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Advances in High Energy Physics
Volume 2012 (2012), Article ID 494031, 18 pages
and Decays in QCD Factorization Approach
Physics Department, Semnan University, P.O. Box 35195-363, Semnan 35351, Iran
Received 10 October 2011; Revised 4 December 2011; Accepted 29 December 2011
Academic Editor: Ira Rothstein
Copyright © 2012 Mahboobeh Sayahi and Hossein Mehraban. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The hadronic and has been analyzed in “QCD factorization” approach and generalized factorization. The effective coefficients have been calculated for three helicity states which give three different contribution of amplitudes. We consider that J/ψ behaves as a light meson in compared to meson. For and , experimental data of branching ratios are (1.8 ± 0.5) × 10−3 and <5 × 10−4, respectively. Our best obtained results are (1.79 ± 0.01) × 10−3 at θ = 58° and (1.42 ± 0.01) × 10−3 at θ = 32° for decay. And we have 4.27 × 10−4 at θ = 58° and 3.45 × 10−4 at θ = 32° for , which are in agreement with experiment.
Recent experimental results sobtained by BABAR, Belle, and CLEO have opened an interesting area of research about production of axial-vector mesons in decays. Two body decays look for CP violation and overconstrain the CKM parameters in the Standard Model. Exclusive modes containing , , and , which have been extensively discussed in the literature have confirmed such expectation . In this research, the two-body hadronic decays of meson into (: vector, : axial-vector meson) are studied in QCD factorization method. First we introduce the structure of axial-vector meson. Then we study the hadronic decays and , particularly. There are two distinct types of axial-vector meson, namely, and . In the quark model, two nonets of axial-vector meson are expected as the orbital excitation of the system. In terms of the spectroscopic notation , there are two types of -wave mesons ( and ). These two nonets have distinctive quantum numbers, and , respectively. Experimentally, the nonet consists of , , , and , while the nonet has , , , and . There are two mixing effects for axial-vector meson: one is the mixing between and states, for example, and , and the other is mixing among or states themselves . The meson behaves similarly to the vector meson this is not the case for the meson. For the latter, its decay constant vanishes in limit. Their light-cone distribution amplitudes are given by using the QCD sum rule method, and the chiral-even two-parton light-cone distribution amplitudes of the () meson are symmetric (antisymmetric) under the exchange of quark and antiquark momentum fractions in the limit due to -parity. The decays involving an axial-vector meson and vector meson in final state have three polarization states. We have studied the two body decays involving axial-vector meson or , and a vector meson in final state. The simplest approach to obtain the hadronic matrix elements in decay amplitudes is naive factorization, where the matrix elements have been parameterized into the product of decay constant and form factors. In generalized factorization, it is well established that nonfactorizable contributions must be present in the matrix elements in order to cancel the energy scale and renormalization scheme dependence of Wilson coefficients. In QCD factorization approach, the nonfactorizable effects such as vertex corrections, hard spectator interactions, and annihilation contributions are calculable in hard scattering approach, and they have been studied in [3, 4] for the decays , , , and .
Using the product expansion, the low-energy effective Hamiltonian for decays can be written generally as [6, 7] where are the product of CKM elements and 's are Wilson coefficients which have been evaluated at next leading order (NLO). The expressions of local operators are .
QCD penguin operators:
Electroweak penguin operators:
Here . For hadronic weak decays, the short distance effects can be calculated by perturbative theory but the computation of nonperturbative long distance effects is difficult. The fundamental problem in computation of hadronic matrix elements is due to nonperturbative effects arising from the strong interactions. The simplest approach to hadronic matrix elements is naive factorization hypothesis, where the hadronic matrix elements have been parameterized into the product of the decay constants and the form factors. Beneke et al. suggested QCD formula to compute the hadronic matrix elements in the heavy quark limit, combining the hard scattering approach with power counting in [9, 10]. In heavy quark limit , up to power corrections of order of , the QCD formula for can be written as and are both light. Which is form factor for to transition and is the light cone distribution amplitude for meson. are momentum fractions of constituent quarks in , and mesons, respectively. and are hard-scattering kernels arising from hard gluon exchanges. Hence, the hadronic matrix elements can be separated into short distance part (hard-scattering kernels) and long distance part. In QCD factorization, nonfactorizable loop effects and spectator scattering contribution have been considered. At leading order, there is a single diagram with no hard gluon interaction, and the contribution to in (1.5) is independent of , and -integral reduces to the normalization condition for the wave function. consequently, the factorization formula (1.5) reproduces naive factorization, if we neglect gluon exchange. In naive factorization, one neglect all corrections of order and of order . In QCD factorization, one computes systematically corrections to higher order in , but still neglects power correction of order . In factorization, we have factorizable and non factorizable contributions. There are soft gluons in the factorizable contributions which are absorbed into the physical form factor. The diagrams also have hard contributions, which go into the short distance coefficient. The diagrams containing gluon exchanges which do not belong to the form factor are called nonfactorizable contributions. At order , these contributions can be divided into four groups: vertex corrections, penguin diagrams, hard spectator interactions, and annihilation diagrams. A detailed discussion of QCD factorization approach can be found in [8–10].
This method works well for the case with two light mesons in which the final-state mesons carry large momenta. When there is a heavy quark in the final state such as , this method still works when a spectator quark of meson is absorbed by meson. However, when the spectator quark is absorbed by a light quark, for example, in , nonfactorizable contributions are infrared divergent, and the factorization breaks down. When we consider and , it looks ambiguous at first sight whether we can apply the same method used in , or , since the spectator quark in the meson goes into light or meson. However, what is special about is that the size of the charmonium is so small (~) that the charmonium has a negligible overlap with the () system .
2. Input Parameters
2.1. Light-Cone Distribution Amplitude (LCDA)
The QCD corrections can change the local quark-anti quark operators into a series of nonlocal operators. If we assume that the behave as a light meson, we can describe the light cone distribution. Note that in QCDF, the LCDAs of the light vector meson are written as  where with . , are twist-2 DAs; , , and are twist-3 ones. is momentum fraction of quark in , . , are vector and tensor decay constants, respectively. Similarly, for axial-vector meson , we have Applying equation of motion to LCDAs, one can obtain the Wandzura-Wilczek relations in which twist-3 LCDAs related to the twist-2 ones . Also are defined as which are transverse components of twist-3 LCDAs. While are longitudinal components of twist-3 ones for vector (axial-vector) meson. We specify the light-cone distribution amplitudes (LCDAs) as for the vector meson, where are Gegenbauer polynomials and are Legendre polynomials. The normalization of LCDAs for vector meson is Also, for axial-vector meson and for axial-vector meson. The normalization conditions are for axial-vector meson and for axial-vector meson.
Here , , and are twist-2 LCDAs; likewise and are twist-3 ones for vector and axial-vector mesons, respectively. The parameters and are defined by the matrix element of a twist-2 conformal operator with conformal spin 3 in . These parameters are unknown for ; then we consider asymptotic forms of amplitudes for . Also, the Gegenbauer moments for and are given in [14, 15].
For the meson, we can write the projection as A detailed dissection of wave function of meson can be found [9, 10, 16–18] with , and the normalization conditions are The wave function corresponds to defined by . is the first inverse moment of the meson's distribution which is of order .
2.2. Decay Constant
Decay constants of vector and axial-vector mesons are defined as Also, the transverse decay constants are defined via the tensor current as In general, the decay constants and are zero in the limit.
2.3. Form Factor
The form factors of transitions are defined as  and for transitions, we have where , and for we have and . is the polarization four-vector, and , , , , and are form factors and The form factors of and transitions have been calculated in the light-cone sum rule (LCSR) . The momentum dependence of form factors is calculated in the LCSR approach which is parameterized in three-parameter form: which , and two parameters and are given in Table 1.
2.4. and Decays in QCD Factorization
Now, we want to calculate the branching ratios of and decays in QCD factorization method. First, we have calculated the branching ratios in generalized factorization which the effective Wilson coefficients are not dependent on renormalization scale . To solve the issue of scale dependence, but not the renormalization scheme dependence , it is proposed in [21, 22], to isolate from the matrix element of four operators the dependence and link with the dependence in the Wilson coefficients to form , effective Wilson coefficients independence of . The formula for effective Wilson coefficients and their numerical values has been given explicitly in . For transition (in naive dimensional regularization): , , , , , , , , , . is the fine structure constant. Wilson coefficients appear in decay amplitudes as linear combinations. So, the coefficients are As it known the physical states and are mixture of states and , whose relations could be parameterized by where and are and axial-vector mesons, respectively, and is the mixing angle. From experimental date on masses and partial ratios of and , it is found two solutions for the mixing angle with a two-fold ambiguity, and . Hence, the physical decaying amplitudes are given by Since the final states of carry spin degrees of freedom, we have calculated the helicity amplitudes, separately. In our work, the longitudinal amplitude and the transverse amplitudes are calculated for and decays . If we assume that behaves as a light meson, due to its size, we can describe the light-cone distribution. For and decays, we have colour suppressed contribution and penguin contributions , , , and . Then, the contributions of vertex corrections for these decays are only for . Hence, three different helicity amplitudes have been written as where the factorizable amplitude is written as The helicity dependencies read as where is . momentum of decays in the rest frame and are the mass of final state mesons. The branching ratio is given by  where is Fermi constant and is life time of meson. If the final states are two light vector mesons, it is expected that for meson decays. (For decays, exchange .) It is because of that the amplitude is suppressed by a factor of , while the amplitude is further suppression in . But for to heavy-light final states, is of order unity . In QCD factorization, the effective coefficients basically are Wilson coefficients in conjunction with short distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. Then, we have  where , the lower (upper) signs apply when is even (odd), are Wilson coefficients (we use the next to leading order coefficients, NLO, obtained in the naive dimensional regularization scheme, NDR, at the energy scale .), with , is emitted meson and have the spectator quark of the meson. is the vertex corrections and is the hard-scattering interactions, obtained due to exchange soft and hard gluons. The superscript denotes the helicity of the final-state meson, that is, . The LCDA is for and for . And . We can take the infinite mass limit of the quark in which goes to infinity while is fixed (). The vertex corrections are given by With we consider asymptotic function for , and also we have and . Hard spectator corrections are given by for , and for . are chirally enhanced parameters. The transverse hard spectator terms are for , and for . In our calculations, the transverse hard spectator term is zero. There are some divergences in other hard spectator terms. We have used the unknown parameter to eliminate these divergences, defined as where is hadronic parameter and which and are phases parameters (, ). We consider in these decays and . This parameter phenomenologically is significant. Hence, the non factorizable contributions (vertex corrections and hard spectator interactions) have been calculated by (2.28)–(2.35).
In the other hand, we know that is a heavy vector meson, although in  has been supposed that quarkonium is light meson relative to the meson. If we consider other limit in which goes to infinity, is held fixed. In this scenario, the amplitude wave function of has represented like meson. Also, to the leading order in , the wave function is  Using , the effect of higher twist for may not converge enough. Because the charmed quark carries a momentum fraction of order ~, the distribution amplitudes of vanish in the end point region. is adapted as the DA of nonlocal vector current of rather as the DA of component since the latter does not vanish at the end point. Comparing (2.1) with (2.37) the leading twist DA of is Then, for vertex corrections , we have with where and . In  by contracting (2.14) for with and applying the equation of motion and (2.13) for , it finds that . Therefore In , the charmed quark carries a momentum fraction of order ~, which is the same fraction momentum , then . The contributions of hard spectator interaction have been written as for , for . The transverse hard spectator terms are for , and for . Hence, the nonfactorizable contributions which are the same vertex corrections and hard spectator interactions have calculated by (2.28)–(2.35) for first scenario and by (2.39)–(2.46) for second one. Finally, the branching ratios of and have calculated at and at mixing angles in Tables 2 and 3.
3. Input Quantities
In generalized factorization, the branching ratios of and decays are small in order to experiment, although they are in range of experimental data for these decays in mixing angles as shown in Tables 2 and 3. In this work, we have been tried to analyze these decays within “QCD factorization” method. Hence, the factorizable amplitudes in (2.24) have been calculated by using the input data of form factors, decay constants, and the non-factorizable contributions corresponding to the vertex corrections, and hard spectator interactions in have been calculated at two scenarios. First, behaves as a light meson in compared to meson (). And in second, we have considered mass of in which is held fixed. The effective coefficients have been calculated at three helicity states , which give three different contributions of amplitude. The branching ratios are calculated at these conditions and for the mixing angles , where given in Tables 2 and 3. As it is shown, the obtained results in QCD factorization for these decays are not suitable assumption for all of the mixing angles. For and , the experimental branching ratios are and , respectively . In first scenario, the best obtained results for decay are at and at ; for , these are at and at , while in second one, as shown in Tables 2 and 3.
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