#### Abstract

The exclusive decay of is studied in the framework of the QCD factorization (QCDF) method and final state interaction (FSI). A direct decay is only occurred via a tree and a penguin based on the quark diagram analysis. The result that is found by using the QCDF method is less than the experimental result, so, the role of FSI is considered. The intermediate states , , , and via the exchange of and are contributed to the decay. The above intermediate states is calculated by using the QCDF method. In the FSI effects the results of our calculations depend on “*η*” as the phenomenological parameter. The range of this parameter are selected from 1 to 2. For the exchanged particles and , it is found that if is selected the numbers of the branching ratio are placed in the experimental range. The experimental branching ratio of decay is , and our prediction number is in the absence of FSI effects, and it becomes when FSI contributions are taken into account.

#### 1. Introduction

The importance of FSI in weak nonleptonic meson decays is investigated by using a relativistic chiral unitary approach based on coupled channels [1–3]. The chiral Lagrangian approach is proved to be reliable for evaluating hadronic processes, but there are too many free parameters which are determined by fitting data, so that its applications are much constrained. Therefore, we have tried to look for some simplified models which can give rise to reasonable estimation of FSI [4, 5]. The FSI can be considered as a rescattering process of some intermediate two-body states with one particle exchange in the *t*-channel and computed via the absorptive part of the hadronic loop level (HLL) diagrams. The calculation with the single-meson-exchange scenario is obviously much simpler and straightforward. Moreover, some theoretical uncertainties are included in an off-shell form factor which modifies the effective vertices. Since the particle exchanged in the *t*-channel is off shell and since final state particles are hard, form factors or cutoffs must be introduced to the strong vertices to render the calculation meaningful in perturbation theory. If the intermediate two body mesons are hard enough, so that the perturbative calculation can make sense and work perfectly well, but the FSI can be modelled as the soft rescattering of the intermediate mesons. When one or two intermediate meson can reach a low-energy region where they are not sufficiently hard, one can be convinced that at this region the perturbative QCD approach fails or cannot result in reasonable values. If the intermediate mesons are soft, one can conjecture that at this region the nonperturbative QCD would dominate, and it could be attributed into the FSI effects. Because all FSI processes are concerning nonperturbative QCD [6], we have to rely on phenomenological models to analyze the FSI effects in certain reactions. In fact, after weak decays of heavy mesons, the particles produced can rescatter into other particle states through nonperturbative strong interaction. In the FSI the decay is a very interesting mode [7]. We calculated the decay according to QCDF method and selected the leading order Wilson coefficients at the scale and obtained the . The FSI can give sizable corrections, and we can utilize it [8]. Rescattering amplitude can be derived by calculating the absorptive part of triangle diagrams. In this case, intermediate states are , , , and . Then, we calculated the decay according to HLL method. Taken FSI corrections into account the branching ratio of becomes and the experimental result of this decay is [9].

This paper is organized as follows. We present the calculation of QCDF for decay in Section 2. In Section 3, we calculate the amplitudes of the intermediate states of decays. Then, we present the calculation of HLL for decay in Section 4. In Section 5, we give the numerical results, and in the last section, we have a short conclusion.

#### 2. Decay in QCD Factorization

A detailed discussion of the QCDF approach can be found in [10, 11]. Factorization is a property of the heavy-quark limit, in which we assume that the *b* quark mass is parametrically large. The QCDF formalism allows us to compute systematically the matrix elements of the effective weak Hamiltonian in the heavy-quark limit for certain two-body final states . In this section, we obtain the amplitude of decay using QCDF method. In factorization approach, there are color-suppressed tree and allowed penguin diagrams to decay. We adopt leading order Wilson coefficients at the scale for QCDF approach. The diagrams describing this decay are shown in Figure 1. According to the QCDF, the amplitude of decay is given by
where
The effective coefficients , which are specific to the factorization approach, and defined as
where the quantities of are effective Wilson coefficients at the renormalization scale for the transition. In the above amplitude the determination of in the current-current transitions has received a lot of attention, the quantities of , and , are the QCD-penguin and electroweak-penguin coefficients, respectively. Numerical values of for representative value of the phenomenological parameter are displayed in Section 5.

#### 3. Amplitudes of Intermediate State

To estimate the decays amplitudes of intermediate states [12], the QCDF method is again used. For decay, Feynman diagrams are shown in Figure 2, and the amplitude comes where There are also some similar diagrams such as and decays, which have polarization vectors. In their amplitude, just contribution of is considered, which has the dominant contribution. So the amplitude of decay, read as where and are form factors for [13], and the amplitude of is given by

#### 4. Final State Interaction of Decay

In our approach, we consider only leading contributions in charmless FSI, when the FSI for decay is calculated, two-body intermediate states, such as, , , , and via the exchange of and are contributed to the decay. The quark model for diagram is shown in Figure 3. This diagram just shows tree level ( contribution), while there are other contributions that were calculated in the last section. The diagrams which determine the HLL effects on the rate of decay are shown in Figure 4. At the HLL, the charge exchange reaction between the and can proceed either by exchange of a or meson. In order to calculate the various Feynman diagrams for the reactions . And so forth (were calculated in the nonrelativistic quark model [14]) we need the construct the effective three mesons vertices, such as, and . We begin with an SU(4) symmetry so as to include charm, and we introduce pseudoscalar and vector meson matrices, where and are pseudoscalar and vector multiplets and the are SU(4) generators. The free meson lagrangian is read [15]
Interaction Lagrangians are generated by replacing the space-time derivative with a gauge covariant one , where . The vector mesons are recognizable as playing roles of gauge bosons. As is usual in effective field theory strategies, to keep gauge consistency, we must collect terms of order . The effective lagrangian is read
The order terms deliver three-point vertices and the order terms are responsible for so-called contact terms or four-point couplings which are necessary for a gauge invariant theory. The interactions of interest for this study are the following [5]:
Here and we define the charm meson iso-doublets as
In this approach there are several coupling constants in (4.3). Methods for constraining them will be discussed below. The terms carry one power of coupling constant for each three-point vertices from which the contact term collapses. The coupling constants , , , and can be derived in the standard framework of the vector meson dominance model (VDM) [16–18]. By using VDM to the coupling of we have
where
The same method can be used for the coupling. Vector dominance gives
For the and couplings the VDM can be applied to the radiative decays of into *D*, that is, . By using the same technique we have
where are calculated in the model prediction based on relativistic potential model that we follow [19]. With the above preparation we can write out the decay amplitude involving HLL contributions with the following formula:
for which both intermediate mesons are pseudoscalar. And
in which both meson are vector. Also involves hadronic vertices factor, which is defined as
The dispersive part of the rescattering amplitude can be obtained from the absorptive part via the dispersion relation [6, 20]:
where is the square of the momentum carried by the exchanged particle and *s* is the threshold of intermediate states, in this case . Unlike the absorptive part, the dispersive contribution suffers from the large uncertainties arising from the complicated integrations. So the amplitudes of the mode , where and mesons are exchanged, respectively, are given by
where is the angel between and , is the momentum of the exchange meson, and
and is the form factor defined to take care of the off-shell of the exchange particles, which introduced as [6, 21]
The form factor (i.e., ) normalized to unity at . and are the physical parameters of the exchange particle and is a phenomenological parameter. It is obvious that for , becomes a number. If then turns to be unity, whereas, as the form factor approaches to zero and the distance becomes small and the hadron interaction is no longer valid. Since should not be far from the and , we choose
where is the phenomenological parameter that its value in the form factor is expected to be of the order of unity and can be determined from the measured rates, and
where
For diagrams (4c) and (4d) in Figure 4 the absorptive part of the amplitude corresponding to the process of , where and mesons are exchanged, respectively, is read as
where
where
For diagrams (4e) and (4f) in Figure 4 the absorptive part of the amplitude corresponding to the process of , where and mesons are exchanged, respectively, becomes
where
where
For diagrams (4g) and (4h) in Figure 4 the absorptive part of the amplitude corresponding to the process of , where and mesons are exchanged, respectively, is given
where
where
As the bridge between the dispersive part of FSI amplitude and the absorptive part, the dispersion relation is
According to the FSI, the amplitude of decay is given by
Concerning the QCDF amplitude and FSI correction, then the total decay amplitude turns to

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

**(h)**

#### 5. Numerical Results

Numerical values of effective coefficients for transition at are given by [22] The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a unitary matrix as [9] The elements of the CKM matrix can be parameterized by three mixing angles and a CP-violating phase The results for the Wolfenstein parameters are We use the central values of the Wolfenstein parameters and obtain The meson masses and decay constants needed in our calculations are taken as (in units of Mev) The Borel mass squares and and continuum thresholds and are auxiliary parameters, hence the physical quantities should be independent of them. The parameters and are determined from the conditions that guarantee the sum rules for form factors to have the best stability in the allowed and region. The working regions for and as well as the values for continuum thresholds are determined in [13]. They choose the values , , , from those working values for auxiliary parameters. The values of the form factors are given by Nonuniversal annihilation phases are [23] The signs of these phases are not predicted. As a result in [23], their predictions for the signs of CP asymmetries must be taken with caution. We use other input parameters as follows: By using the input parameters and according to the QCDF method of decay in Section 2, we get We note that our estimate of branching ratio of decay according to QCDF method seems less than the experimental result. Before calculating the decay amplitude via FSI, we have to compute the intermediate state amplitude, for the decay amplitude we get Now, using the above amplitude and according to the FSI, we are able to calculate the branching ratio of