Flavor SU(3) Topological Diagram and Irreducible Representation Amplitudes for Heavy Meson Charmless Hadronic Decays: Mismatch and Equivalence
Abstract
Flavor SU(3) analysis of heavy meson ( and ) 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 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.
I Introduction
Hadronic charmless decays provide an ideal platform to extract the CKM matrix elements, test the standard model description of CP violation and look for new physics effects beyond the standard model (SM). Experimentally, quite a number of physical observables like branching fractions, CP asymmetries and polarizations have been precisely measured by experiments at the electronposition colliders and hadron colliders. For a collection of these results, please see Refs. Amhis:2016xyh (); Patrignani:2016xqp (). On the other hand, theoretical calculations of the decay amplitudes greatly rely on the factorization ansatz. Depending on the explicit realizations of factorization, several QCDbased dynamic approaches have been established, such as QCDF Beneke:2001ev (), PQCD Keum:2000ph (); Keum:2000wi (); Lu:2000em (), SCET Bauer:2000yr (); Bauer:2001cu (). Apart from factorization approaches, the flavor SU(3) symmetry has been also wildly used in twobody and threebody heavy meson decays Zeppenfeld:1980ex (); Savage:1989ub (); Deshpande:1994ii (); He:1998rq (); He:2000ys (); Hsiao:2015iiu (); Chau:1986du (); Chau:1987tk (); Chau:1990ay (); Gronau:1994rj (); Gronau:1995hm (); Cheng:2014rfa (). An advantage of this method is its independence on the detailed dynamics in factorization. Since the SU(3) invariant amplitudes can be determined by fitting the data, the SU(3) analysis also provides a bridge between the experimental data and the dynamic approaches.
In the literature, the SU(3) analysis has been formulated in two distinct ways. One is to derive the decay amplitudes correspond to various topological diagrams (TDA) Chau:1986du (); Chau:1987tk (); Chau:1990ay (); Gronau:1994rj (); Gronau:1995hm (); Cheng:2014rfa (), and another is to construct the SU(3) irreducible representation amplitude (IRA) by decomposing effective Hamiltonian according to irreducible representations Savage:1989ub (); Deshpande:1994ii (); He:1998rq (); He:2000ys (); Hsiao:2015iiu (). These two methods should give the same physical results in the SU(3) limit when all relevant contributions are taken into account. However, as we will show we find 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 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 and and draw our conclusions.
Ii Basics for IRA and TDA approaches
ii.1 Su(3) Structure
We start with the electroweak effective Lagrangian for hadronic charmless meson decays in the SM. The Hamiltonian responsible for such kind of decays at one loop level in electroweak interactions is given by Buchalla:1995vs (); Ciuchini:1993vr (); Deshpande:1994pc ():
(1) 
where is a fourquark operator or a moment type operator. The fourquark operators are given as follows:
(2) 
In the above the denotes a quark for the transition or an quark for the transition, while .
At the hadron level, QCD penguin operators behave as the representation while tree and electroweak penguin operators can be decomposed in terms of a vector , a traceless tensor antisymmetric in upper indices, , and a traceless tensor symmetric in upper indices, . For the decays, the nonzero components of the effective Hamiltonian are Savage:1989ub (); He:2000ys (); Hsiao:2015iiu ():
(3) 
and all other remaining entries are zero. For the decays the nonzero entries in the , , can be obtained from Eq. (3) with the exchange corresponding to the exchange.
The above Hamiltonian can induce a meson to decay into two light pseudoscalar nonet mesons. There are three mesons which form a flavor SU(3) fundamental representation . The light pseudoscalar mesons contain nine hadrons:
(4) 
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 axialvector mesons.
ii.2 Irreducible Representation Amplitudes
To obtain irreducible representation amplitudes for ( is an element in ) decays, one takes the various representations in Eq. (3) and uses one and light meson to contract all indices in the following manner
(5)  
There also exist the penguin amplitudes which can be obtained by the replacements , , and ().
Expanding the above , one obtains amplitudes in the first two columns in Tables 1 and 2. Notice that the amplitude can be absorbed into and with the following redefinition:
(6) 
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 twobody decays. If one also considers (or ) mixing, one more parameter, the mixing angle , is requested making total 36 independent parameters.
ii.3 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. Cheng:2014rfa (), they are classified as follows:

denoting the colorallowed tree amplitude with emission;

, denoting the colorsuppressed tree diagram;

denoting the exchange diagram;

, corresponding to the QCD penguin contributions;

, being the flavor singlet QCD penguin;

, the electroweak penguin.
In addition, there exists annihilation diagrams, usually abbreviated as . 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 colorfavored contribution and the colorsuppressed one . The electroweak penguin operators can be reexpressed as:
(7) 
where the second part can be incorporated into the penguins transforming as a of SU(3). The contribution from is similar to tree operators, and thus we will use the symbol and to denote this electroweak penguin contribution. The is a flavor triplet whose contribution , as far as flavor SU(3) structure is concerned, can be absorbed into penguin contribution. We can write
(8) 
The three penguin type of amplitudes , and , can be grouped together. We can redefine by .
Actually these TDAs can be derived in a similar way as done for IRAs earlier by indicating (omitting the Lorentz indices ) by . For , the nonzero elements are and for , . The penguin contribution (including , and ) is an SU(3) triplet with for the transition and for the transition. Eq. (7) implies that the loop induced term proportional to has both and . Note that is no longer traceless.
The tree amplitude is given as
(9) 
while the penguin amplitude is given as:
(10)  
Expanding the above equations, we obtain the decay amplitudes for in the third column in Tables 1 and 2. It is necessary to point out that the singlet contribution in the form requires multigluon exchanges. One might naively think that its contributions are small compared with other contributions because more gluons are exchanged. However, at energy scale of decays, the strong couplings are not necessarily very small resulting in nonnegligible contributions. One should include them for a complete analysis.
Iii Mismatch and Equivalence
From previous discussions, one can see that the total decay amplitudes for decays for IRA and TDA can be written as
(11) 
For the amplitudes given in the previous section, it is clear that for both and , 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.
channel  IRA  TDA 

channel  IRA  TDA 

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:
(12)  
(13)  
The mismatch problem can be partly traced to the fact that defined in the TDA analysis is not traceless, that is . Because of this fact, and the two can contract with to form SU(3) invariant amplitudes and also the trace for is not zero when 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 1 and 2. With these new amplitudes at hand, one can derive the relation between the two sets of amplitudes:
(14) 
Here we have absorbed the into and . In the appendix, we give a direct derivation of relations between IRA and TDA amplitudes, in which the amplitude is kept.