 and Ξ + c → Σ + K 0 are Cabibbo-favored decays, their branching fractions are small especially for the former due to large destructive interference between factorizable and nonfactorizable amplitudes.
• We have compared our results with the approach of SU(3) flavor symmetry. Excluding those predictions of α with the uncertainty greater than the central value, we find that both approaches agree on the signs of decay asymmetries except the three modes: Λ + c → nπ + , Ξ + c → Σ + K 0 and Ξ + c → Ξ 0 K + . We also agree on the hierarchy B(Ξ 0 c → Ξ − π + ) ≫ B(Ξ + c → Ξ 0 π + ).
for U -spin ladder operators, for V -spin ladder operators, and  Table VIII. For Λ + c → pη8, we have assumed that form factors are dominated by the (cd) quark content.

Axial-vector form factors
In the MIT bag model the axial form factor in the static limit can be expressed as g A(P ) Based on Eq. (D5), the axial-vector form factors related to CF processes are 11