Analysis of Hadronic Weak Decays of Charmed Baryons in the Topological Diagrammatic Approach

We perform a global fit to the experimental data of two-body charmed baryon decays based on the topological diagrammatic approach (TDA) and take into account the phase shifts between $S$- and $P$-wave amplitudes as inspired by the recent BESIII measurement of the decay asymmetry in the decay $\Lambda_c^+\to \Xi^0K^+$. The TDA has the advantage that it is more intuitive, graphic and easier to implement model calculations. The measured branching fractions and decay asymmetries are well accommodated in the TDA except for a few modes, in particular, the predicted ${\cal B}(\Xi_c^0\to \Xi^-\pi^+)=(2.83\pm0.10)\%$ is larger than its current value. The equivalence of the TDA and the irreducible SU(3) approach (IRA) is established. We show that the number of the minimum set of tensor invariants in the IRA and the topological amplitudes in the TDA is the same and present their relations. The predicted magnitudes of $S$- and $P$-wave amplitudes and their phase shifts are presented for measured and yet-to-be-measured modes in both the TDA and IRA which can be tested in the near future. Besides the decay $\Lambda_c^+\to \Xi^0 K^+$, there exist several modes which proceed only through $W$-exchange. In particular, the observed channel $\Xi_c^0\to \Sigma^+ K^-$ should have phase shifts similar to that in $\Lambda_c^+\to \Xi^0 K^+$ and its decay asymmetry is predicted to be $-0.21\pm0.17$ which can be used to test our theoretical framework. In contrast, the TDA leads to a large $\alpha$ of order $-0.93$ for the decay $\Xi_c^+\to \Xi^0\pi^+$ even after the phase-shift effect is incorporated in the fit.

The first analysis of two-body nonleptonic decays of antitriplet charmed baryons B c ( 3) → B(8)M (8 + 1) within the framework of the TDA was performed by Kohara [25].A subsequent study was given by Chau, Cheng and Tseng (CCT) in Ref. [26].Among the recent analyses in the TDA [18,[27][28][29], 1 there are 19 TDA amplitudes and 7 topological diagrams in Ref. [18].Authors of Ref. [27] followed the Kohara's scheme closely with 8 topological diagrams and 8 TDA amplitudes, but did not distinguish between S-and P -wave contributions and hence the topological amplitudes were fitted to the branching fractions only.Therefore, unlike the IRA, global fits to the rates and decay asymmetries are still absent in the TDA.
Although the TDA has been applied very successfully to charmed meson decays [30][31][32], its application to charmed baryon decays is more complicated than the IRA.As stressed in Ref. [18], it is easy to determine the independent amplitudes in the IRA, while the TDA gives some redundancy.Some of the amplitudes are not independent and therefore should be absorbed into other amplitudes.Nevertheless, the TDA has the advantage that it is more intuitive, graphic and easier to implement model calculations.The extracted topological amplitudes by fitting to available data will enable us to probe the relative importance of different underlying decay mechanisms, and to relate one process to another at the topological amplitude level.In this work, we are going to show that the TDA is applicable to charmed baryon decays as well and it has the same number of independent amplitudes as that of the IRA.
The Cabibbo-favored mode Λ + c → Ξ 0 K + which proceeds only through W -exchange deserves special attention.Early studies in 1990's indicated that its S-and P -wave amplitudes are very small due to strong cancellation between various terms (see e.g.Ref. [33]).For example, the use of current algebra implies a vanishing S-wave in the SU(3) limit.Consequently, the calculated branching fraction is too small compared to experiment and the predicted α is zero owing to the vanishing S-wave amplitude.It is thus striking that the approach based on the IRA tends to predict a large decay asymmetry close to unity [19][20][21].This long-standing puzzle was finally resolved by a recent BESIII measurement [34].Not only the decay asymmetry α Ξ 0 K + = 0.01 ± 0.16 was found to be consistent with zero, but also the measured Lee-Yang parameter β Ξ 0 K + = −0.64 ± 0.69 was nonzero, implying a phase difference between S-and P -wave amplitudes, δ P − δ S = −1.55 ± 0.25 or 1.59 ± 0.25 rad.Since | cos(δ P − δ S )| ∼ 0.02, this accounts for the smallness of α Ξ 0 K + .This first direct evidence supporting the existence of strong phases in the partial-wave amplitudes of hadronic charmed baryon decays plays a pivotal role in a further exploration of CP violation in the charmed baryon sector.
Recently, a new analysis of charmed baryon decays based on the IRA that takes into account the phase shifts of the partial-wave amplitudes has been put forward in Ref. [35].In this work we shall perform a similar study within the framework of the TDA.Since the TDA has been applied very successfully to charmed meson decays, it is conceivable that the same approach is applicable to the charmed baryon sector.
The layout of this work is as follows.In Sec.II we first discuss the choice of octet-baryon wave wave functions.After writing down the general expression of the decay amplitudes in the TDA, we show that the number of independent amplitudes can be reduced through the Körner-Pati-Woo theorem and the removal of redundancy.The equivalence of the TDA and IRA is explicitly demonstrated in Sec.III.Sec.IV is devoted to the numerical analysis and fitting results.Sec.
V comes to our conclusions.The relevant experimental results are collected in the Appendix.A short version of this work has been presented in Ref. [36].

II. FORMALISM
Since baryons are made of three quarks in contrast to two quarks for the mesons, the application of TDA to the baryon case will inevitably lead to some complications.For example, the symmetry of the quarks in flavor space could be different.As stated in the Introduction, there exist two seemingly different analyses of two-body nonleptonic decays of antitriplet charmed baryons within the framework of the TDA: one by Kohara [25] and the other by Chau, Cheng and Tseng (CCT) in Ref. [26].The difference between Kohara and CCT lies in the choice of the wave functions of octet baryons: with |a| 2 + |b| 2 = 1 in Ref. [26], and in Ref. [25], where χ m (1/2) A,S are the spin parts of the wave function defined in Eq. ( 23) of Ref. [26] and are the octet baryon states that are antisymmetric and symmetric in the first two quarks, denoted by [ ] and {}, respectively.As shown explicitly in Ref. [37], physics is independent of the convention one chooses.The TDA amplitudes expressed in the schemes with B m,k (8) and Bm,k (8) are equivalent.Nevertheless, we prefer to use the bases ψ k (8) A 12 and ψ k (8) S 12 as they are orthogonal to each other, while ψ k (8) A 12 and ψ k (8) A 23 are not.
To construct the decay amplitudes of B c ( 3) → B(8)M (8 + 1) decays in the TDA, we first specify the building blocks.The antitriplet charmed baryons are usually presented by For the purpose of constructing the TDA amplitudes, it is more convenient to group them into an antisymmetric matrix The superscripts i and j also stand for the light quark flavors.For example, (B c ) 12 refers to the charmed baryon state Λ + c with the quark content cud.The lowest-lying octet baryons B(8) are normally represented in the matrix form one can parameterized the decay amplitudes of B c → B n M process as (2.6) By expanding Eqs.(2.6), decay amplitudes can be expressed by coefficients related to the irreducible representations and 16 IRA parameters.

TDA amplitudes
For topological diagram approach, the topological diagrams are given in Fig. .., with the analysis for different diagrams and quark lines, we find 24 TDA parameters to express the -3 - 8)M (8 + 1) decays.
However, for the TDA purpose it is more convenient to introduce the anti-symmetric tensor ϵ ijk to write (B 8 The quark content of the baryon can be read from the subscript ijk.For example, (B 8 ) 121 = (B 8 ) udu = p and (B 8 ) 122 = (B 8 ) udd = n.Tensor form of the nonet pseudoscalar mesons M (8 + 1) reads or with The physical states η and η ′ are given by where the mixing angles θ and ϕ are related through the relation θ = ϕ−arctan −1 √ 2. For ϕ = 40 • , one will have θ = −15 • .In Eqs. ( 9) and (10), the superscript i stands for the quark flavor, while the subscript j for the antiquark flavor.
In terms of the baryon and meson states, we follow Ref. [18] to write down the general expression of the decay amplitudes in the TDA: where the flavor indices of weak interactions are related to the H matrix with the non-vanishing elements: The corresponding topological diagrams are depicted in Fig. 1: the external W -emission, T ; the internal W -emission C; the inner W -emission C ′ ; W -exchange diagrams E 1 , E 2 , E 3 and the hairpin diagram E h .We shall see shortly that there exist two different types of E 1 and E 2 diagrams.Notice that A TDA in Eq. ( 13) is the same as Eq.(80) of Ref. [18] except that we add one more term to T and one to C for the reason of completeness.
At first sight, it appears that there are 21 amplitudes and 7 topological diagrams.However, because the two light quarks of the antitriplet charmed baryon are antisymmetric in flavor, so are the two spectator quarks q i and q j in diagrams T and C (see Fig. 1).This implies that b 3 = −b 2 and b 6 = −b 5 .Moreover, the final-state quarks q l and q k in topological diagrams C ′ , E 3 and E h must be antisymmetric in flavor owing to the Körner-Pati-Woo (KPW) theorem which states that the quark pair in a baryon produced by weak interactions is required to be antisymmetric in flavor in the SU(3) limit [38].This amounts to having  [18].Hence, all of them can be absorbed into the topological amplitudes T , C, C ′ , E 3 and E h , respectively.Consequently, the number of independent TDA amplitudes is reduced from 21 to 11.
In this work we shall employ the octet baryon wave function B m,k (8) given by Eq. ( 3) and keep in mind that physics is independent of the choice of the baryon wave function, B m,k (8) or Bm,k (8).Under this convention, we are forced to have b 12 = b 11 and b 15 = b 14 .This means that the topological diagram E 1 is decomposed into two: E 1A and E 1S which are antisymmetric and symmetric in the quark pair q j and q k , respectively.Likewise, the topological diagram E 2 is also decomposed into E 2A and E 2S (see Fig. 1).Absorbing the coefficients into the TDA amplitudes, the decay amplitudes of B c ( 3) → B(8)M (8 + 1) in the TDA thus have the expressions: where E 1A = b 10 E 1 and E 1S = b 11 E 1 and likewise E 2A = b 13 E 2 and E 2S = b 14 E 2 .However, the final-state quarks q l and q k in the diagrams E 1A,1S and E 2A,2S are required by the KPW theorem be flavor antisymmetric in the SU(3) limit.Consequently, we are led to As a result, the number of independent topological diagrams depicted in Fig. 1 and the TDA amplitudes in Eq. ( 15) is 7.
If we choose Bm,k given in Eq. ( 4) as the octet baryon wave function, we will have b 11 = b 14 = 0. Consequently, the topological diagrams E 1S and E 2S in Fig. 1 are replaced by E ′ 1A and E ′ 2A , respectively.They are antisymmetric in the quark pairs (q k , q m ) and (q l , q m ), respectively.In this case, the TDA amplitudes read Just like the previous case, the final-state quarks q l and q k in the W -exchange diagrams E 1A,2A and E ′ 1A,2A are required be flavor antisymmetric.This implies that Note that ÃTDA is the starting point of Ref. [27] for analyzing charmed baryon decays in the TDA.However, the relations shown in Eq. ( 18) were not utilized by the authors.Working out Eq. ( 15) for B c ( 3) → B(8)M (8 + 1) decays, we obtain the TDA decay amplitudes listed in Tables I and II.It is straightforward to show that the expressions of the TDA amplitudes agree with CCT [26] through the following relations: where and C ′ S are the TDA amplitudes defined in Ref. [26].The agreement is non-trivial in view of the different methodologies adopted in [26] and here.Among the 7 TDA amplitudes given in Eq. ( 15), there still exist 2 redundant degrees of freedom through the redefinitions [26]: A closer look of the TDA amplitudes of Cabibbo-favored, singly-Cabibbo-suppressed and doubly-Cabibbo-supprerssed decays given in Tables I and II shows that E 1S can be absorbed by T , C, C ′ , E 1 and E h , as shown in the above equation.Hence, the redundant E 1S can be eliminated.Also the amplitude E 3 is always accompanied by E 1A + E 1S .Consequently, it can be absorbed by the combination of E 1A + E 1S .As a result, among the seven topological amplitudes T , C, C ′ , E 1A , E h , E 1S and E 3 , the last two are redundant degrees of freedom and can be omitted through redefinitions.
In Tables I and II we shall use TDA to denote the tilde TDA amplitudes.By now it is clear that the minimum set of the topological amplitudes is 5.This is in agreement with the number of tensor invariants found in the IRA [35] (see also Sec.III).
For the η and η ′ final states, we have exhibited the TDA amplitudes in Tables I and II in the η 8 and η 1 basis.Of course, one can also work in the η q and η s basis.For example, Applying Eq. ( 11), we obtain We see that the hairpin diagram E h [39][40][41] contributes to the SU(3)-singlet η 1 but not to η 8 as it should be.Many sum-rule relations can be derived from Table I, for example, The first relation was first derived in Ref. [35].It is very useful to constrain the branching fraction of Ξ 0 c → Ξ − π + .

III. EQUIVALENCE OF TDA AND IRA
In this section we will consider the general SU(3)-invariant decay amplitudes in the IRA.To demonstrate the equivalence between the TDA and IRA, we need to show that the number of the minimum set of tensor invariants in the IRA and the topological amplitudes in the TDA is the same.If the number of minimum independent amplitudes is different, the global fit to the data will yield different fitting results for branching fractions, decay asymmetries and phase shifts.
Consider the effective Hamiltonian responsible for the ∆C = 1 weak transition where and the sum rule This sum rule is identical to the first one in Eq. ( 23) except for the left-hand-side term.However, both the relation (25) and the sum rule (26) are not borne out by experiment.This indicates that the contributions from H eff (15) cannot be neglected.We follow Ref. [18] to write down the general SU(3) invariant decay amplitudes in the IRA: For the explicit expressions of (H 6 ) ij k and (H 15 ) ij k , see Ref. [18].The first five terms associated with H 6 are not totally independent as one of them is redundant through the redefinition.It should be stressed that the redefinition is not unique.For example, we will consider the following redefinitions which amounts to deleting the a 5 term.There is another set of redefinitions adopted in Ref. [18] a 2 The relations between the IRA amplitudes in Ref. [18] and ours are: It is straightforward to check that the IRA amplitudes given in Tables 14-16 of Ref. [18] can be expressed in terms of a ′ 1 , a ′ 2 , a ′ 3 , a ′ 4 or a ′′ 1 , a ′′ 2 , a ′′ 3 , a ′′ 5 together with a 6 , • • • , a 10 .As for the five terms associated with H 15 in Eq. ( 27), four of them are prohibited by the KPW theorem, namely, a 6 = a 7 = a 8 = a 10 = 0 [35,42].To see this, we follow the argument presented in Ref. [42] 15), The H matrix given in Eq. ( 14) is related to H 6 and H 15 via where the expression of H( 6) kl can be found in e.g.Ref. [42].Projecting out the factorizable contributions from H(15), we obtain Comparing this with Eq. ( 27) leads to In the IRA we thus have five independent SU(3) tensor invariants a ′ 1 , a ′ 2 , a ′ 3 , a ′ 4 and a ′ 9 = a 9 , in agreement with Ref. [35].In the TDA we find the minimum set of the topological amplitudes: T , C, C′ , Ẽ1 and Ẽh .By comparing the TDA amplitudes in Tables I and II with the IRA amplitudes given in Tables 14-16 of Ref. [18], we arrive at the relations and hence T Therefore, we have the correspondence in terms of the redefinitions given in Eq. (28).The equivalence between the TDA and IRA is thus established.
There is another set of the IRA amplitudes given in Ref. [35] A The equivalence between TDA, IRAa and IRAb leads to the relations: and It should be stressed that since we will only fit to the observed branching fractions and decay asymmetries, the amplitudes are subject to sign ambiguity.Hence, the relations (36), (38) and ( 39) also hold if all the relations are flipped in sign simultaneously.For example, are also valid.

IV. NUMERICAL ANALYSIS AND RESULTS
As there are 5 independent tilde TDA amplitudes given in Eq. ( 20), we have totally 19 unknown parameters to describe the magnitudes and the phases of the respective S-and P -wave amplitudes, namely, , where the subscripts S and P denote the S-and P -wave components of each TDA amplitude.Since there is an overall phase which can be omitted, we shall set δ T S = 0. Hence, we are left with 19 parameters.Likewise, for the tidle IRA amplitudes given in Eq. ( 37), we also have We shall aslo set δ f a S = 0. Of course, physics is independent of which phase is removed.Indeed, one can check that the phase differences δ X i P − δ X i S or δ S remain the same no matter which phase is set to zero.Notice that the number of available experimental observables has increased to 30 by the end of 2023.To pursue a set of proper parameters, the χ 2 function in the following maximum likelihood analysis is defined as in which c i are the fitted 19 input parameters, O theor,expt stand for the 30 theoretical and experimental observables.The 30-dimensional general error matrix Σ can be taken diagonal by neglecting correlations among different observables and only incorporating pure experimental errors here.In terms of the S-and P -wave amplitudes given in Eq. ( 1) and their phases δ S and δ P , respectively, the decay rate and Lee-Yang parameters read where p c is the c.m. three-momentum in the rest frame of initial baryon and the auxiliary parameter κ is defined as The phase shift between S-and P -wave amplitudes can be deduced explicitly as in terms of the measured Lee-Yang parameters α and β.
In practice, we shall make use of the package iminuit [45,46] to search for χ 2 min together with its corresponding fitting parameters c i , and generate the covariance matrix among parameters which further helps predict physical observables.The fitting 19 TDA and IRA parameters given in Eqs. ( 41) and (42), respectively, are exhibited in Table III.Note that our results for the IRA amplitudes f x are numerically different from that given in Eq. ( 9) of Ref. [35].This may be partially ascribed to the fact that we use 30 instead of 29 data points for fit and employ the Belle's result (1.80 ± 0.52)% [47] for B(Ξ 0 c → Ξ − π + ) rather than the PDG value of (1.43 ± 0.32)% [43].Nevertheless, our predicted branching fractions, decay asymmetries and phase shifts are in gross agreement with Ref. [35].
We see from Eq. ( 44) that both the branching fraction and the longitudinal decay asymmetry α remain insensitive to the sign of the phase shift.Consequently, the current global fit, which incorporates 30 experimental inputs including solely branching fractions and the decay asymmetry α, lacks the ability to discern phase-shift signs.In contrast, the transverse asymmetry β, being proportional to the sine of the phase shift, aids in determining the sign of δ P − δ S .Although the measurement of β has been carried out in the Λ + c → Ξ 0 K + channel [34], a determination of the phase-shift sign proves challenging owing to its considerable uncertainty, which aligns with the presence of two phase-shift solutions at the current stage (see Eq. ( 48) below).In our numerical analysis, we indeed obtain two sets of solutions, distinct from a global sign difference.These solutions await a further discrimination by the forthcoming β measurements in the near future.
To obtain the numerical values in Table III, we have set δ T S = 0 and δ f a S = 0. We notice that no matter which phase δ X i S or δ X i P is set to zero, the phase difference δ X i P − δ X i S remains the same except for a sign ambiguity.Indeed, this is the so-called Z 2 ambiguity in Ref. [35], namely, (δ X i S , δ . Since β is proportional to sin(δ P − δ S ), this sign ambiguity can be resolved by the measurement of β as just noticed in passing.
Since the phase differences δ P − δ S for Ẽ1 and f c are −2.14 ± 0.57 and −2.62 ± 1.55 rad, respectively, it is clear from Table III that the magnitudes of S-wave or P -wave component of Ẽ1 and f c are consistent with each other within errors.Hence, the relation Ẽ1 = f c is numerically satisfied within errors.Likewise, the relation Ẽh = f a is also verified.
In the TDA, there are many modes receive contributions only from one of the tidle TDA amplitudes: Eq. ( 23) and the measured data collected in Table VII, we find B(Ξ 0 c → Ξ − π + ) = (2.85 ± 0.30)% in excellent agreement with the aforementioned prediction.
As for the ratio R Ξ − K + , we see from Table I that in the SU(3) limit, one will have R Ξ − K + = sin 2 θ C which is equal to 0.045 after taking into account the phase-space difference between Ξ 0 c → Ξ − K + and Ξ 0 c → Ξ − π + .The current measurement is 0.0275 ± 0.0057 which is away from the SU(3) expectation by 2σ.Since both modes proceed through the topological diagrams T and E 1S with the combination 2T − 2E 1S = 2 T , it is conceivable that SU(3) breaking in the external Wemission T and especially in W -exchange E 1S could account for the discrepancy between theory and experiment.
If the pseudoscalar meson in the final-state is an SU(3) flavor-singlet η 1 , it will receive an additional contribution from the hairpin diagram E h in Fig. 1.Numerically, we find that the combination Ẽh = E h + 2E 1S which contributes only to η 1 is sizable (see Table III).Recall that in the charmed meson sector there is a strong indication of the hairpin effect in the decay D + s → ρ + η ′ (see e.g.Ref. [32]).
It should be stressed that although the BESIII's measurement of α Ξ 0 K + is in good agreement with zero, it does not mean that the theoretical predictions in the 1990s with vanishing or very small S-wave amplitude are confirmed.We have checked that if we set δ X i S = δ X i P = 0 from the outset and keep the measured α Ξ 0 K + as an input, the fit B(Λ + c → Ξ 0 K + ) of order 1 × 10 −3 will be too small compared to experiment because of the smallness of the S-wave contribution.On the contrary, if the input of (α Ξ 0 K + ) exp is removed, the fit α Ξ 0 K + will be of order 0.95 .Hence, we conclude that it is inevitable to incorporate the phase shifts to accommodate the data.It is the smallness of | cos(δ P − δ S )| ∼ 0.02 that accounts for the nearly vanishing α Ξ 0 K + .
Besides the decay Λ + c → Ξ 0 K + , we have noticed in passing that the following modes Ξ 0 c → Σ + K − , Σ + π − , pK − , pπ − , nπ 0 and Ξ + c → pπ 0 , nπ + also receive contributions only from the topological W -exchange amplitude Ẽ1 .In the absence of strong phases in S-and P -wave amplitudes, they are expected to have large decay asymmetries.For example, α Ξ 0 c →Σ + K − was found to be 0.79 +0.32  −0.33 , 0.81 ± 0.16 and 0.98 ± 0.20, respectively, in Refs.[19][20][21].Once the phase shifts are incorporated in the fit, the above-mentioned modes should have δ P − δ S similar to that in Λ + c → Ξ 0 K + and their decay asymmetries will become smaller (see Tables IV-VI).In particular, for the CF channel Ξ 0 c → Σ + K − whose branching fraction has been measured before, we predict that α Ξ 0 c →Σ + K − = −0.21± 0.17 which can be used to test our theoretical framework.It is worth emphasizing again that when the strong phases δ X i S and δ X i P are turned off for the global fit to the data, the decay asymmetry α is found to be close to unity in magnitude for some of the modes, e.g., Λ 20,21].As noticed in passing, the nearly vanishing α Ξ 0 K + measured by BESIII implies the necessity of incorporating phase shifts for global fits, which will also help explain the measured value of α Σ 0 K + .The measurement of α Ξ 0 c →ΛK S and α Ξ 0 c →Ξ 0 η in the future will also help understand the phase shifts which are predicted to be 1.89 ± 0.25 and 1.48 ± 0.24 (both in rad), respectively.In contrast, it is also important to measure the decay asymmetry of Ξ + c → Ξ 0 π + to see if its largeness of order −0.93 is not affected by the phase shift which is expected to be δ P − δ S = 2.79 ± 0.23 rad (see Table IV) and hence | cos(δ P − δ S )| ∼ 0.94 .
For the Cabibbo-favored (CF) modes involving a neutral K 0 S or K 0 L , it was customary to use the relation Γ(K 0 ) = 2Γ(K 0 S ).However, this relation can be invalidated by the interference between CF and doubly Cabibbo-suppressed (DCS) amplitudes.Using the phase convention K 0 S = 1 Since B c → BK 0 is doubly Cabibbo-suppressed, it is expected that Γ(B c → BK 0 L ) ≈ Γ(B c → BK 0 S ).For singly Cabibbo-suppressed (SCS) channels, Ξ + c → pK S/L and Ξ 0 c → nK S/L receive contributions only from K 0 , while Ξ 0 c → Ξ 0 K S/L proceeds only through K 0 .

V. CONCLUSION
Inspired by the recent BESIII measurement of the decay asymmetry and the phase shift between S-and P -wave amplitudes in the decay Λ + c → Ξ 0 K + , we have performed a global fit to the experimental data of charmed baryon decays based on the TDA which has the advantage that it is more intuitive, graphic and easier to implement model calculations.Our main results are as follows.
• In order to draw the topological diagrams and construct the relevant amplitudes in the TDA, we employ the anti-symmetric matrix (B c ) ij for the charmed baryon and (B 8 ) ijk for the octet baryon, where the indices i, j, k stand for light quark flavors.
• The wave functions of octet baryons can be represented in several different manners, but physics is independent of the convention on chooses.We use the bases ψ(8) A 12 and ψ(8) S 12 to assign different topological diagrams and amplitudes.After applying for the KPW theorem, the number of independent topological diagrams and amplitudes is reduced to 7. At this stage, there still exist 2 redundant degrees of freedom through redefinition.We conclude that the minimum set of the topological amplitudes in the TDA is 5.
• To demonstrate the equivalence between the TDA and IRA, we have shown that the number of the minimum set of tensor invariants in the IRA and the topological amplitudes in the TDA is the same and presented their relations.
• As there are 5 independent tilde TDA amplitudes, we have totally 19 unknown parameters to describe the magnitudes and the phases of the respective S-and P -wave amplitudes.The measured branching fractions and decay asymmetries are well accommodated in the TDA except for three modes: Ξ 0 c → Ξ − π + , Λ + c → Ξ 0 K + and the ratio R Ξ − K + .The χ 2 value is 2.0 per degree of freedom.The predicted B(Ξ 0 c → Ξ − π + ) = (2.83 ± 0.10)% is larger than its current value, but it is in agood agreement with the sum rule derived in both the TDA and IRA.This needs to be tested in the near future.
• The phase difference δ P −δ S between S-and P -wave amplitudes is subject to a sign ambiguity which can be resolved by the measurement of the transverse asymmetry β.Hence, more accurate measurements of β in Λ + c → Ξ 0 K + and other modes are called for.
• When the strong phases δ X i S and δ X i P are turned off for the global fit to the data, the decay asymmetry α is found to be close to unity in magnitude for some of the modes, e.g., Λ + c → Σ + η, Ξ 0 K + , Σ 0 K + , Ξ + c → Ξ 0 π + and Ξ 0 c → Λ 0 K 0 S , Ξ 0 η.The nearly vanishing α Ξ 0 K + measured by BESIII implies the necessity of incorporating phase shifts in order to accommodate the data.The measurement of α Ξ 0 c →ΛK S and α Ξ 0 c →Ξ 0 η in the future will also help understand the phase shifts which are predicted to be 1.89 ± 0.25 and 1.48 ± 0.24 (both in rad), respectively.In contrast, it is also important to measure the decay asymmetry of Ξ + c → Ξ 0 π + to see if its largeness of order −0.93 is not affected even after the phase-shift effect is incorporated.
• The fit results for the decay asymmetry and the phase shift δ P − δ S for Λ + c → Ξ 0 K + are consistent with the BESIII measurements, though our fit of its branching fraction is slightly smaller than the measured one.Nevertheless, our TDA results are very close to those obtained in the IRA.
• For yet-to-be-measured modes, we have presented the fitting magnitudes of S-and P -wave amplitudes and their phase shifts in both the TDA and IRA which can be tested in the near future.

b 9 =
−b 8 , b 18 = −b 17 and b 20 = −b 19 .Furthermore, we notice that the combinations of the coefficients b 1 + b 2 , b 4 + b 5 , b 7 + b 8 , b 16 + b 17 and b 21 − b 19 always appear in the decay amplitudes Under flavor SU(3) symmetry, the operators O − and O + transform as the irreducible representations of 6 and 15, respectively.Notice that O + (O − ) is symmetric (antisymmetric) in the color indices of light quarks.Since the Wilson coefficients c − ≫ c + , it has been assumed the sextet 6 dominance over 15 in the literature.Under this hypothesis, one would have the relation[15] closely.Consider the decay amplitude of B c → B 8 M induced by H 15 or the operator O + .For the nonfactorizable contributions of O + , the relevant matrix elements are ⟨B * |O + |B c ⟩ and ⟨B 8 |O + |B * ⟩ after considering the pole contributions from the intermediate baryon pole B * .Since O + is symmetric in color indices while baryons are antisymmetric, we are led to ⟨B i |O + |B j ⟩ = 0, which is one of the ingredients for the derivation of the KPW theorem.Hence, we are left with the factorizable contributions of ⟨B 8 M |O + |B c ⟩, which can be inferred from the factorizable terms T and C given in Eq. (

TABLE II :
Same as Table I except for doubly Cabibbo-suppressed charmed baryon weak decays.