Flavor SU(3) Topological Diagram and Irreducible Representation Amplitudes for Heavy Meson Charmless Hadronic Decays: Mismatch and Equivalence

Flavor SU(3) analysis of heavy meson ($B$ and $D$) hadronic charmless decays can be formulated in two different ways. One is to construct the SU(3) irreducible representation amplitude (IRA) by decomposing effective Hamiltonian according to the SU(3) transformation properties. The other is to use the topological diagrams (TDA). These two methods should give equivalent physical results in the SU(3) limit. Using $B \to PP$ decays as an example, we point out that previous analyses in the literature using these two methods do not match consistently in several ways, in particular a few SU(3) independent amplitudes have been overlooked in the TDA approach. Taking these new amplitudes into account, we find a consistent description in both schemes. These new amplitudes can affect direct CP asymmetries in some channels significantly. A consequence is that for any charmless hadronic decay of heavy meson, the direct CP symmetry cannot be identically zero.

TDA approach for a heavy meson decaying into two light pseudoscalar SU(3) octet (or U(3) nonet) mesons. In this work, we carry out a systematic analysis and identify possible missing amplitudes in order to establish the consistence between the RRA and TDA approaches. We find that these new amplitudes are sizable and may affect direct CP asymmetries in some channels significantly. An important consequence of the inclusion of these amplitudes is that for any charmless hadronic decay of heavy mesons, the direct CP symmetry cannot be identically zero, though in some cases it is tiny.
The rest of this paper is organized as follows. In Sec. II, we introduce the SU(3) analysis using the TDA and IRA approaches. We summarize those amplitudes already discussed in the literature. In Sec. III, we first point out the mismatch problem, and then identify those missed amplitudes. The complete sets of SU(3) independent amplitudes in both IRA and TDA approaches will be given to establish equivalence of these two approaches. In Sec. IV, we include the missing amplitudes to discuss the implications for hadronic charmless decays of B and D and draw our conclusions.

II.
BASICS FOR IRA AND TDA APPROACHES

A. SU(3) STRUCTURE
We start with the electroweak effective Lagrangian for hadronic charmless B meson decays in the SM. The Hamil- tonian H ef f responsible for such kind of decays at one loop level in electroweak interactions is given by [21][22][23]: where O i is a four-quark operator or a moment type operator. The four-quark operators O i are given as follows: In the above the q denotes a d quark for the b → d transition or an s quark for the b → s transition, while q ′ = u, d, s.
The first term forms an SU(3) octet and the second term is a singlet. Grouping them together it is a nonet of U(3).
It is similar for other light mesons, like the vector or axial-vector mesons.

B. Irreducible Representation Amplitudes
To obtain irreducible representation amplitudes for B → P P (P is an element in M i j ) decays, one takes the various representations in Eq. (3) and uses one B i and light meson M i j to contract all indices in the following manner There also exist the penguin amplitudes A IRA p which can be obtained by the replacements ). Expanding the above A IRA t , one obtains B → P P amplitudes in the first two columns in Tables I and II. Notice that the amplitude A T 6 can be absorbed into B T 6 and C T 6 with the following redefinition: Thus we have 18 (tree and penguin contribute 9 each) SU(3) independent complex amplitudes. Since the phase of one amplitude can be freely chosen, there are 35 independent parameters to describe the two-body B → P P decays.

C. Topological Diagram Amplitudes
The topological diagram amplitudes are obtained by diagrams which connect initial and final states by quark lines as shown in Fig.1 with vertices determined by the operators in Eq. (2). As shown in many references for instance Ref. [20], they are classified as follows: (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) P EW , the electroweak penguin.
In addition, there exists annihilation diagrams, usually abbreviated as A. In Fig. 1, we have only shown the diagrams for tree operators, and those for penguin operators can be derived similarly.
The electroweak penguins contain the color-favored contribution P EW and the color-suppressed one P C EW . The electroweak penguin operators can be re-expressed as: where the second part can be incorporated into the penguins transforming as a3 of SU(3). The contribution from qbūu is similar to tree operators, and thus we will use the symbol P T and P C to denote this electro-weak penguin contribution. Theqb q ′q ′ q ′ is a flavor triplet whose contribution P ′ , as far as flavor SU (3) structure is concerned, can be absorbed into penguin contribution. We can write The three penguin type of amplitudes P , P ′ and P ′ C , can be grouped together. We can redefine P by P + P ′ + P ′ C .
Actually these TDAs can be derived in a similar way as done for IRAs earlier by indicatingquūb (omitting the Lorentz indices ) byH ij k . For ∆S = 0, the non-zero elements areH 12 1 = 1 and for ∆S = −1,H 13 1 = 1. The penguin contribution (including P , P ′ and P ′ C ) is an SU(3) tripletH i withH 2 = 1 for the b → d transition and H 3 for the b → s transition. Eq. (7) implies that the loop induced term proportional to V * tq V tb has bothH ij k andH i . Note that H ij k is no longer traceless. The tree amplitude is given as while the penguin amplitude is given as: Expanding the above equations, we obtain the decay amplitudes for B → P P in the third column in Tables I and II. It is necessary to point out that the singlet contribution in the form M j j requires multi-gluon exchanges. One might naively think that its contributions are small compared with other contributions because more gluons are exchanged.
However, at energy scale of B decays, the strong couplings are not necessarily very small resulting in non-negligible contributions. One should include them for a complete analysis. (a)

III. MISMATCH AND EQUIVALENCE
From previous discussions, one can see that the total decay amplitudes for B → P P decays for IRA and TDA can be written as For the amplitudes given in the previous section, it is clear that for both A i t and A i p , the amplitudes do not have the same number of independent parameters: there are 18 independent complex amplitudes in the IRA, while only 9 amplitudes are included in the TDA. There seems to be a mismatch between the IRA and TDA approaches. However since both approaches are rooted in the same basis, the same physical results should be obtained. It is anticipated that some amplitudes have been missed and must be added.
A close inspection shows that several topological diagrams were not included in the previous TDA analysis. For the tree amplitudes we show the relevant diagrams in Fig. 2. The missing penguin diagrams can be obtained similarly.
Since there are electroweak penguin operator contributions, as far as the SU(3) irreducible components are concerned, the effective Hamiltonian have the same SU(3) structure as the tree contributions. Taking these contributions into account, we have the following topological amplitudes: The mismatch problem can be partly traced to the fact thatH ij k defined in the TDA analysis is not traceless, that is H lj l = 0. Because of this fact, B i and the two M i j can contract withH lj l to form SU(3) invariant amplitudes and also the trace for M i j is not zero when η 1 is included in the final states. While in the previous discussions, these terms are missed.
One can expand the above new terms to obtain the results for tree amplitudes in Tables I and II. With these new amplitudes at hand, one can derive the relation between the two sets of amplitudes: Here we have absorbed the A T 6 into B ′T 6 and C ′T 6 . In the appendix, we give a direct derivation of relations between IRA and TDA amplitudes, in which the amplitude A T 6 is kept. Naively there are total 10 tree amplitudes and 10 penguin amplitudes defined in Eq. (9,12). However, only 9 of the 10 tree amplitudes are independent. Choosing the option to eliminate the W-exchange E, we can express the TDA amplitudes in terms of the IRA ones: The analysis of penguin contributions is similar with the replacement for TDA amplitudes: From the above discussions we see that the two sets of amplitudes in IRA and TDA can be mutually expressed by each other. The IRA and TDA approaches are completely equivalent. As long as all amplitudes are taken into account in the analysis, they give the same results for B → P P decays, and we expect the equivalence for other decays 1 .

IV. DISCUSSIONS AND CONCLUSIONS
We now make a few remarks about our results obtained.
Several missing terms in the TDA analysis involve the trace M j j . The trace actually singles out the singlet in the nonet representation M i j . To have a color singlet in the diagram shown in Figs. 1 and 2, the single M j j need to exchange two or more gluons. As pointed out earlier that these contributions are expected to be small compared with other contributions. However, at energy scale of B decays, the strong couplings are not necessarily very small resulting in non-negligible contributions. Terms associated with the traceH lj l actually can be thought of as turning the tree operator into penguin operator with u quark exchange in the loop whose Wilson coefficient contains the large logarithms ln(µ/m u ) which can also make non-negligible contributions. One should include them for a complete analysis.
Recently, Ref. [14] has performed a fit of B → P P decays in the IRA scheme. Depending on various options to use the data, four cases are considered in Ref. [14]. As an example, we quote the results in their case 4: where the magnitudes and strong phases relative to C P 3 have been given. From Eq. (14), one can see that the C T 3 is a combination of color-allowed tree T , color-suppressed tree amplitude C and others while the B T 15 corresponds to (T ES + T T S) /8 in TDA approach. The fitted result in Eq. (17) indicates that compared to C T 3 , the B T 15 can reach 20% in magnitude, and more importantly, the strong phases are different significantly. The B T 15 , equivalently T ES and T T S , have non-negligible contributions supporting our call for a complete analysis. With more and more accurate data for B → P P from experiments, one can now carry out a more careful analysis to obtain the amplitudes and derive implications for model calculations of the relevant amplitudes.
Without the new contributions in the TDA analysis, some of the amplitudes only have terms proportional to This implies that CP violation in these two decays are identically zero. However, these two decay modes receive contributions from the new terms T P + 2T P A which is multiplied by V * uq V ub . In principle they can have non-zero CP violation. Therefore if one takes into account the missing tree and penguin amplitudes, an important consequence is that no charmless and hadronic B decay channel has a vanishing direct CP asymmetry.
Flavor SU(3) symmetry is an approximate symmetry, and symmetry breaking sources exist in QCD, mostly caused by the unequal masses for the light u, d, s quarks. How SU(3) breaking effect manifest itself is not completely clear.
Experimental data [2,25] for B 0 → K − π + and B s → K + π − agree with relation predicted for these two modes under SU(3) symmetry [12,26]. Therefore the use of SU(3) symmetry for B decays might be justified. With new data from BELLE II and LHCb, one can study SU(3) symmetry in B decays with a high precisions. One should keep in mind that for an appropriate analysis of SU(3) symmetry breaking, one must take into account all the above amplitudes, otherwise, the missing amplitudes will be disguised as symmetry breaking effects. As we have shown above, the modification due to the missing amplitudes can reach 20%, which is comparable with the generic SU (3) symmetry breaking effects. Thus the additional TDAs must be treated carefully to correctly interpret the data.
Our analysis is also applicable to other decay channels of heavy mesons and baryons. In the appendix, we give a discussion on the D meson decays. For charm quark decay, penguin operators are often negligible and the3 representation does not contribute either. So there are five independent tree amplitudes, while in TDA only four amplitudes, T, C, E, A, are used for the global fit.
In summary, we have carried out an analysis comparing two different approaches, the irreducible representation amplitude and topological diagram amplitude, to study B → P P decays. We find that previous analyses in the literature using these two methods do not match consistently in several ways. A few SU(3) independent amplitudes have been overlooked in the TDA approach. Taking these new amplitudes into account, we find a consistent description in both approaches. These new amplitudes can affect direct CP asymmetries in some channels significantly. A consequence is that for any charmless hadronic decays of heavy mesons, the direct CP symmetry cannot be identically zero. With more data become available, we can have a a better understanding of the role of flavor SU(3) symmetry in B decays.
Substituting this expression into the amplitude T for instance, we have From the expansion of IRA amplitudes, one can notice that the A T 6 can be absorbed into B T 6 and C T 6 .