Unification of Flavor SU(3) Analyses of Heavy Hadron Weak Decays

Flavor SU(3) analyses of heavy mesons and baryons hadronic charmless decays can be formulated in two different forms. One is to construct the SU(3) irreducible representation amplitude (IRA) by decomposing effective Hamiltonian, and the other is to draw the topological diagrams (TDA). We study various B/D → PP, V P, V V , Bc → DP/DV decays, and two-body nonleptonic decays of beauty/charm baryons, and demonstrate that when all terms are included these two ways of analyzing the decay amplitudes are completely equivalent. We clarify some confusions in drawing topological diagrams using different ways of describing beauty/charm baryon states.


I. INTRODUCTION
Weak decays of heavy mesons and baryons carrying a bottom and/or a charm quark are of great interests and have been studied extensively on both experimental and theoretical sides. These decays provide useful information about the strong and electroweak interactions in the standard model (SM). Rare decays are ideal to look for new physics effects beyond SM, and recent measurements of lepton flavor universality have shown notable deviations from the standard model (see Ref. [1] for a recent brief review on the anomalies in B decays). Quite a number of physical observables like branching fractions, CP asymmetries and polarizations have been precisely measured by experiments [2][3][4]. On the other hand, due to our poor understanding of QCD at low energy regions, theoretical calculations of decay amplitudes are not well understood. Most of the current calculations rely on the factorization methods. Among them, many available studies are conducted at leading power in 1/m b , while recent analyses of semileptonic and radiative processes have indicated the importance of next-to-leading power corrections [5,6]. Apart from factorization approaches, the flavor SU(3) symmetry is a powerful tools frequently used in two-body and threebody heavy meson decays [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Although flavor SU(3) symmetry is approximate, yet it still provides very useful information about the decays. Since the SU(3) invariant amplitudes can be determined by fitting the data, the SU(3) analysis bridges experimental data and the dynamical approaches.
Among different realizations of carrying out SU (3) analysis for decay amplitudes there are two popular methods. One of them is topological diagram amplitude (TDA) method, where decay amplitudes are represented by connecting quark lines flows in different ways and then relate them by SU (3) symmetry, and another way is to construct the SU(3) irreducible representation amplitude (IRA) by decomposing effective Hamiltonian. While the TDA approach gives a better understanding of dynamics in the different amplitudes, the IRA approach shows a convenient connection with the SU(3) symmetry. These two methods looks very different in formulations, one may wonder whether they will obtain the same results. In Ref. [21], two of us have explored two-body B/D meson decays, B → P P and D → P P and pointed out the two methods are consistent when all contributions are included. However as we have pointed out, this equivalence is nontrivial: a few amplitudes were not included in the TDA; among the known diagrammatic amplitudes, one of them is not SU(3) independent, and should be removed. In this work, we extend our analysis to several other types of two body decays of B/D and B c mesons and also beauty/charm baryons to show the equivalence of the TDA and IRA methods. For two-body decays of beauty/charm baryons, we clarify some subtle points including the description of baryon representation, one or two indices for3, in relation to TDA. As we will show, it is easy to determine the independent amplitudes in IRA while TDA gives some redundancy. A few amplitudes are not independent and therefore should be removed. On the other hand, an advantage of TDA approach is that based on its topological nature, diagrams are helpful for understanding the internal dynamics in a more intuitive way.
The rest of this paper is arranged as follows. In Section II, we briefly summarize the SU(3) properties of various inputs. In Section III, we give the results for B/D → P P in the TDA and IRA methods to set up the notation. Then we provide results for B/D → P V, V V and discuss some points specialized to these decays. In section IV, we carry out a similar analysis for B c → DP, DV . In section V, we discuss beauty/charm baryon decays into an octet baryon and an octet pseudo-scalar meson. The expanded amplitudes and relations given these sections are useful for a global analysis when enough data is available in the future. In section VI we summarize our results. In the Appendix, we give the relations for different parametrizations in TDA and IRA methods for bottom and charmed baryon decays.

A. Hadron Multiplets
Several classes of heavy hadron, containing at heavy quark b or c, will be considered in this work. The involved processes include heavy SU(3) triplet mesons B and D decays into P P, P V, V V , and a B c meson decays into DP, DV . For heavy baryons, the decay processes include a heavy anti-triplet T c3 or a T b3 decay into a baryon in the decuplet T 10 plus a light meson, and decay into a baryon in the octet T 8 plus a light meson. We display the hadron SU are flavor SU (3) anti-triplets. The light pseudoscalar P and vector V mesons are mixture of octets and singlets so that each of them contain nine hadrons: where ω and φ mix in an ideal form, while the η and η ′ are mixtures of η 8 and η 1 with the mixing angle θ: η = cos θη 8 + sin θη 1 , η ′ = − sin θη 8 + cos θη 1 .
Since η 8 and η 1 are not physical states, optionally one can choose the η q and η s basis for the η mixing, which are defined so that the pseudoscalar octets P has the same form of parametrization as vector octets V : with An advantage of the parametrization in Eq. (4) is that there is a one-to-one correspondence between the decay amplitudes of channels with vector final state and that of channels with pseudoscalar final state in the SU(3) limit. A charmed or a bottom baryons with two light quarks can form an anti-triplet or sextet. Most members of the sextet can decay through strong interaction or electromagnetic interactions. The only exceptions are Ω b and Ω c [22]. We will concentrate on anti-triplet weak decays. For the anti-triplet bottom and charmed baryons, we have the following matrix expressions: One can also contract the above matrix with the anti-symmetric tensor ǫ ijk (ǫ 123 = +1) to have T3 ,i = ǫ ijk T jk One can also contract the above with ǫ ijk to have (T 8 ) ijk ≡ ǫ ijn (T 8 ) n k . The light baryon decuplet is given as: B.

SU (3) properties of effective Hamiltonian
Effective Hamiltonian for charmless b decays In the SM weak decays of charmless b decays are induced by the following electroweak effective Hamiltonian [23][24][25]: Here G F is the Fermi constant, and the V uq and V tq are CKM matrix elements. The O i is a four-quark operator with C i as its Wilson coefficient. The explicit forms of O i s are given as follows: . For the ∆S = 0(b → d) decays, the non-zero components of the effective Hamiltonian are [8,11,12]: For the ∆S = −1(b → s) decays the nonzero entries in the H3, H 6 , H 15 can be obtained from Eq. (12) with the exchange 2 ↔ 3 corresponding to the d ↔ s exchange. QCD penguin operators O 3−6 behave as the3 representation. For the magnetic moment operators, the color magnetic moment operator O 8g = (g s m b /4π)sσ µν T a G a µν (1 + γ 5 )b is an SU(3) triplet, while the electromagnetic moment operator O 7γ = em b 4πs σ µν F µν (1 + γ 5 )b can be effectively incorporated into the O 7−10 . Thus both of them are not included in Eq. (10) and the above decomposition is complete.
The irreducible representation amplitude (IRA) method of describing related decays is to decompose effective Hamiltonian according to the above mentioned representations and construct the amplitudes accordingly. On the other hand the the topological diagrams (TDA) method is to take the effective Hamiltonian with two light antiquarks and a light quark H ij k to representquūb with i =ū, k = u and j =q (omitting the Lorentz indicies), and then contract the indices with initial and final hadron states. In this way the decays are represented by diagrams following the quark line flows. Note that in the TDA method, the indices i and j ordering matters which are neither symmetry nor anti-symmetric. They are not traceless neither.
The effective Hamiltonian have both tree and loop contributions. When strong penguin and electroweak penguin are all included the tree and loop contributions have3, 6 and 15 representations. The independent amplitudes have the same numbers, except that one can make one of the tree or penguin amplitude real and the rest all in principle complex. Using the unitarity property of the CKM matrix V ub V * uq + V cb V * cq + V tb V * tq = 0, one can also rewrite c loop induced penguin contributions into amplitude proportional to V ub V * uq and V tb V * tq , For simplicity, we refer to A u as "tree" amplitude since it is dominated by tree contributions with modifications from u and c loop contributions. A t is "penguin" amplitude with c and t loop contributions. It is necessary to stress that not all contributions in A u are tree diagrams, and the same for A t . Since both A u and A t have similar amplitudes with SU (3) representations. In our later discussions we will concentrate on the A u amplitudes. One can easily obtain the A t amplitudes by just changing the labels.

Effective electroweak Hamiltonian for c hadronic decays
For weak interaction induced c hadronic decays, the effective Hamiltonian with ∆C = 1 and ∆S = 1 is given as: where we have neglected the highly suppressed penguin contributions, and For the doubly Cabbibo suppressed c → dus transition, we have amplitudes to be proportional to V cd V * us and the Hamiltonians are: For decays proportional to V cs V * us , we have: and for decays proportional to V cd V * ud , we have: For singly Cabbibo suppressed decays, c → udd and c → uss transitions have approximately equal magnitudes but opposite signs: As a result, the contributions from thē 3 representation vanish, and one has the nonzero components contributed only by 6 and15 representations.
For the singly Cabibbo-suppressed transition, there are also loop contributions proportional to V cb V * ub . Such loop contributions are small so that we will concentrate on the dominant amplitude proportional to V cs V * us . However, one can include these contributions by adding a 3 representation in the Hamiltonian.
Again, we use the above SU (3) decompositions for IRA analysis and use the effective Hamiltonian H ij k with i =s, j =ū and k = q for TDA analysis to trace the quark line flows.

III. CHARMLESS TWO-BODY B DECAYS
A. B → P P decays Let us start with the B → P P decays. The generic amplitude is decomposed according to CKM matrix elements: The IRA and TDA amplitudes should be equivalent, though as we have shown [21] this equivalence is not obvious.
To obtain IRA, one takes various representations in Eq. (12) and contracts all indices in B i and light meson P i j with various combinations: In the TDA decomposition, one has: According to this decomposition, topological diagrams for B → P P decays can be found in Fig. 1. Apart from the ordinary T, C, A, E, we have also included the other SU(3) irreducible amplitudes, most of which come from loop diagrams, and/or the flavor singlet diagram. Expanding Eqs. (21,22), one obtains B → P P amplitudes in Table I. Since we have decomposed the effective Hamiltonian into irreducible representations, one may expect that there are 10 independent amplitudes for A u and similarly 10 amplitudes for A t . A careful examination shows that the A T 6 can be absorbed into B T 6 and C T 6 with a redefinition: This combination can also be found explicitly from Table I. After eliminating the redundant amplitude, there are actually only 18 (A u and A t contribute 9 each) SU(3) independent amplitudes. An overall phase convention is not an observable, thus there are only 17 independent real parameters for decay amplitudes. We list all TDA amplitudes in Table I. It is necessary to point out that the last 6 diagrams in Fig. 1 are often omitted in the SU(3) TDA analysis. However only by including them the complete equivalence of IRA and TDA can be established. One of the 10 TDA amplitudes must be redundant. Such redundancy can be understood through the following relations between the IRA and TDA amplitudes: We have adopted the choice in which E is always in companion with another amplitude. It is also possible to replace the role of E by one of the amplitudes A, C or even T . One can also reversely obtain: Similar analysis for the A t contributions can be obtained with the replacement for the IRA: while for TDA, we have: 1. Impact of the new TDA amplitudes The new TDA amplitudes in Fig. 1 may play an important role in understanding CP violation (CPV) phenomena. Without the new TDA amplitudes, some decays only have terms proportional to V * tq V tb , such as B 0 → K 0K 0 and B 0 s → K 0K 0 . For instance, in Ref. [18], the amplitudes for B 0 → K 0K 0 read: This would imply the CP violating asymmetry is identically zero. However, as we have shown, these two decays receive contributions from the P u + 2P u A multiplied by V * uq V ub : Therefore a non-vanishing direct CP asymmetry is obtained. This would certainly affect the search for new physics in a precise CP violation measurement.
Most new TDA amplitudes in Fig. 1 arise from higher order loop corrections, and thus they are likely small in magnitude. However, sometimes they can not be completely neglected. In Ref. [12], the authors have performed a fit of B → P P decays in the IRA framework. Depending on different choices of data, four cases were considered in their analysis [12]. Here for illustration, we give their results in case 4: where the magnitudes and strong phases are defined relative to the amplitude C P 3 . From Eq. (25), one can find that the C T 3 is a mixture of T , C and others, while the B T 15 equals (E u S + A u S )/8. The fitted results in Eq. (30) indicate, compared to C T 3 , the B T 15 could reach 20% in magnitude, and more notably, the strong phases are sizably different. The fact that the B T 15 , namely E u S and A u S , have non-negligible contributions supports our call for a complete analysis.

Comparison with QCDF amplitudes
The topological amplitudes in B → P P decays can be compared to the QCDF amplitude in Ref. [26]. Such a comparison requests two remarks. Firstly, in our decomposition, we adopt the CKM matrix elements V ub V * uq and V tb V * tq , while Ref. [26] used V ub V * uq and V cb V * cq . The unitarity of CKM matrix guarantees the equivalence of the two approaches. So we will directly compare the "tree" A u and "penguin" A t amplitudes, though some of them might be recombined in order to have the same CKM factors. Secondly, we have decomposed one part of the electroweak penguin into the QCD penguin as shown in Sec. II, and we will do so for QCDF amplitudes too.
We have the following relations for "tree" amplitudes: where the notations α i and β i are from Ref. [26]. For "penguin" ones, one can derive the relation:

U-Spin relations
Some decay channels shown in Table I with ∆S = 0 and ∆S = 1 are related by U-spin, the d ↔ s exchange symmetry. The relations will be discussed explicitly in the following. These pairs of channels include: In the past years, there have been extensive examinations on the U-spin symmetry. One of the interesting features of these U-spin pairs is that there are CP violating relation among them. Here we consider two U -spin related decays with the same "tree" A u and and "penguin" A t 1 : Through the relation Im This leads to a relation between branching ratio and CP asymmetry A i CP (∆S) = ∆(B i → P P, ∆S)/B(B i → P P ): Here B(B i → P P ) is the branching ratio of B i → P P and τ i is the lifetime of B i . One of the most prominent example is the case of the U-spin pair B 0 s → π − K + and B 0 → π + K − . Here we will comment on the experimental situation for this case and introduce a parameter r c to account for the deviation from 1 For baryonic decay modes to be discussed in the following, there are non-trivial Clebsch-Gordon coefficients, such that Eq. (33) is modified as:

The relation in Eq. (35) is changed to:
In the SU(3) symmetry limit r c = 1.
Using the experimental data from PDG [3,4]: one finds: where all errors have been added in quadrature. The resulting r c value indicates that the U-spin symmetry is well in the case of this decay pair. The exploration in more decay pairs is helpful for further investigation on this symmetry. Similar U-spin relations existing in other decays will be studied in the following sections. We will comment on them when specific decay channels are be discussed.
Decay amplitudes for B → V V channels can be obtained similarly by replacing the pseudo-scalar multiplet P by the vector multiplet V in Eq. (21) and in Eq. (22).
• Since we have chosen the same parametrization for pseudoscalar and vector mesons, the expanded amplitudes for the B → V V channels can be obtained directly from the B → P P .
• There are three sets of amplitudes for B → V V decays, corresponding to different polarizations. For convenience, one can choose the helicity amplitudes A 0 , A + , A − defined as: with ǫ 0123 = 1, and Thus there are in total 3 × 9 = 27 complex amplitudes for both tree and penguin, where "9" is the number of the polarization combination of final two vectors. These amplitudes correspond to 2 × 54 − 1 = 107 real parameters in theory. Two phases can not be measured through direct measurements of individual B andB decays, but one of the two can be obtained through the time-dependent analysis.
• In principle, all these 107 parameters could be determined through the angular distribution studies in experiment. Each B → V (→ P 1 P 2 )V (→ P 3 P 4 ) channel can provide 10 observables. The angular distribution is given as: . Here θ 1 (θ 2 ) is defined by the flight direction of P 1 (P 3 ) in the rest frame of V 1 (V 2 ) and the flight direction of V 1 (V 2 ) in the B meson rest frame. φ is the relative angle between the two decay planes.
• Unfortunately, due to the large amount of input parameters, it is a formidable task to perform a global fit, and in particular only limited data is available [3]. A realistic analysis at this stage will pick up only a limited amount of amplitudes. In this direction, the weak annihilations and hard scattering amplitudes were extracted by fitting relevant data in Ref. [28], while the authors in Ref. [29] have performed a factorization-assisted TDA analysis. This allows one to remove some suppressed amplitudes at the leading order approximation. In Ref. [30], the authors have adopted the dynamical analysis in the SCET and performed a flavor SU(3) fit of B → V V decays.
On the other hand, recent dynamical improvements exist in Refs. [31,32] using the perturbative QCD approach and Refs. [33] in QCDF.

C. B → V P Decays
We now study the B → V P decays, whose amplitudes can be obtained by replacing one of the P in Eq. (21) and in Eq. (22) by V to obtain the IRA and TDA amplitudes. There are two ways to replace one of the P , therefore the amplitudes will be doubled compared with B → P P . We have IRA and TDA for B → V P decays as follows: The expanded amplitudes are given in Tab. II and Tab. III. Relations between the two sets of amplitudes are derived as: The inverse relations are solved as: Unlike the B → P P and B → V V case, we are not able to find any redundant amplitude. Thus in total, we have 18 complex amplitudes for "tree" and "penguin", respectively. It corresponds to 2 × 36 − 1 = 71 real parameters in theory. A fit with all parameters is not available again, and most of the current analyses have made approximations by neglecting some suppressed amplitudes [18,19,34,35].
The B → V P channels related by the U-spin include: However on the experimental side, there are not enough measurements to examine these relations, in particular the CPV in B s sector has received less consideration. We expect the situation will be improved when a large amount of data is available at LHCb, and Belle-II.
Using the effective Hamiltonina in Eqs. (16) and (17), one can easily obtain the SU (3) decay amplitudes in a similar fashion as that for B → P P, V V, P V . In this case there is only tree contributions which we write as IV: Decay amplitudes for two-body D → P P decays. Decay amplitudes for two-body D → P P decays. The CKM factor should be multiplied: VcsV * ud for Cabibblo-Allowed decays; VcsV * us for singly Cabibbo-suppressed modes and V cd V * us for doubly Cabibbo-suppressed modes.
For D → P P we have: The expanded amplitudes are given in Tab. IV. The amplitudes A T 6 can be incorporated in B T ′ 6 and C T ′ 6 , and then we have 5 independent amplitudes for D → P P : with the inverse relation: Since one amplitude is redundant, fits with all 6 complex amplitudes should not be resolved in principle. This has been indicated by the strong correlation of parameters in the fits in Ref. [36]. Again for D → V V decays there are three sets of amplitudes similar as the D → P P , and thus we have 15 independent amplitudes in total.
The IRA and TDA for D → V P decays are given as: The expanded amplitudes are collected in Tab. V, Tab. VI and Tab. VII for the different transitions. Relations   between the two sets of amplitudes are derived as: The inverse relations are solved as:  It is interesting to explore the useful relations for decay widths from the amplitudes listed in Tab. V, Tab. VI and Tab. VII. For Cabibblo Allowed channels, we find Γ(D + s → ρ + π 0 ) = Γ(D + s → ρ 0 π + ). For singly Cabibblo suppressed channels, one has: We refer the reader to Refs. [36][37][38][39] for some explorations of the implications on decay rates and CP asymmetries, and Refs. [40,41] for the experimental analyses. We should point out that since the quark mass effects in charm decays might play an important role when analyzing the D decays, and the SU(3) symmetry is less impressive for D meson decays [36].

IV. Bc → DP, DV DECAYS
The effective Hamiltonian for b quark decays can induce B c → DP, DV transitions. The corresponding topological diagrams are given in Fig. 3. The IRA and TDA for B c → DP decays are given as: The expanded amplitudes can be found in Tab. VIII. Relations between the two sets of amplitudes are given as: "Penguin" amplitudes are obtained similarly: Including the "penguins", one has 8 complex amplitudes in total. Again decay amplitudes for B c → DV can be obtained by replacing the pseudoscalars by their vector counterparts. The U-spin related channels include: In Ref. [42], the LHCb collaboration has measured the product: where the f (B c ) and f (B + ) are the production rates of B + c and B + , respectively. With the measured ratio [43]: one can obtain an estimated branching fraction: On theoretical side, model-dependent analyses give 1.3×10 −7 [44], and 6.6×10 −5 [45], while a phenomenological study [46]. Since this transition is induced by b → s, the large branching fraction may imply a large penguin amplitude P . Such a scenario can be tested by measuring the corresponding (B + c → D + K 0 ), which has the same penguin amplitude. Model-dependent calculations of other B c decays can be found in Refs. [47,48].
Some recent SU(3) analyses of two-body B c decays can be found in Ref. [49,50]. Compared to these studies, we have included all penguin amplitudes.
For the B − c meson, the charm quark can also decays, with the final state BP or BV [51]. Since the heavy bottom quark plays as a spectator, the decay modes are simpler. For example, for Cabibbo-allowed decay modes, there are only two channels: s . Thus we expect that the SU(3) symmetry will not provide much information in these decays.
It is necessary to point out that the charmless two-body B c decays are purely annihilation, and the typical branching fractions are below the order 10 −6 [52][53][54]. Since there are not too many channels, it is less useful to apply the flavor SU(3) symmetry to these modes.

V. ANTITRIPLET BOTTOM BARYON DECAY INTO A BARYON AND A MESON
In this and next sections, we discuss weak decays of baryons with a heavy b and c quark. Charmed or bottom baryons with two light quarks can form an anti-triplet or a sextet. Most members of the sextet can decay via strong interactions or electromagnetic interactions. The only exceptions are Ω b and Ω c . In the following we will concentrate on the anti-triplet baryons, whose weak decays are induced by the effective Hamiltonian H b ef f and H c ef f .
Decay into a decuplet baryon T10 and a light meson The IRA amplitudes for the T b decays into a decuplet baryon and a light meson can be parametrized as: b3 (H 6 ) [kl] The TDA amplitudes are shown in Fig. 4 with the parametrization: We find relations between the two sets of amplitudes as: The expanded amplitudes for individual decay modes can be found in Tab. IX. A few remarks are given in order.
• As the two light quarks in the initial state are antisymmetric in the flavor space while they are symmetric in the final state. An overlap of wave functions vanishes [55], unless hard scattering interactions occur [56]. In other words, there is no "factorizable" contribution in the transition. In addition, all diagrams in Fig. 4 are suppressed by powers of 1/N c compared to the T b → T 8 P . This will indicate that branching fractions for these decays are likely smaller than the relevant B decays and T b → T 8 P decays, where T 8 represents the octet baryon.
• For the T b → T 10 P , one can construct the amplitudes with the spinors, and a general form is: where A and B are two nonperturbative coefficients containing the CKM factors, and have the same flavor structure with A u,t . Thus in total, one has 6 × 2 × 2 = 24 complex amplitudes in theory.
• Since the initial baryon and final baryons are polarized, it is convenient to express the decays with helicity amplitudes: where S in and S f 1 , S f 2 are polarizations of initial and final states. The two sets of helicity amplitudes for T b → T 10 P can be derived using the parametrization in Eq. (64): Here E T10 and p cm are the energy and 3-momentum magnitude of T 10 in the rest frame of T b . N T10 and N T b are normalization factors of T 10 and T b spinors: • For T b → T 10 V , one can construct the amplitudes with the spinors and polarization vector 2 : There are six different polarization configurations. The helicity amplitudes are given as: The definitions of E T10 , p cm , N T10 and N T b are the same as Eq.(68) except replacing m P by m V . Again all these amplitudes can be determined from the angular distributions of the four-body decays T b → T 10 (→ T 8 P 1 )V (→ P 2 P 3 ).
• Branching fractions for T b decays into a proton with three charged pion/kaons are found at the order 10 −5 in Ref. [57]. A plausible scenario is that the T b → T 10 V contribute significantly to the T b decaying into a proton and three charged light mesons. If this is true, we expect that with more data in future, a detailed analysis will determine the decay widths of T b → T 10 V . Then the flavor SU(3) symmetry can be examined, and meanwhile it will also shed light on the CP and T violation in baryonic transitions by using the triplet product asymmetries [58,59].
• Through the results in Tab. IX, we can find the relations both for decays into T 8 P and T 8 V . Here only the channels with one vector octet in final states can be listed (71), (72). For channels with one pseudoscalar in final states the relations are almost the same, obtained by replacing the vector multiplets V by the pseudo-scalar multiplets P . However, η q and η s are unphysical states so that the decay width relations involving them should be removed.
For b → d transitions, one has: For b → s transition, we have the relations for decay widths: • As discussed in the previous section, charmless b → d and b → s transitions can be connected by U-spin.
In Table X, we collect the T b → T 10 V decay pairs related by U -spin, while results for the final state with a light pseudoscalar meson can be obtained similarly. CP asymmetries for these pairs satisfy relation in Eq. (35). Inspired from B decay data [3,4], we expect CP asymmetries for these decays are at the order 10%. Experimental measurements of these relations are important to test flavor SU (3) symmetry and the CKM description of CP violation in SM.

into an octet baryon and a meson
If the final state contains a baryon octet, the topological diagrams are shown in Fig. 5 where ten diagrams can be found. However unlike the decuplet baryon, the octet baryon is not fully symmetric or antisymmetric in flavor space. Thus each of the diagrams can provide more than one amplitudes. In total, one can have 26 independent TDA amplitudes: In the IRA approach, one can construct 14 amplitudes: It should be noticed that in the IRA approach the antitriplet baryon and octet baryon are expressed in SU(3) representation3 and 3 ⊗3 respectively which are different from the representation 3 ⊗ 3 and3 ⊗3 ⊗3 used for TDA. Superficially this different representation contains less indexes so that it reduces the number of amplitudes from   amplitudes are related as follows: However, even after such reduction, there still exists one independent degree of freedom among the 14 IRA ampli- tudes. The redundant amplitude can be made explicit with the redefinitions: In addition, this redundancy can be understood more explicitly. In this work as well as the previous work Ref. [21] we use the irreducible representation operators for IRA as (H 6/15 ) ij k . Actually there exists a simpler H 6 representation introduced by Ref [74], where H 6 has only two lower indexes (H 6 ) ij . With the use of (H 6 ) ij we do have only 13 IRA amplitudes. However, Since the IRA operators (H 6/15 ) ij k have the same index structure as the TDA operators. They make the derivation of IRA/TDA correspondence more directly so we will keep the use of them.
The expanded amplitudes can be found in Tab. XI for the b → d transition and XII for the b → s transition, respectively. Again if the final state is a vector meson, the amplitudes can be derived similarly.
A few remarks are given in order.
• At first sight, the diagrammatic approach, as depicted in Fig. 5, is more intuitive, however as we have shown in the above, it is very hard for us to determine the independent amplitudes in this approach. This will introduce subtleties to the global fit in the diagrammatic approach.
• Without including the polarization, one can see from the IRA approach, there exist 13 independent complex amplitudes with CKM factor V ub V * uq and another 13 amplitudes accompanied by V tb V * tq . • Two polarization configurations exist for decays into a pseudoscalar meson, while there are four possibilities for decays into a vector meson.
• The U-spin related decay pairs are given in Tab. XIII, which completely fits with the results given by Ref. [60].
Here only the case for T b → B 8 P is listed. Since no unphysical states η q and η s exist in Tab. XIII. The U-spin pairs for T b → B 8 V are similar by replacing pseudoscalar octets by vector octets.
Thus measuring the branching fractions and CP asymmetries for Ξ 0 b → π − Σ + and Ξ 0 b → K − Σ + will help us to understand the U-spin in baryonic decays.  subtleties. Using two-body B/D meson decays, we have demonstrated that a few SU(3) independent amplitudes have been overlooked in TDA. Most of these amplitudes arises from higher order loop corrections, but they are irreducible in the flavor SU(3) space, and thus can not be neglected in principle. Taking these new amplitudes into account, we find a consistent description in both approaches. We have pointed out that these new amplitudes can affect direct CP asymmetries in some channels significantly. An interesting observation is that, with these new amplitudes, the direct CP symmetries for charmless nonleptonic b decays cannot be identically zero. For heavy baryon decays, we pointed out though the TDA approach is very intuitive, it suffers the difficulty in providing the independent amplitudes. On this point, the IRA approach is more helpful.
All results derived in this paper can be used to study the heavy meson and baryon decays in the future when sufficient data become available. Then one can have a better understanding of the role of flavor SU(3) symmetry in heavy meson and baryon decays.
For charm quark decays, we did not include the penguin contributions, which can also be studied in a similar manner. It is also necessary to notice that the flavor SU(3) symmetry has been applied to study weak decays of doubly heavy baryons [80][81][82], and multi-body Λ c decays [78]. The equivalence between the TDA and IRA approaches in these decay modes can be studied similarly.