A self-consistent framework of topological amplitude and its $SU(N)$ decomposition

We propose a systematic theoretical framework for the topological amplitudes of the heavy meson decays and their $SU(N)$ decomposition. In the framework, the topological amplitudes are expressed in invariant tensors and classified into tree- and penguin-operator-induced diagrams according to which four-quark operators, tree or penguin, being inserted into their effective weak vertexes. By decomposing the four-quark operators into irreducible representations of $SU(N)$ group, one can derive the $SU(N)$ irreducible amplitudes from the tensor form of the topology. Taking the $D\to PP$ decay ($P$ denoting a pseudoscalar meson) with $SU(3)_F$ symmetry as an example, we show our framework in detail. The fact that some topologies are not independent in the $SU(3)_F$ limit is explained by group theory. It is found that there are only nine independent topologies in all tree- and penguin-operator-induced diagrams contributing to the $D\to PP$ decays in the Standard Model. If a large quark-loop diagram is assumed, the large $\Delta A_{CP}$ and the very different $D^0\to K^+K^-$ and $D^0\to \pi^+\pi^-$ branching fractions can be explained with a normal $U$-spin breaking. Moreover, our framework provides a simple and systematic way to analyze the $SU(N)$ breaking effects. As examples, the linear $SU(3)_F$ breaking and the high order $U$-spin breaking in charm decays are re-investigated in our framework, which are consistent with literature. We propose the concepts of splitting and degeneracy of topologies, and use them to describe the charm-less bottom decay. We find $SU(3)_F$ analysis for the charm-less $B$ decays is different from the $D$ decays because the charm-quark loop is beyond the $SU(3)$ symmetry and should be investigated in the symmetry breaking chain of $SU(4)\to SU(3)$.


I. INTRODUCTION
Heavy quark non-leptonic decays provide an ideal platform to test the Standard Model (SM) and search for new physics. A tremendous amount of data on the heavy hadron (especially B/D meson) decays have been collected by experiments in the last few decades [1]. In particular, the LHCb Collaboration observed the CP violation in the charm sector with 5.3σ in 2019 [2]. It is a milestone of heavy flavor physics since it fills the last piece of the puzzle of the Kobayashi-Maskawa (KM) mechanism [3,4]. In theory, several QCD-inspired approaches are established to calculate the non-leptonic B meson decays, such as QCD factorization (QCDF) [5][6][7][8], perturbative QCD approach (PQCD) [9][10][11][12], and soft-collinear effective theory (SCET) [13,14]. However, the QCDinspired approaches do not work well in the D meson decays because the expansion parameters α s (m c ) and Λ QCD /m c are bigger than the ones in the B meson decays. It remains to be seen if the heavy quark expansion (HQE) can be applied to the charm sector [15,16].
An alternative way to investigate the heavy meson decays is the flavor symmetry analysis. This method bypasses form the dynamic details, widely used in studying charm/bottom meson , baryon  and even stable tetraquark [83][84][85][86] decays. There are two popular approaches based on the flavor symmetry. One is the topological diagram amplitude (TDA)  approach, in which the topological diagrams are classified according to the topologies in the flavor flow of weak decay diagrams, with all strong interaction effects induced implicitly. It is intuitive and helpful for understanding the internal dynamics of hadron decays, providing a framework in which we cannot only do the model-dependent data analysis but also make evaluations of theoretical model calculations. The other method is the SU (3) irreducible representation amplitude (IRA) [47][48][49][50][51][52][53] approach which is blind to the dynamic mechanics. The SU (3) irreducible representation amplitudes are expressed in the tensor form [47] and the Wigner-Eckhart theorem [87,88] ensures that there is one constant for each invariant tensor. Both the TDA and IRA approaches can include the flavor symmetry breaking effects. The first order flavor SU (3) breaking has been analyzed in the irreducible amplitude [52] and the topological amplitude [89,90] approaches.
The TDA and IRA approaches seem to be equivalent in the SU (3) F limit. The equivalence between them was discussed as early as in 1980 [22][23][24], and followed by other literature such as Refs. [91,92]. But in some literature, such as Refs. [27,91], the relation between topological amplitudes and irreducible amplitudes is extracted by expanding the decay amplitudes of some channels in the TDA and IRA approaches and comparing them. The equivalence of TDA and IRA is used like a priori presumption rather than a derived conclusion. An instructive attempt to solve the relation between the topological amplitude and the SU (3) irreducible amplitude was done in Refs. [93,94]. In their work, a bridge between topological diagrams and invariant tensors constructed by the four-quark operators and the initial and final states was built. It shows that the difference between the SU (3) irreducible amplitude and the topological amplitude is whether the four-quark operators are decomposed into the SU (3) irreducible representations or not, and the equivalence relation between them can be derived. However, there are some mistakes in the SU (3) decomposition, resulting in an incorrect relation between topological and irreducible amplitudes [93,94]. Their treatment of charm-quark loop and classification of topologies in the B meson decays are ambiguous. They did not explain why not all topologies are independent. A complete and self-consistent framework of the topologies and their SU (N ) decomposition for the heavy hadron decays has not been established yet.
The goal of this work is to propose a systematic theoretical framework for the topological amplitudes of heavy meson decays. For this purpose, a one-to-one mapping between the topological diagram and the invariant tensor is set up. Then some mathematical techniques can be introduced to study the topological amplitudes. For example, the number of possible topologies contributing to one type of decay is counted in permutations and combinations. And the SU (N ) irreducible amplitudes are derived from the tensor form of the topologies by decomposing the four-quark operators into irreducible representations. Taking the D → P P decay as an example, we will show our framework in detail. And Fig. 1 is a sketch.
An attractive achievement of our framework is the linear correlation of the topological amplitudes. In some of earlier literature, such as Refs. [22][23][24], it has been noticed that one of the topologies in the D and B meson decays is not independent in the SU (3) F limit. This conclusion still holds when the diagrams with quark loop are included [93,94]. Moreover, Ref. [89] pointed out that matrix linking T , C, E and A diagrams to the physical amplitudes has only rank three in the case of only the D → P P modes without η ( ) being analyzed. But it is no longer correct when η and η are taken into account [33,95]. In this work, it is found that above conclusions can be be explained coherently in group theory, in which some specialities of SU (3) group play a crucial role.
In order to match the tensor form of topology, we suggest to classify the topologies in the Standard Model into tree-and penguin-operator-induced diagrams according to which operators, tree or penguin, being inserted into the effective vertexes, no matter whether the topologies involving quark loop or not. It is found that once the tree-operator-induced amplitudes are given, the penguin-operator-induced amplitudes are completely determined. There are nine independent treeoperator-induced diagrams contributing to the D → P P decays. Five of them, namely T , C, E, A and T LP , are not suppressed by the hard gluon exchanges. If we assume the quark-loop diagram T LP is comparable to other four diagrams, the large ∆A CP and the very different D 0 → K + K − and D 0 → π + π − branching fractions can be explained together with a normal U -spin breaking.
Analogous to the charm meson decays, we deduce that a sizeable CP violation might exist in the Ξ + c → pK − π + mode. In addition, our framework provides a simple and systematic way to analyze the SU (N ) F breaking effects. The linear SU (3) F breaking [89] and the high-order U -spin breaking [96,97] in charm decays are re-investigated in the tensor form of topology, which are consistent with the original literature. Analogous to the degeneracy and splitting of energy levels, we propose the concepts of degeneracy and splitting of topological diagrams. As an application, we analyze the charmless B/strange-less D decays in the SU (4) F /SU (3) F symmetry breaking into SU (3) F /SU (2) F symmetry.
The rest of this paper is organized as follows. In Sec. II, we introduce our theoretical framework with a model-independent analysis of the D → P P decays. The linear correlation of topologies under SU (3) symmetry will be clarified. In Sec. III, we shall discuss the topologies in the Standard Model and analyze the observed CP violation. In Sec. IV, we will generalize our framework to the broken flavor symmetry. And Sec. V is a short summary. Besides, the topological and irreducible amplitudes in the D → P V decays will be presented in Appendix A. And the SU (3) decomposition of operators in the b-quark decays and the decay amplitudes of the charmless B → P V modes will be discussed in Appendixes B and C respectively.

II. MODEL-INDEPENDENT ANALYSIS
In this section, we study the topological amplitudes and the SU (3) irreducible amplitudes model-independently, taking the D → P P decay as an example.

A. Topological amplitude
The weak Hamiltonian of charm decay in a general effective theory can be written as in which is labeled by (H (p) ) ij k , the effective Hamiltonian of charm decay can be written as In above notation, (H (p) ) is a 3 × 3 × 3 complex matrix and (H (p) ) ij k is a component of matrix. In the rest of paper, we will call (H (p) ) ij k as the "CKM component". To illustrate above convention more clearly, we re-write the effective Hamiltonian of charm decay in the SM into the form of Eq. (3). The effective Hamiltonian of charm decay in the SM is written as [98]: where the tree operators are with α, β being color indices, and q 1,2 being the d or s quark. The QCD penguin operators are The chromomagnetic penguin operator is The magnetic-penguin contributions can be included into the Wilson coefficients for the penguin operators following the substitutions [5][6][7][8] C 3,5 (µ) → C 3,5 (µ) + αs(µ) 8πNc 2m 2 c l 2 C eff 8g (µ), C 4,6 (µ) → C 4,6 (µ) − αs(µ) 8π 2m 2 c l 2 C eff 8g (µ), with the effective Wilson coefficient C eff 8g = C 8g + C 5 and l 2 being the averaged invariant mass squared of the virtual gluon emitted from the magnetic penguin operator.
In the notation (3), the tree and penguin operators can be written as, The corresponding CKM components of operators O and the other (H (0,1) ) ij k are zero. We use the general effective Hamiltonian Eq. (3) to construct the model-independent amplitude of the D → P P decay. To achieve this goal, an algebraic tool, tensor analysis, is needed. According to Ref. [99], an arbitrary state in the tensor product space can be written as Tensor v is a "wave-function", because one can get tensor component v j 1 ...jn i 1 ...im by taking the matrix element of |v with the tensor product state, Applying to physics, a light pseudoscalar meson state is expressed as in which |P i j is the quark composition of meson state, |P i j = |q iqj . (P α ) j i is the coefficient of the quark composition |P i j . In the SU (3) picture, pseudoscalar meson notet |P i j is expressed as where i is row index and j is column index. According to Eq. (14), one can derive The bar state of Eq. (13) is Since (P α ) j i is a real number, (P α ) j i = (P α ) j i . A charmed meson state is expressed as and The decay amplitude of D γ → P α P β can be constructed to be in which Per. present summing over all the possible full contractions of P m n P r s |O (p)l jk |D i . Under the flavor symmetry, decay amplitude is a complex number without flavor indices, i.e., a SU (N ) invariant. Then P m n P r s |O (p)l jk |D i is a invariant tensor in which all the indices either contract with each other [99]. Once the contraction of P m n P r s |O (D γ ) i (H (p) ) jk l (P α ) n m (P β ) s r is determined too, vice versa. For more simplicity, the decay amplitude of D γ → P α P β in p-order is expressed as [52] where ω labels the different contractions of the SU (3) indices. X (p) ω is the reduced matrix element According to the Wigner-Eckhart theorem [87,88], X (p) ω is independent of decay channels, i.e., indices α, β and γ. All the information of initial/final states is absorbed into the Clebsch-Gordan coefficient If the index-contraction is understood as quark flowing, the reduced matrix element X (p) ω is a topological amplitude. The contraction maps the topological diagram by following rules.
• The contraction between the final-state meson P and the four-quark operator indicates that the quark or anti-quark produced in one effective vertex of operator O (p)k ij enters the final state-meson P .
• The contraction between the initial-state D meson and the four-quark operator indicates that the light anti-quark in D meson annihilates in the vertex of four-quark operator.
• The contraction between the initial-state D meson and the final-state meson P indicates that the light anti-quark in D meson, as a spectator quark, enters the final state meson P .
• The contraction between two indices of the four-quark operator presents the quark loop. O According to these rules, one can set up a one-to-one mapping between the topological diagram and the invariant tensor. For example, the reduced matrix element P j i P l k |O T diagram in literature. The schematic description of index-contraction in T diagram is shown in There are four upper/lower indices in P m n P r s |O (p)l jk |D i . The number of all possible contractions is N = A 4 4 = 24. Considering that we cannot distinguish the two pseudoscalar nonet in the D → P P decay, some repeated count should be subtracted and then N = A 4 4 − 2 × A 3 3 + 2 = 14. Amplitude of the D → P P decay can be written as Each term and the corresponding interchange α ↔ β in Eq. (22) present one topological amplitude and the coefficient is calculated by the product of (D γ ) i (H) jk l (P α ) n m (P β ) s r . In the case of P α = P β , the decay amplitude A TDA (Dγ →PαPα) has to time 1/ √ 2 due to the symmetry factor of 1/2 appearing in the decay rate. As we have mentioned above, it is the Wigner-Eckhart theorem [87,88] ensures that topological amplitude is independent of initial and final states. In Eq. (22), we do not write the order of perturbation of four-quark operators explicitly. But notice that the same contractions with different p present different topological amplitudes. For example, p = 0 denotes the diagrams induced by tree operators in the SM, while p = 1 denotes the diagrams induced by penguin operators. In fact, perturbation order p provides a natural way to classify the topologies. We will discuss this question in detail in Sec. III.
The topological diagrams contributing to the D → P P decays are showed in Fig. 3. The first four diagrams, T , C, E and A, have been analyzed in plenty of literature. T ES and T AS are the singlet contributions which requires multi-gluon exchanges. The last eight diagrams are quark-loop contributions. We call them T X because that tree operators can be inserted into their effective vertexes. In principle, all the topological diagrams listed in Fig. 3 should contribute to the D → P P decays. But some diagrams always disappear when some operators are inserted into their effective vertexes. If the operator with three same indices (for instance (uu)(uc)) is inserted, all the 14 diagrams in Fig. 3 contribute to the D → P P decays. If the operator with two same indices (for instance (ud)(dc)) is inserted, only the first ten diagrams in Fig. 3 contribute. If the operator without same indices (for instance (ud)(sc)) is inserted, only the first six diagrams contribute.
That is why topologies T QP , T QC , T QA and T QS always disappear when tree operators in the SM are inserted. But T QP , T QC , T QA and T QS are necessary to derive the correct relation of the topological diagram amplitudes and the SU (3) irreducible amplitudes, see II B. In [93,94], the last four topological diagrams in Fig. 3 are overlooked, which results in an incorrect relation.

B. SU (3) irreducible amplitude
Operator O k ij defined in Eq. (2) can be regarded as a (2, 1)-rank tensor representation of SU (N ) group. There are two covariant (lower) indices and one contravariant (upper) index in O k ij . Indices i and j transform according to the foundational representation N of SU (N ) group, and index k transforms according to the complex conjugate representation N [99]. Now let us discuss how to decompose the general (2, 1)-rank tensor T k ij into the irreducible representations of SU (N ) group. Firstly, we study a simple tensor, T ij with two covariant indices i and j. T ij is decomposed as N ⊗ N . In group theory, the foundational representation of SU (N ) group can be expressed as one square in young's tableaux. The decomposition of N ⊗ N is in which presents symmetrization of indices i, j and presents anti-symmetrization of indices i, j. The number of possible combination of antisymmetric i, j is C 2 N = N (N − 1)/2. And the number of possible combination of symmetric i, j is N 2 − C 2 N = N 2 − N (N − 1)/2 = N (N + 1)/2. Thereby, the first term in Eq. (23) presents a N (N + 1)/2 representation and the second term presents a N (N − 1)/2 representation of SU (N ) group. If N = 3, we have 3 ⊗ 3 = 6 ⊕ 3. Secondly, we analyze another simple tensor, T j i with one covariant index i and one contravariant index j. In group theory, a mixed tensor with contraction of one covariant index and one contravariant index is known as trace tensor. For any mixed tensor, it can be decomposed into trace tensor and traceless tensor. Both the subspaces composed by trace tensor and traceless tensor are invariant subspaces of SU (N ) group. So T j i can be decomposed as The trace tensor is a trivial representation and the traceless tensor is ( There are two steps to decompose T k ij into the direct sum of irreducible representations: extracting the trace tensors and symmetrizing/anti-symmetrizing indices of the remaining traceless tensor. The result is in which the two trace tensors are dimension-N since only one free index left. T k {ij} and T k [ij] are traceless with the dimensions of ( If operator O k ij is the representation of SU (3) group, it can be decomposed as 3 ⊗ 3 ⊗ 3 = 3 p ⊕ 3 t ⊕ 6 ⊕ 15. The explicit decomposition is [52] which is consistent with Eq. (25). The coefficients 1/4 and 1/8 before 6-and 15-dimensional representations are used to match most literature. All irreducible presentations are listed following.
3 p presentation: 3 t presentation: There are nine operators in irreducible representation 6, but only six of them are independent If operator O k ij is decomposed into irreducible representations, the CKM component (H) ij k should be decomposed correspondingly: To obtain the SU (3) irreducible amplitude of the D → P P decay, one can contract all indices in the following manner Eq. (37) declares the equivalence between the topological amplitudes and SU (3) irreducible amplitudes.
From above discussions, one can find the sole difference between the TDA and IRA approaches is whether the four-quark operators (or equivalent, (H) ij k ) are decomposed into the SU (3) irreducible representations or not. In Sec. III, the topological and SU (3) irreducible amplitudes of the D → P P 1 Taking T diagram as an example, contributing to ) decays in the SM will be presented to verify Eq. (37). The equivalence of TDA and IRA approaches is also verified in the D → P V decays, see Appendix A for details.
The SU (3) decomposition of tensor operator O k ij can be generalized to non-leptonic b decays. Since the sole difference of O k ij in b decay and charm decay is the heavy quark, the decomposition should be the same. The decomposition of b decay is discussed in Appendix B, in which some mistakes in Refs. [93,94] are cleared.

C. Linear correlation of topologies
In this subsection, we discuss the linear correlation of topologies. Because of (H(6)) ij k = ijl (H(6)) lk and the symmetric lower indices in (H(6)) lk , the terms constructed by (H(6)) ij k in Eq. (34) can be written as in which the equations are used. The last term in Eq. (40), c 6 (D γ ) [jl] (H(6)) ki (P α ) i j (P β ) k l , cancels with its α-β interchanging term c 6 (D γ ) [jl] (H(6)) ki (P β ) i j (P α ) k l because the indices j, l are antisymmetric and indices k, i are symmetric. Thereby, there are only two SU (3) irreducible amplitudes associated with 6 representation contributing to the D → P P decays. According to Eqs. (38)∼(40), parameter c 6 can be absorbed into a 6 and b 6 by following redefinition: This redefinition is not sole. One can also get rid of a 6 or b 6 via the redefinition of or Since the topological amplitudes are equivalent to the SU (3) irreducible amplitudes, one of the topological amplitudes in the D → P P decays is not independent.
From above analysis, it is found the fact that one of the topological diagrams is not independent is only associated with 6 representation. According to Eq.
and hence c 6 can be absorbed into a 6 via a 6 = a 6 − c 6 . Thereby, there are three independent parameters corresponding to the irreducible representations 15 and 6 if η 1 is not included.
Let us look at the prerequisites of Eqs. (38)∼ (40). In the SU (3) irreducible amplitudes, indices of (H) ij k transform according to SU (3) group. The decomposition of 3 ⊗ 3 ⊗ 3 is written as As mentioned above, the first step of Eq. The topological amplitude of K → ππ decay with Isospin symmetry is the same with the one in Eq. (22) (except for those diagrams involving singlet), The Notice that there is no irreducible representation of (H) ij k in the decomposition of 2⊗2⊗2 = 2⊕2⊕4 corresponding to the 6 representation in the decomposition of 3 The SU (2) irreducible amplitude of K → ππ decay can be constructed by replacing 3 p , 3 t and 15 in Eq. (34) with 2 p , 2 t and 4 respectively, Notice that there are only six SU (2) irreducible amplitudes in Eq. (50). Thereby, two of the topologies are not independent in the K → ππ decays. According to Eq. (49), the relations between topological diagrams and the irreducible amplitudes in the K → ππ decays are The covariant index and contravariant indices of the tensor representation of SU (N ) group can transform to each other via the completely antisymmetric tensor (2, 1)-rank mixed tensor can be written as a tensor only containing upper indices via To intuitively understand the difference of the SU (2), SU (3) and SU (4) groups, we compare the Young's tableaux of decomposition (26) in the cases of N = 2, 3 and 4:  In summary, the linear correlation of topologies depends on the symmetry of the physical system.
For different symmetry group, the linear correlation of topologies is different, and can be explained in group theory.

III. TOPOLOGIES OF D → P P DECAYS IN THE SM
In this section, we present the amplitude decompositions of the D → P P decays in the Standard Model and discuss the applications.

A. Topologies in the SM: classification and linear correlation
According to Eq. (4) and Eq. (10), the CKM components in the SM, (H (p) ) ij k , can be obtained from the map (ūq 1 )(q 2 c) → V * cq 2 V uq 1 in current-current operators and (qq)(ūc) → −V * cb V ub in penguin operators and the others are set to be zero. The non-zero CKM components induced by tree operators in the topological amplitude include The non-zero CKM components induced by penguin operators in the topological amplitude include The superscripts (0) and (1) differentiate tree and penguin contributions. The non-zero CKM components induced by the tree operators in the SU (3) irreducible representations are The non-zero CKM components induced by the penguin operators in the SU (3) irreducible representations are In general, the topologies in the SM are classified into two types: tree diagram and penguin diagram. The quark-loop contributions induced by tree operators are absorbed into the Wilson coefficients of penguin operators [5][6][7][8], with the averaged invariant mass squared of the virtual gluon emitted from the quark loop l 2 and the function The penguin operator induced quark loop contributions are negligible. Thereby, the penguin diagrams include those diagram induced by penguin operators without quark-loop and quarkloop diagrams induced by tree operators. But this classification is not convenient to analyze the topologies in the tensor.
We suggest to put the traditional tree diagrams and the quark-loop diagrams induced by tree operators (O 1,2 ) together, named after "tree-operator-induced diagrams", and put the diagrams induced by penguin operators (O 3−6 ) together, named after "penguin-operator-induced diagrams".
The new classification of topologies is according to which operators (O 1,2 or O 3−6 ) being inserted into the diagrams, indifferent to the diagrams with quark-loop or not. Because the Wilson coefficients C 1,2 are larger than C 3−6 , the new classification is also based on magnitude of the Wilson coefficients, or in other word, the perturbation order p introduced in Eq. (1). The advantage of the new classification is that it is convenient to build the one-to-one mapping between topology and invariant tensor, just like we have done in Sec. II. And then some mathematical techniques, such as group theory, can be induced to study the topological amplitudes.
Revisit the topological diagrams listed in Fig. 3. There are ten tree-operator-induced diagrams contributing to the D → P P decays in the SM. The diagrams T QP , T QC , T QA , T QS vanish because there is no tree level FCNC transition in the SM. But for penguin-operator-induced diagrams, all the fourteen topologies contribute to the D → P P decays. In the rest of paper, notations P T , P C, P E, P A, P ES , P AS , P LP , P LC , P LA , P LS , P QP , P QC , P QA and P QS are used to label the penguin-operator-induced amplitudes corresponding the topologies in Fig. (3) orderly. And all the SU (3) irreducible amplitudes induced by penguin operators are added P before their notations to differentiate the amplitudes induced by tree operators.
In order to compare with other literature, we give the relation between our classification of the topologies and the traditional one (see Ref. [100] for example). The tree-operator-induced diagrams without quark loop defined in this work (left) and the tree diagrams defined in Ref. [100] (right) have one-to-one correspondence: T this work = T Ref. [100] , C this work = C Ref. [100] , E this work = E Ref. [100] , A this work = A Ref. [100] , T ES this work = SE Ref. [100] , T AS this work = SA Ref. [100] .
The tree-operator-induced diagrams with quark-loop in this work are the quark-loop contributions of the penguin diagrams in Ref. [100]. The relations between them are in which the subscript "loop" is used to distinguish them from the contributions induced by penguin operators. The penguin-operator-induced diagrams without quark loop in this work are the penguin diagrams proportional to C 3−6 in Ref. [100]. The relations between them are P C this work = (P pen ) Ref. [100] , P E this work = (P E pen ) Ref. [100] , P A this work = (P A pen ) Ref. [100] , P T this work = (S pen ) Ref. [100] , P ES this work = (SP E pen ) Ref. [100] , P AS this work = (SP A pen ) Ref. [100] , in which the subscript "pen" represent the O 3−6 contributions. The topologies defined in Ref. [100], P and P E, as well as S and SP E, always appear as P +P E and S +SP E. It is easy to understand since our definition of topology only consider quark line flowing into and out of hadrons but does not care the gluon exchanges. The only difference between P and P E (as well as S and SP E), actually, is the different gluon exchanges.
The linear correlation of topological diagrams in the SM is beyond the model-independent analysis in II C because some characters of the effective Hamiltonian of the SM. In the IRA approach, Eq. (57) shows that there are no penguin-operator-induced amplitudes in the 15-and 6-dimensional irreducible representations. From Eqs. (56) and (57), it is found the non-zero CKM only contain the first components. Because of the unitarity of the CKM matrix, we have can be absorbed into four parameters with following redefinition: Notice that Eq. (63) only holds in the Standard Model but not a general conclusion. According to Eq. (63) and Eq. (37), the tree-operator-induced diagrams with quark loop and all the penguinoperator-induced diagrams can be absorbed into four parameters with following redefinition: All the penguin-operator-induced amplitudes are determined if the tree-operator-induced amplitudes with quark loop are known. There is no degree of freedom of the penguin-operator-induced diagrams in the SM. It is understandable since a penguin operator can be regard as a "quark-loop" induced by "tree operator" in high energy scale. According to II C, one of the ten tree-operatorinduced topological diagrams of the D → P P decays is not independent. Thereby, there are only nine degrees of freedom for all the tree-and penguin-operator-induced amplitudes in the D → P P decays in the SM.  (63) and (64) are used for simplification.
The tree-and penguin-operator-induced amplitudes of the D → P P decays in the TDA and IRA approaches are listed in Tables. I and II According to this result, Ref. [102] proposed a ∆U = 0 rule in the charm physics: the ratio of ∆U = 0 over ∆U = 1 amplitudes is For the ∆U = 0 rule, there are two different arguments: it arises from new physics [103,104], or non-perturbative QCD enhancement [105][106][107][108]. On the other hand, a long-standing puzzle in charm decays is the very different D 0 → K + K − and D 0 → π + π − decay rates. In general, the SU (3) breaking is expected to be around 30%. For example, amplitude T of the D decaying into KK and ππ in the factorization approach has the expressions as and T KK /T ππ ≈ 1.3 [37]. Such a SU (3) breaking is not enough to explain the branching fractions and In the following, we will show that assuming a large quark-loop diagram T LP could be a better choice to solve the puzzles of large ∆a dir CP and the very different branching fractions in the D 0 → K + K − and D 0 → π + π − decays simultaneously.
In the SU (3) F limit, the amplitudes of D 0 → K + K − and D 0 → π + π − decays are in which the penguin-operator-induced amplitudes with quark-loop are neglected. Considering the U -spin breaking, the amplitude of D 0 → K + K − decay can be written as where Similarly, the amplitude of D 0 → π + π − decay can be written as In the effective Hamiltonian (4), the Wilson coefficients C 3−6 are much smaller than C 1,2 [98].
The penguin-operator-induced amplitudes are smaller than the tree-operator-induced ones. On the other hand, topology T LA is suppressed by the OZI rule [109][110][111]. Thereby, we have following pattern about topologies in the D 0 → K + K − and D 0 → π + π − decays: Then the decay amplitudes of D 0 → K + K − and D 0 → π + π − are simplified to be where subscript d of T LP is removed for convenience.
Considering the strong phases, the situation will be more complicated. But it does not affect the order estimation. If we assume a normal U -spin breaking in T LP diagram: we get which is consistent with the value extracted from the CP violation in charm given in Eq. (66).
Thereby, the KK −ππ puzzle and the large CP violation in charm can be explained simultaneously if a large T LP diagram is assumed. The similar idea was proposed in Refs. [35,37,89,90,112]. But the measured CP violation in several years ago was too large [113][114][115] and hence the reliability was questioned.
D meson decay is dominated by topologies T , C, E, A. In the other diagrams listed in Fig. 3, only T LP cannot be separated into two disconnected parts by removing the internal gluon lines and does not suppressed by the OZI rule [109][110][111]. For the other diagrams, they need the hard gluon exchanges to emit a color singlet, or connect initial and final states. It is plausible that T LP is enhanced by strong non-perturbative final-state interaction, such as re-scattering and resonance.
The authors of Ref. [106] and Ref. [108] argued that which topology, P or P E (P is called P C in Ref. [106]), leads to the large CP violation in charm. But since P and P E always appear as P + P E and T LP include main contributions of P and P E, assuming a large T LP diagram does not conflict to both Ref. [106] and Ref. [108]. On the other hand, we cannot rule out the possibility that the large T LP arises from new physics.
Similarly to the D 0 → K + K − and D 0 → π + π − decays, T LP break can be used to explain the branching fraction differences of other D → P V modes, such as D 0 → π − ρ + and D 0 → K − K * + , The amplitudes of the D → P V decays are listed in Appendix A. In Refs. [40,41], a Glauber strong phase associated with π [116][117][118] is introduced to solve KK − ππ puzzle. To test which effect, T LP break or Glauber phase, is the dominate source of U -spin breaking, we suggest to measure the branching fractions of D + → K 0 S K * + and D + s → K 0 S ρ + , since there is no π meson in the final states. The factorizationassistant topological amplitude approach [41] predicts the branching fractions of D + → K 0 S K * + and D + s → K 0 S ρ + are approximately equal. The amplitudes of D + → K 0 S K * + and D + s → K 0 S ρ + can be written as Analogy to D 0 → K + K − and D 0 → π + π − , the difference of Br(D + → K 0 S K * + ) and Br(D + s → K 0 S ρ + ) might be large. If the ratio of Br(D + → K 0 S K * + ) and Br(D + s → K 0 S ρ + ) is beyond the normal SU (3) breaking, it might be an evidence of a large T LP P, break . The branching fraction of D + → K 0 S K * + is poorly measured so far [1]: And the branching fraction of D + s → K 0 S ρ + has not been measured yet. The precise measurements of Br(D + → K 0 S K * + ) and Br(D + s → K 0 S ρ + ) are desirable. Above discussion can be generalized into the charmed baryon decay modes Λ + c → Σ + K * 0 and Ξ + c → pK * 0 . In Ref. [74], we find that if two singly Cabibbo-suppressed decay modes of charmed hadrons are associated by a complete interchange of d and s quarks, their decay amplitudes are connected by a complete interchange of λ d and λ s in the U -spin limit. As a consequence, the tree-operator-induced amplitudes of Λ + c → Σ + K * 0 and Ξ + c → pK * 0 under the U -spin symmetry can be parameterized to be in which T A , T B and T L are not the specific topological amplitudes but the sum of the topological amplitudes proportional to λ d , λ s and (λ d + λ s ) respectively. Neglecting the small quark-loop contributions proportional to λ b , we have However, the experimental data of branching fractions [1,78], show that the ratio between the decay amplitudes A(Λ + c → Σ + K * 0 ) and A(Ξ + c → pK * 0 ) is Such a ratio, at least its central value, is larger than |A(D 0 → K + K − )/A(D 0 → π + π − )| ≈ 1.67.
Considering the U -spin breaking, the tree-operator-induced amplitudes of Λ + c → Σ + K * 0 and Ξ + c → pK * 0 are Just like the D 0 → K + K − and D 0 → π + π − modes, we can introduce a large T L break to explain the large ratio in Eq. (86).
So we predict that CP violation in the Ξ + c → pK − π + mode can reach to be O(10 −3 ). Since all the final-state particles in the Ξ + c → pK − π + decay are preferable in experiments, it is a promising mode to search for CP violation of the charmed baryon decays.

IV. SYMMETRY BREAKING AND SPLITTING OF TOPOLOGIES
Our framework provides a simple way to formulate the flavor symmetry breaking effects. In this section, we will use some examples to illustrate how the flavor symmetry breaking effects are included in the tensor form of topology. In the D → P P decays without η and η mesons, neglecting the penguin-operator-induced amplitudes, only six terms in Eq. (22) left: Considering the first order of H SU (3) F , amplitude of the D → P P decay can be obtained by summing all possible invariant tensors in which index 3 (presenting s quark) is written explicitly: where the flavor symmetric part and other SU (3) F breaking terms of T LP diagram are ignored because they are proportional to V * cd V ud + V * cs V us = −V * cb V ub . Following [89], topology T LA is also neglected in Eq. (90). Comparing Eq. (90) with the Table. II in Ref. [89], one can find the topological amplitudes defined in this work match to the ones defined in Ref. [89] one by one: 1 , The emergence of topologies T 1 , T 2 ... is analogous to the splitting of energy levels. In the flavor SU (3) symmetry, some diagrams, for instance T , T 1 , T 2 , T 3 , are degenerate, When the SU (3) F symmetry breaks into its SU (2) subgroup, the original T diagram splits into four different diagrams.

B. High order U -spin breaking
In this subsection, we study the U -spin symmetry and its breaking, taking D 0 → K − π + , Mesons π − and K − form a U -spin doublet, (P ) u i |P i u . Mesons K + and π + form another U -spin doublet, (P ) i u |P u i . Under the U -spin symmetry, the amplitude of D 0 decay is expressed as These results are consistent with the results in flavor SU (3) symmetry if A = T + E and A L = T LP +2T LA . Considering the approximation of V * cs V ud cos 2 θ C , V * cs V us −V * cd V ud cos θ C sin θ C and V * cd V us − sin 2 θ C , our results are consistent with Eq. (3) in Ref. [96].
By comparing the last two equations of Eq. (96) with Eqs. (72) and (73), one can find 2(ε 1 +ε 2 )A is the difference between T KK + E KK and T ππ + E ππ , and 2ε (1) 3 A is (T LP break + 2T break ). The first order U -spin breaking induced by the s − d spurion mass operator and the linear SU (3) F breaking are equivalent in the U -spin multiplet. If we "translate" the m s s − m d d to m s s , the amplitudes of D 0 → K − π + , D 0 → K + K − , D 0 → π + π − and D 0 → K + π − with the 0th and first order U -spin breaking are 3 ), Compared to the results given in linear SU (3) F breaking [89], the relations of the two methods in The D 0 decay amplitude with the second order U -spin breaking can be constructed as And the corrections to the 3 + 2ε (2) 4 + 2ε (2)

Strange-less charm decay
For the strange-less charm decay, the flavor symmetry is the isospin symmetry. D 0 and D + form an isospin doublet |D i = (|D 0 , |D + ) and π + , π 0 , π − , η q form a quartet To find all the topological amplitudes contributing to the strange-less charm decay, the first step is to find a appropriate assemble of the four-quark operators. One might use O k ij to describe the strange-less charm decay, just like we have done in Sec. II. But O k ij is not enough. O k ij means that all the indices i, j and k transform as the foundational or conjugate representations of SU (2) group. So O k ij cannot contain s-quark loop contributions. To give a complete description to the strange-less charm decay, O s sj and O s js , i.e., the s-quark loop contributions, should be included. In analogy with the linear SU (3) F breaking, the amplitude of the strange-less charm decay, in which index 3 = s is written explicitly, is In Eq. (102) The operators O s js and O s sj are the SU (2) irreducible representations themselves, labeled by 2 and 2 , respectively. The SU (2) irreducible amplitude of the strange-less charm decay is expressed as By substituting Eq. (49) into the amplitudes of T , C, E..., the relations between topological diagrams and the irreducible amplitudes in the strange-less charm decay are derived to be According to Eq. (4), the non-zero CKM components induced by the tree operators in the SM are (H (0) ) 12 The non-zero components induced by the penguin operators are In the SU (2) irreducible amplitudes, the non-zero CKM components induced by the tree operators The non-zero CKM components induced by the penguin operators are  (107) can be absorbed into four parameters with following redefinition: So there are only seven independent parameters in the strange-less charm decay in the SM. The

tree-operator-induced amplitudes with quark loop and all penguin-operator-induced amplitudes in
Eq. (102) can be absorbed into four parameters with following redefinition: As an example of the strange-less charm decays, we write down the decay amplitude of D 0 → π + π − . The SU (2) irreducible amplitude of D 0 → π + π − is The topological amplitude of D 0 → π + π − reads as If the difference between the s-quark loop and u/d-quark loop is neglected, Eq. (116) returns to the result in the SU (3) F symmetry:

Charm-less bottom decay
In the charm-less B decay, the SU (4) F symmetry breaks into the SU (3) F symmetry. Analogy to the strange-less charm decay, the index 4 = c is written explicitly in the amplitude. The B meson The amplitude of the charm-less B decay is constructed by The SU (3) decomposition of O k ij in the B decay is presented in Appendix B. The SU (3) irreducible amplitude of the charm-less B decay is expressed to be By substituting Eq. (B1) into the amplitudes of T , C, E..., the relations between topological diagrams and the SU (3) irreducible amplitudes in the charm-less B decay are derived to be The non-zero CKM components (H (0,1) ) ij k , (H (0,1) ) ci c , (H (0,1) ) ic c in the SM and their SU (3)  presentations can be absorbed into four parameters with following redefinition: The tree-operator-induced topological amplitudes with quark loop and all the penguin-operator-induced topological amplitudes contributing to ∆S = 0 transition can be absorbed into four parameters with following redefinition: For ∆S = −1 transition, λ u,c,t in Eqs. (121) and (122) are replaced by λ u,c,t and λ u = V ub V * us , λ c = V cb V * cs , λ t = V tb V * ts . After the redefinitions, there are ten parameters left in the charm-less B (2a 6 + 10a 15 + 3b 6 + 15b 15 + c 6 + 5c 15 )λu  Tables. III   and IV. As an example of the charm-less B decays, we write down the amplitude of B 0 → π + π − decay.
The topological amplitude of B 0 → π + π − decay is − λ t (P C + P E + 2P A + P LP + 2P LA + 3P QP + 6P QA + P QP c + 2P QA c ). (124) In above formula, the charm-quark loop amplitudes are written explicitly. If the difference between the c-quark loop and u/d/s-quark loop is neglected, Eq. (124) is simplified to the result under the flavor SU (4) symmetry: In [94], the topological diagrams are classified according to CKM matrix element: The c-quark loop is absorbed into A u and A t with unitarity of the CKM matrix V ub V * uq + V cb V * cq + V tb V * tq = 0. A u is called as "tree" amplitude and A t is "penguin" amplitude since A u is dominated by tree contributions but A t is not. This classification of topologies is ambiguous. In our scheme, the topologies is classified according to which operators, tree or penguin, being inserted into the diagrams, do not care the diagram with quark-loop or not. This classification is definite and convenient.
The strange-less charm decay and the charm-less bottom decay are two examples of the degeneracy and splitting of topologies. In the strange-less charm decay, the u, d-quark loops and s-quark loop are degenerate in the flavor SU (3) symmetry. When the SU (3) F symmetry breaks into isospin symmetry, the identical u, d, s-quark loops split into the unequal u, d-quark loops and s-quark loop. The similar situation also exist in the charm-less B decay and the sole difference is that the SU (4) group breaking into SU (3) group. The operator O k ij with its indices transforming according to one symmetry group might not include all the contributions. Operators beyond the given flavor symmetry should be included to get a complete description of topological amplitudes.
And then the corresponding irreducible amplitudes should be modified to match the topological amplitudes. If not, the topological amplitudes and the irreducible amplitudes are not equivalent.
Refs. [93,94] did not introduce O c ci and O c ic to describe the charm-less B decay, which leads to some confusion.

V. CONCLUSION
In this work, we proposed a systematic theoretical framework for the topological and SU (N ) irreducible amplitudes in the two-body non-leptonic heavy meson decays. Some model-independent conclusions are listed following. 6. The linear correlation of topologies depends on the symmetry of the physical system. Applying our framework to the D → P P decays in the Standard Model, we drew some useful conclusions following.
1. The topological diagrams can be classified into tree-and penguin-operator-induced diagrams according to which operators, tree or penguin, being inserted into the effective vertexes, no matter whether the diagrams involving quark-loop or not.
2. Once the tree-operator-induced amplitudes in one decay channel are determined, the penguin-operator-induced amplitudes are determined too.
3. There are ten tree-operator-induced and fourteen penguin-operator-induced diagrams contributing to the D → P P decays in the SM, but only nine of the twenty-four diagrams are independent in the SU (3) F limit.
4. Assuming a large quark loop diagram T LP could explain the large CP violation in charm and the very different branching fractions of the D 0 → K + K − and D 0 → π + π − decays with a normal U -spin breaking.

Ξ +
c → pK − π + might be a promising mode to search for CP violation in the charmed baryon decays.
Our framework can include the flavor SU (N ) breaking effects naturally. Some conclusions are listed following.
1. The linear SU (3) F breaking and the high-order U -spin breaking in the charm decays can be reformulated as tensor form of topology, being consistent with literature.
2. The degeneracy/splitting of topologies in the heavy quark decays is similar to the degeneracy/splitting of energy levels. Our theoretical framework can be generalized into other decay modes, which we leave for future work. In this appendix, we present the topological amplitudes of the D → P V decays. The vector meson nonet is In the D → P V mode, there are N = A 4 4 = 24 possible topological diagrams. Amplitude of the D → P V decay can be written as The topological diagrams in the D → P V decays are showed in Fig. 4. The SU (3) irreducible amplitude of the D → P V decay is Similar to the D → P P decay, the irreducible amplitude c 6 in the D → P V decay can be absorbed into a 6 , b 6 , d 6 , e 6 , f 6 with following redefinition: Similarly to Eqs. (63) and (64), all the penguin-operator-induced amplitudes of the D → P V modes in the SM can be absorbed into six parameters in both IRA and TDA approaches with following redefinitions. IRA: TDA: A LC P = T LC P + P T P + P ES P + P LC P + 3P QC P , A LP P = T LP P + P C P + P E P + P LP P + 3P QP P , The tree-and penguin-operator-induced amplitudes of all the D → P V modes are listed in Ta-bles. V, VI and VII.
in which O(6) k ij = ijl O(6) kl . To compare with literature, we use the convention that index i of O k ij presents quark q i produces in the effective vertex connecting with b quark line, and indices j and k present quark q j and anti-quark q k produce in the other effective vertex. Notice that this convention is different from the convention in the charm decay. All components of the SU (3) irreducible presentation are listed following.    [47,51]. In Ref. [93], the SU (3) decomposition of O k ij is given by However, this general formula is not consistent with Eq. (B6). If we set i = 1, j = 2, k = 1 in Eq. (B8), the coefficient of O(3) is zero but not −1/8 since δ 1 2 = 0. Similarly, Eq. (B8) is not consistent with Eq. (B7) either. Thereby, Eq. (B8) is incorrect. Because of the mistake in SU (3) decomposition, the relation of topological amplitudes and SU (3) irreducible amplitudes in [93,94] is incorrect.
IRA (b → d): e 3 = − λ u λ t e 3 − λ c λ t e 3 + P e 3 + 3P e 3 + P e 3 , TDA (b → d): A LA = − λ u λ t T LA − λ c λ t T LA c + P A P + P A V + P LA + 3P QA + P QA c , λ c λ t T LC P c + P T P + P ES P + P LC P + 3P QC P + P QC P c , λ c λ t T LP P c + P C P + P E P + P LP P + 3P QP P + P QP P c ,  Tables. VIII∼XI.