Unification of flavor SU(3) analyses of heavy Hadron weak decays

Analyses of heavy mesons and baryons hadronic charmless decays using the flavor SU(3) symemtry can be formulated in two different forms. One is to construct the SU(3) irreducible representation amplitude by decomposing effective Hamiltonian, and the other is to draw the topological diagrams. In the flavor SU(3) limit, we study various B/D→PP,VP,VV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B/D\rightarrow PP,VP,VV$$\end{document}, Bc→DP/DV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_c\rightarrow DP/DV$$\end{document} 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. Furthermore we clarify some confusions in drawing topological diagrams using different ways of describing beauty/charm baryons.


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 limited 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 a e-mail: hexg@sjtu.edu.cn b e-mail: shiyuji@sjtu.edu.cn (corresponding author) c e-mail: wei.wang@sjtu.edu.cn 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].
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. The TDA approach gives a better understanding of decay dynamics especially in the bottom hadron decays, where the power expansion of 1/m b can be properly conducted so that each TDA amplitude can be related with matrix elements of SCET operators [21]. On the other hand, 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. This equivalence for some considered decays has been discussed in Refs. [7,16,20], in particular for charmed meson decays, Ref. [20] has shown the equivalence in the presence of SU(3) symmetry breaking effects. In Ref. [22], two of us have explored two-body B/D meson decays, B → P P and D → P P and pointed out that in the exact SU(3) symmetry limit, the two methods are consistent when all contributions are included. However this equivalence is nontrivial: a few amplitudes are suppressed and thus were not included in some TDA analysis; among the known diagrammatic amplitudes, one of them is not SU(3) independent, and should be absorbed into other amplitudes. Actually, in charm meson decays, the fact that one of the known amplitudes, T, A, C, E, is not independent has already been noticed in Ref. [20].
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. We will also work in the exact flavor SU(3) limit throughout this work. For two-body decays of beauty/charm baryons, we clarify some subtleties 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 absorbed into other amplitudes. Despite this disadvantage, the topological nature of TDA is still helpful for understanding the internal dynamics underlying in b decays in a more intuitive way.
The rest of this paper is arranged as follows. In Sect. 2, we briefly summarize the SU(3) properties of various inputs. In Sect. 3, we give the results for B → P P in the TDA and IRA methods to set up the notation. Then we provide results for B → PV, V V and discuss some points specialized to these decays. In Sect. 4, we give the results for D → P P, V V, PV in the TDA and IRA methods. In Sect. 5, we carry out a similar analysis for B c → D P, DV . In Sects. 6 and 7, we respectively discuss beauty and charm baryon decays into an octet baryon and an octet pseudo-scalar meson, while for beauty baryon decays we also consider the final states containing a decuplet baryon. The expanded amplitudes and relations given these sections are useful for a global analysis when enough data is available in the future. In Sect. 8 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.

Hadron multiplets
Several classes of heavy hadron, containing at heavy quark b or c, will be considered in this work. The involved processes include decays of heavy SU(3) triplet mesons B and D into P P, PV, V V , and the B c meson into D P, DV . For heavy baryons, the decay processes include a heavy anti-triplet T c3 or a T b3 decays into a baryon in the decuplet T 10 plus a light meson, and decay into a baryon in the octet T 8 are flavor SU(3) anti-triplets. Light pseudoscalar P and vector V mesons are mixtures of octets and singlets so that each of them contain nine hadrons: where ω and φ mix in an ideal form. The η and η are mixtures of η 8 and η 1 with the mixing angle θ : 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 b and c [24]. 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 antisymmetric tensor i jk ( 123 = +1) to have T3 ,i = i jk T jk 3 with The lowest-lying baryon octet is given by: One can also contract the above with i jk to have (T 8 ) i jk ≡ i jn (T 8 ) n k . The light baryon decuplet is given as:

Effective Hamiltonian for charmless b decays
In the SM weak decays of charmless b decays are induced by the following electroweak effective Hamiltonian [25][26][27]: 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: 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 anti-quarks and a light quark H i j 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 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 in topology, and the same for A t . In Ref. [22], two of us have shown that both A u and A t have similar form of amplitudes with SU (3) representations. Thus in our later discussions we will concentrate on the A u amplitudes. One can easily obtain the A t amplitudes by just changing the amplitude labels.

Effective electroweak Hamiltonian for c hadronic decays
For hadronic decays of charmed hadrons, the effective Hamiltonian with C = 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 i j k with i =s, j =ū and k = q for TDA analysis to trace the quark line flows.

B → P P decays
Let us start with the B → P P decays. The generic amplitude is decomposed according to CKM matrix elements: where the amplitudes expressed by IRA and TDA should be equivalent. Although in fact this equivalence is concrete and obvious, as we argued in [22], it has not been thoroughly discussed in the literature.
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: Fig. 1 Topological diagrams for the amplitudes with CKM factor V ub V * uq in B → P P and B → V V decays. We work in the effective field theory at m b scale, and thus there exists four-quark interaction operators as shown in the above. The quark flavors corresponding to these operators are explicitly given. In these diagrams, when theūu annihilate, two or more gluons are needed to create one pair of quarks with flavor u, d, s, which are denoted by the unspecified lines For the TDA decomposition, one can classify the different topologies of diagram as [18]: (i) T , denoting the color-allowed tree amplitude with W emission; (ii) C, denoting the color-suppressed tree diagram; (iii) E denoting the W -exchange diagram; (iv) P, corresponding to the QCD penguin contributions; (v) S, being the flavor singlet QCD penguin; (vi) A, annihilation diagrams.
With each coefficient denoted by the above topologies, one has the TDA decomposition as: According to this decomposition, topological diagrams for B → P P decays can be found in Fig. 1. Since we work in the effective field theory at m b scale, as shown in this figure there exists four-quark interaction operators. In these diagrams, the quark flavors corresponding to these operators are explicitly given. When theūu annihilate, two or more gluons are needed to create one pair of quarks with flavor u, d, s, which are denoted by the unspecified lines. 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 1, where the IRA amplitudes are consistent with Ref. [12] and an earlier work [23]. 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 1.
After eliminating the redundant amplitude, there are actually only 18 (A u and A t contribute 9 each) SU(3) independent amplitudes. Since one overall phase can be chosen free, there are 35 independent real independent parameters. If one consider η 1 − η 8 mixing, the mixing angle θ should also be introduced.
We list all TDA amplitudes in Table 1. It is necessary to point out that the last six diagrams in Fig. 1 are omitted in some SU(3) TDA analysis [16,18], while other amplitudes are consistent with Refs. [16,18]. 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: It should be pointed out that the amplitudes generated Q 7,8,9,10 operators have the form (2/3)ūu − (1/3)(dd +ss) and these amplitudes can be can again be as a sum of a treelike operatorūu and a penguin-like operator −(1/3)(ūu + dd +ss). Penguin-like contributions have been absorbed into P, P A , S, P SS , while tree-like amplitudes are denoted as P T , , P C , P T A , P T E , P E S . It should be noticed that the tree-like amplitudes P T and P C are also denoted as P EW , P EW,C in some references.

Impact of the new TDA amplitudes
The new TDA amplitudes in Fig. 1 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, as noticed in many references for instance Refs. [28][29][30]. 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 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. [31]. 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. [31] 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 Sect. 2, and we will do so for QCDF amplitudes too.
We have the following correspondence between the SU(3) TDA amplitudes and the QCDF amplitudes for "tree" amplitudes: where the notations α i and β i are from Ref. [31]. The correspondence for "penguin" ones is given as:

U-Spin relations
Some decay channels shown in Table 1 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 : This leads to a relation between branching ratio and CP asym- 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 Uspin pair B 0 s → π − K + and B 0 → π + K − . Their CP asymmetry have been studied in Ref. [33]. Here we will comment on the experimental situation for this case and introduce a parameter r c to account for the deviation from SU(3) sym- 1 For 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:
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.

B → V V decays
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 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 and B 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.
channel can provide 10 observables. The angular distribution is given as: Here 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. [34], while the authors in Ref. [35] have performed a factorization-assisted TDA analysis. This allows one to remove some suppressed amplitudes at the leading order approximation. In Ref. [36], 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. [37,38] using the perturbative QCD approach and Ref. [39] in QCDF.

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 Table 2 and Tab. 3. 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 [18,19,40,41].
Decay amplitudes for B → V P decays in Eqs. (42) and (43) are organized in a symmetric way. Taking the B − → ρ −K 0 and B − → K * 0 π − as an example, the IRA ampli- , and C T 1 15 ↔ C T 2 15 , while for the TDA amplitudes, the correspondence is: A 1 ↔ A 2 and P u1 ↔ P u2 . Other relations can be found in a similar way.
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.

D → P P, V V, P V decays
Using the effective Hamiltonian 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, PV . For D decays we will only discuss tree contributions which are written The penguin amplitudes are suppressed and thus neglected in this work, however it is necessary to stress penguin amplitudes are mandatory for the CP violation. The SU(3) amplitudes for D → P P are parametrized as The expanded amplitudes are given in Table 4. The amplitudes A T 6 can be incorporated in B T 6 and C T 6 , and then we have five independent amplitudes for D → P P: with the inverse relation: Since one amplitude is redundant, fits with all six complex amplitudes should not be resolved in principle. This has been indicated by the strong correlation of parameters in the fits in Ref. [42]. 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 (Fig. 2).
The IRA and TDA for D → V P decays are given as: The above amplitudes are also expanded in a symmetric way. The expanded amplitudes are collected in Tables 5, 6 and 7 for the different transitions. Relations between the two sets of amplitudes are derived as:  Table 4 Decay amplitudes for two-body D → P P decays. Decay amplitudes for two-body D → P P decays. The CKM factor should be multiplied: V cs V * ud for Cabibblo-Allowed decays; V cs V * us for singly Cabibbo-suppressed modes and V cd V * us for doubly Cabibbo-suppressed modes The inverse relations are solved as: It is interesting to explore the useful relations for decay widths from the amplitudes listed in Tables 5, 6 and 7. For Cabibblo Allowed channels, we find (D + s → ρ + π 0 ) = (D + s → ρ 0 π + ). For singly Cabibblo suppressed channels, one has: We refer the reader to Refs. [42][43][44][45] for some explorations of the implications on decay rates and CP asymmetries, and Refs. [46,47] for the experimental analyses. It is necessary to notice that since the QCD scale is comparable to charm quark mass, finite quark mass difference in s and d quarks may lead to sizable SU(3) symmetry breaking effect. A notable example to explore SU(3) symmetry breaking effects is the D 0 → K 0K 0 . This channel has vanishing branching fraction if exact SU(3) symmetry holds and the3 contribution proportional to V cb V * ub is neglected, which can also been seen from Table 4. The experimental data for branching ratios of D 0 → K 0K 0 and D 0 → K + K − are given as [3,4]: The decay amplitude of D 0 → K + K − is E + T , where T is color-allowed and could be estimated using the factorization framework. As |V cb V * ub /V cs V * sd | < 1.3 × 10 −3 , it is unlikely that the above non-zero branching ratio is caused by the3 penguin contribution. The above data shows that the SU(3) symmetry breaking effects in some channels can be as large as 30% at the amplitude level. Though the above estimate is channel-dependent, it indicates that symmetry breaking effects must carefully treated in D meson and charmed baryon decays in a systematic way [20,42,44,45,48,49].

B c → D P, DV decays
The effective Hamiltonian for b quark decays can induce B c → D P, DV transitions. The corresponding topological diagrams are given in Fig. 3. The IRA and TDA for B c → D P decays are given as: The expanded amplitudes can be found in Table 8. 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. [50], 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 [51]: one can obtain an estimated branching fraction: On theoretical side, model-dependent analyses give 1.3 × 10 −7 [52], and 6.6 × 10 −5 [53], while a phenomenological study implies the B(B + c → D 0 K + ) ∼ [4.4 − 9] × 10 −5 [54]. 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. [55][56][57][58]. Some recent SU(3) analyses of two-body B c decays can be found in Refs. [59,60]. 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 B P or BV [61]. 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 [62][63][64]. Since there are not too many channels, it is less useful to apply the flavor SU(3) symmetry to these modes.

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 b and c . In the following we will concentrate on the anti-triplet baryons, whose weak decays are induced by the effective Hamiltonian H b e f f and H c e f f .

into a decuplet baryon T 10 and a light meson
The IRA amplitudes for the T b decays into a decuplet baryon and a light meson can be parametrized as: 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 Table 9.
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 of the initial and final baryons is zero [65], which lead to vanishing decay amplitudes unless hard scattering interactions occur [66]. 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:   Fig. 4 Topology diagrams for the bottom baryon decays into a decuplet baryon and a light meson. We have explicitly specified the quark flavors for the four-quark interaction vertex, while the unspecified quarks can be u, d, s Table 9 Decay amplitudes for T b → T 10 P decays. Only those amplitudes proportional to V ub V * uq are shown, while "penguin" amplitudes proportional to V tb V * tq are similar Table 9 continued 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 can be 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. (65): Here E T 10 and p cm are the energy and 3-momentum magnitude of T 10 in the rest frame of T b . N T 10 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: 2 One may expect a term which looks like μναβ * μūν ( p T10 )(G σ αβ + H σ αβ γ 5 )u( p b ). Actually such term cam be absorbed into the term * μū μ ( p T10 )(E + F γ 5 )u( p b ) by using the fact that the spinor-vector u μ ( p T10 ), as a irreducible representation of 1/2 ⊗ 1, must satisfy γ μ u μ ( p T10 ) = 0.
The definitions of E T 10 , p cm , N T 10 and N T b are the same as Eq. (69) 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. [67]. 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 C P and T violation in baryonic transitions by using the triplet product asymmetries [68,69]. • Through the results in Table 9, 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 (72), (73). 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:

Fig. 5
Topology diagrams for the bottom baryon decays into an octet baryon and a light meson. Since the octet baryon is not fully symmetric or antisymmetric in flavor space, there are more than one amplitudes corresponding to one topological diagram. Actually the ten topological diagrams correspond to 26 amplitudes shown in Eq. (74). As in Fig. 4 In the IRA approach, one can construct 14 amplitudes: It should be noticed that in the above hadrons have been written in different forms for the same SU(3) multiplet. For illustration, we use the heavy bottom anti-triplet as the example.
Since it is an anti-triplet, it is most straightforward to use the (T b3 ) i to represent this particle multiplet, as in IRA approach. The advantage of this form is its compactness, however, with this form it is not easy to understand the quark flows in the decays. Instead there are two anti-symmetric light quarks in the anti-triplet heavy bottom baryon, and thus it is viable to use (T b3 ) i j (with i j anti-symmetric) to denote this multiplet. The second form contains two SU(3) indices, less compact, but it can reflect the quark flows in the decays. Thus the second form is suitable for drawing Feynman diagrams, and thus adopted in the above TDA amplitude. These two forms are equal, and one can establish relations between the SU(3) parameters corresponding to these two forms. The explicit discussions in IRA approach are given in "Appendix A". The 14 IRA amplitudes and 26 TDA 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. [22] we use the irreducible representation operators for IRA as (H 6/15 ) i j k . Actually there exists a simpler H 6 representation introduced by Ref. [70], where H 6 has only two lower indexes (H 6 ) i j . With the use of (H 6 ) i j we do have only 13 IRA amplitudes. However, Since the IRA operators (H 6/15 ) i j 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 Table 11 for the b → d transition and Table 12 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 a first sight, the diagrammatic approach, as depicted in Fig. 5, is more intuitive, however there are more TDA amplitudes than the corresponding diagrams. For an octet baryon in the final state, there are three light quarks. In the same diagram, the symmetry of the quarks in flavor space could be different. For example, in the third diagram of Fig. 5, the u quark and another light quark could be flavor anti-symmetric, or the two unspecified quarks could be flavor anti-symmetric. These different combinations will lead to different TDA amplitudes. Thus it is very hard to determine the independent amplitudes in this approach, which will introduce subtleties to the global fit in the diagrammatic approach. The mismatch between Feynman diagrams and TDA amplitudes will not happen for the decays into decuplet baryons, since all three quarks are symmetric in flavor space. • 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 Table 13, which completely fits with the results given by Ref. [73].
• The flavor SU(3) symmetry in charmed baryon decays and the symmetry breaking effects have been extensively explored in Refs. [70,[83][84][85][86][87][88][89][90], and we refer the reader to these references for detailed discussions. • On the experimental side, BESIII collaboration has given the first measurement of decay branching fractions for the W -exchange induced decays [91]: It indicates that the decays into a decuplet baryon might not be power suppressed compared to those decays into an octet baryon. This introduces a theoretical difficulty to understand the charmed baryon decays. • One can find some relations between the different channels listed in Tables 14, 15 and 16. For the charmed baryon two-body decay, there is only one relation for decay width:

Fig. 6
Topology diagrams for the charmed baryon decays into an octet baryon and a light meson. Due to the same reason as that of bottom baryon decays, one topological diagrams correspond to more than one TDA amplitudes. Here the 7 topological diagrams correspond to 19 TDA amplitudes as given in Eq. (80) This relation fits well with the data in Ref. [3]: • In Ref. [70], a global fit was conducted for charmed baryon decays, particularly inspired by the recent BESIII data [71,72,91]. In that work the sextet contribution was expressed in a different representation. Relating the four coefficients in [70] with our notations, we have: 3 −C T 6 − D T 6 = a 2 = (0.115 ± 0.014) GeV 3 , 3 In Ref. [70], these parameters are denoted as a 1 , a 2 , a 3 , h. Here we add primes in order to distinguish them with the parameters used in this work.
Such a fit was conducted with the neglect of the H 15 terms, which might be challenged in interpreting the c → pπ 0 [84,87,89].

Discussions and conclusions
In this work, we have carried out a comprehensive analysis comparing two different realizations of the flavor SU(3) symmetry, the irreducible operator representation amplitude and topological diagram amplitude, to study various bottom/charm meson and baryon decays. We find that previous analyses in the literature using these two methods do not match consistently in several ways. The TDA approach provides a more intuitive understanding of the decays, however it also suffers from a few subtleties. Using two-body B/D meson decays, we have demonstrated that a few SU(3) independent amplitudes (the last six diagrams in Fig. 1) are sometimes overlooked in TDA (for instance Refs. [16,18]). 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. In addition, using B and D decays we have found that one of the A, C, E, T amplitudes should be absorbed into others, which has been pointed out in Ref. [20]. 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 [92][93][94], and multi-body c decays [90]. The equivalence between the TDA and IRA approaches in these decay modes can be studied similarly. No. 15DZ2272100. X.G. He was also supported in part by MOST (Grant Nos. MOST104-2112-M-002-015-MY3 and 106-2112-M-002-003-MY3).

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: There is no associated data for our paper.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .

Appendix A: Relations for bottom antitriplet baryon decay amplitudes
In Sect. 5, we have argued that due to different forms to represent the hadron SU(3) multiplet, the IRA amplitudes can be constructed in different ways. In Eq. 75, the antitriplet baryons and octet baryons are used as (T b3 ) i and (T 8 ) i j . Instead it is possible to use another set of forms, (T b3 ) i j and (T 8 ) i jk , which gives the IRA amplitudes with 26 terms: The relation between the two sets of baryon SU(3) representation is: These two sets of IRA are related to each other by: , E T 6 = 2B 1 + 2B 4 +B 2 −B 3 +B 9 +B 10 −B 11 , E T 15 = 2C 1 +C 2 +C 3 +C 6 +C 7 +C 8 . (A3) In addition, the relation between the coefficients of 26 amplitudes in A I R A T b →PT 8 (u) and A T D A T b →PT 8 (u) is: (−ā 1 + 3ā 2 − 3ā 3 +ā 5 ) +b 1 , The relation between the two sets of IRA is given as:  can be obtained as: