Studies of Three-Body Decay of to and to *
We investigate the and decay by using the Dalitz plot analysis. As we know there are tree, penguin, emission, and emission-annihilation diagrams for these decay modes in the factorization approach. The transition matrix element is factorized into a form factor multiplied by decay constant and also a form factor multiplied by decay constant. According to QCD factorization approach and using the Dalitz plot analysis, we calculate the branching ratios of the and three-body decay in view of the mixing and obtain the value of the , while the experimental results of them are and , respectively. In this research we also analyze the decay which is similar to the previous decay, but there is no experimental data for the last decay. Since for calculations of the decay we use assumptions of the decay, we hope that if this decay will be measured by the LHCb in the future, the experimental results will be in agreement with our calculations.
The three-body decay of was originally measured by the BABAR  and BELLE  collaborations and later on tabulated by the Particle Data Group . A long time ago, the decay was studied with the Dalitz plot analysis . According to this technique, we can find many articles such as [5–9] which do not assume that the mass of the -meson is heavy in front of the pi-meson one. So the momentum of the -meson is sizable against the momentum of the pi-mesons. In these kinds of decay, when three particles are light, given that the theoretical momentums of the output particles are not directly calculable, momentums and form factors are written in terms of the and [10, 11] and the Dalitz plot analysis should be used for calculation of the decay rate integral from , to , . In our selected decay with the experimental values of and , we obtain by using the Dalitz plot analysis.
Note that, to implement the mixing, we will use the two-mixing-angle formalism proposed in [12, 13], in which one has where and are, respectively, the flavor SU-octet and SU-singlet components. In the quark basis they are given by The relations for the pseudoscalar decay constants in this mixing formalism involving the axial-vector currents and are In order to get four unknown parameters (, , , and ) allowed values one has to use as constraints the experimental decay widths of : On the other hand [14, 15] If is that large, the radiative decay may be dominated by a contribution where the pair runs from the to the meson instead of being annihilated. On that supposition the width of that process can be calculated along the same lines as that one for the decay. The ratio of the two decay widths reads  The best-fit values of the mixing parameters yield which are used to calculate the decay rates in which and/or are involved. In the meson decay into states with an , in the case of , since the meson is described by the , , and combination, there are two different color-suppressed internal W-emission Feynman diagrams; the decay mode contains and pairs while the decay mode includes and pairs of the components. In addition, there is a penguin diagram where the and pairs are considered for both decay modes. The diagrams in which is emitted via three-gluon exchange are called “hairpin” diagrams so that pairs of the components are used. Before giving the matrix elements for the decay, we discuss the parametrization of the decay constants and form factors which appear in the factorized form of the hadronic matrix elements. In the tree and penguin levels, the and mesons are placed in the form factors and the meson is placed in the decay constant  in which the vector meson’s decay constant, such as , is expressed in terms of the matrix element . We also have a form factor multiplied by decay constant in the emission diagram with an emission-annihilation level.
The present analysis contains nonfactorizable effects, whereas the hadronic physics governing the transition and the formation of the emission particle is genuinely nonperturbative; nonfactorizable interactions connecting the two systems (hard-scattering kernels) are dominated by hard gluon exchange. The hard-scattering kernels are calculable in perturbation theory, which starts at tree level and, at higher order in , contains nonfactorizable corrections from hard gluon exchange.
The decay channels can also receive contributions through intermediate resonances and , namely, . Therefore, in order to get a reliable estimation on the branching fraction, it is important to have an estimate of the resonant contributions.
2. Amplitudes of the and Decay
2.1. The Dalitz Plot Analysis
2.1.1. Nonresonant Background
In the factorization approach, the Feynman diagrams for three-body and decay are shown in Figure 1; the meson is produced from three , , and components; according to Figure 1 to draw the Feynman diagrams of the decay, two and components are used for tree level and just component is considered for penguin contribution. These topics are shown in (a), (c), and (e) panels. For decay, the and components can be used for tree and pairs for penguin contributions. The panels of (b), (d), and (f) show the mentioned content. Panels (g)–(k) show the emission and emission-annihilation diagrams in which is emitted via three-gluon exchange, which are the so-called hairpin diagrams; as we can see, both decay modes have the same amplitudes. Under the factorization approach, the and decay amplitudes consist of three distinct factorizable terms: (i) the tree and penguin processes, , (ii) the meson emission process, , and (iii) the emission-annihilation process, , where denotes an transition matrix element. Here denotes two-meson transition matrix element. The leading nonfactorizable diagrams in Figure 2 should be taken into account. To this, we employ the QCD factorization framework, which incorporates important theoretical aspects of QCD like color transparency, heavy quark limit, and hard scattering and allows us to calculate nonfactorizable contributions systematically.
The matrix elements of the decay amplitude are given by For the current-induced process, the two-meson transition matrix element has the general expression as  and the decay constant is defined as  The direct three-body decay of mesons in general receives two distinct contributions: one from the point-like weak transition and the other from the pole diagrams that involve three-point or four-point strong vertices . So the , , and form factors are computed from point-like and pole diagrams; we also need the strong coupling of , , and vertices. These form factors are given by  The other two-body matrix element can be related to the and matrix element of the weak interaction current where is the to transition form factor and needs to be determined from experiment . This transition occurs by two gluons where both of the gluons are off shell or by two-photon decay widths of the meson in the decay . The contribution of gluonic wave function to the transition form factor has been tested in [20, 21], and the form factor is defined as follows : where . The emission-annihilation matrix element is assumed to be  and the form factor is parameterized as where Mev is the chiral-symmetry breaking scale. Then the matrix elements read where under the Lorentz condition . The meson polarization vectors become
Consider the decay of meson into three particles of masses , , and . Denote their 4 momenta by , , , and , respectively. Energy-momentum conservation is expressed by
Define the following invariants:
The three invariants , , and are not independent; it follows from their definitions together with 4-momentum conservation that We take and , so we have . In the center of mass of and , according to Figure 3, we find and the cosine of the helicity angle between the direction of and that of reads
With these definitions, we obtain multiplying of the 4-momentum conservation as
In this framework, nonfactorizable contributions (hard-scattering and vertex corrections) to can be obtained by calculating the diagrams in Figure 2. Each of the diagrams in Figure 2 contains a leading-power contribution relevant to power-suppressed terms, which are not factorized in general. An important class of such power-suppressed effects is related to certain higher-twist meson distribution amplitudes. Fortunately, it turns out that ratios of the different hard-scattering contribution have very small uncertainties so we just put the vertex corrections in the calculation of the Wilson coefficients.
Now we can derive the nonresonant amplitude with mixing angle as The vertex corrections to the decay, denoted as in QCDF, have been calculated in the NDR scheme and can be adopted directly. Their effects can be combined into the Wilson coefficients associated with the factorizable contributions : where and is the meson asymptotic distribution amplitude which is given by 
2.1.2. Resonant Contributions
According to Figure 1, the decay channels of can also receive contributions through intermediate resonances and . Resonant effects are described in terms of the usual Breit-Wigner formalism where where denote , , and is the c.m. momentum. In determining the coupling of , we have used the partial widths Mev, Mev, and Mev measured by PDG . Then the decay amplitude through resonance intermediate reads Finally by using the full amplitude, the decay rate of is then given by  where where .
3. Amplitudes of the and Decay
The Feynman diagrams for three-body decay of and , in the factorization approach, are shown in Figure 4 and types of these decay modes can be obtained from the following options.(1)For choices of and(1-1), decay mode becomes ,(1-2), decay mode becomes ,(1-3), decay mode becomes ,(1-4), decay mode becomes .(2)For selection of and(2-1), decay mode becomes ,(2-2), decay mode becomes .For this decay, according to Figure 2, the decay constant is defined as Then the nonresonant amplitude can be obtained by In the decay modes we use instead of (also within the ). Note that when the final states contain meson, the decay amplitudes multiplied by . In addition, several intermediate resonant states involving , , , , , and resonances are used in the calculations .
4. Numerical Results
The theoretical input parameters used in our analysis, together with their respective ranges of uncertainty, are summarized below.
The Wilson coefficients have been calculated in different schemes. In this paper we will use consistently the naive dimensional regularization (NDR) scheme. The values of at the scales , , and at the next to leading order (NLO) are shown in Table 1.
There is a potentially quite large error that could come from the uncertainty in the parameter available on the form factors. This parameter is determined from the decay and we use  For the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, we use the values of the Wolfenstein parameters and obtain The meson masses and decay constants needed in our calculations are taken as (in units of Mev)  and the form factors at zero momentum transfer are taken as 
Using the parameters relevant to the and decay, we calculate the branching ratios of this decay, which are shown in Table 2. Note that, as we mentioned before, both decay of and decay of have similar amplitudes. Since the masses of the and and also and mesons are very close to each other, both branching ratios are the same.
In this work, we have calculated the branching ratios of the and decay by using the Dalitz plot analysis. In this calculation we have used factorizable terms, nonfacorizable effects, and mixing. According to QCD factorization approach, we have obtained while the experimental results of them are and , respectively. The branching ratios obtained by applying the Dalitz plot analysis are compatible with the experimental results.
Moreover we have analyzed the decay which is similar to the decay, but there are no experimental data for the last decay. Since for calculations of the decay we have used assumptions of the decay, we hope that if this decay will be measured by the LHCb in the future, the experimental results will be in agreement with our calculations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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