Research Article | Open Access
Qin Chang, Xiaohui Hu, Zhe Chang, Junfeng Sun, Yueling Yang, "Study of Decays with QCD Factorization Approach", Advances in High Energy Physics, vol. 2016, Article ID 3863725, 9 pages, 2016. https://doi.org/10.1155/2016/3863725
Study of Decays with QCD Factorization Approach
Motivated by the -physics experiments at running LHC and upcoming SuperKEKB/Belle-II, the nonleptonic ( and ) weak decays are studied within QCD factorization framework. The observables of these decay modes are first predicted. It is found that the tree-dominated and CKM-favored decays have the largest branching fractions and thus are hopefully to be measured. The decays are dominated by the longitudinal polarization states. In addition, associating with the relevant meson decays, some interesting phenomena and relations are discussed in detail; for example, , and .
Thanks to the fruitful running of BABAR, Belle, CDF, and D0 experiments in the past decade, most of the meson decays with branching fractions are measured precisely, which provide a very fertile testing ground for the Standard Model (SM) pictures of flavor physics, charge-parity (CP) asymmetry, and QCD mechanism. As the particle physics enter a new era of precision, more experimental information about -physics will be explored at ongoing LHC and forthcoming SuperKEKB/Belle-II. In the system, besides the bound state meson, some excited states decays are also hopefully to be well measured in the future.
The meson with quantum number of and is the first excited state in the spectra of heavy-light system [1–4]. In the past years, the masses (or the differences ) are well measured by many collaborations [5–10]. However, the experimental information about the meson decays is very limited due to the following facts: (i) At collider, and mesons are produced mainly through resonance decays, while the past -physics experiments (BABAR and Belle) run mainly around resonance for producing mesons, and the collected data sample of collisions in the vicinity of resonance is not sufficient enough to probe rare decays; (ii) the decays of unstable mesons are dominated by the electromagnetic transition , and thus the other decay modes are too rare to be detected soon. Fortunately, such situation will possibly be improved by the running LHC and upgrading Belle-II experiments in the near future.
For the Belle-II experiment at SuperKEKB, one of the main goals of the physics program is to study the decays of meson. Meanwhile, it also should be noted that a plenty of samples would be produced simultaneously. With the target luminosity , the annual integrated luminosity is expected to be after 2018 . Using the cross section of production  and the branching fractions of decays related to final states , we find that about and samples could be collected per year. As a result, the decays with branching fractions are hopefully to be observed by Belle-II. Moreover, because of the much larger beauty production cross section of collisions [14–16], the LHC experiments may also provide a lot of experimental information for decays. For instance, as analyzed in , even though decay having branching fraction is very rare and obviously out of the scope of Belle-II, it is still possible to be measured by LHC after high-luminosity upgrade (Run-III). So, with the rapid development of experiment, the theoretical studies of weak decays are worthwhile then.
Recently, a few interesting theoretical studies of decays have been performed. For instance, in [18–20], the theoretical estimates of the semileptonic decays are made within QCD sum rules framework. In , the pure leptonic and decays are studied, and the detectability of LHC on these decays is analyzed in detail. Besides the (semi-)leptonic decays, the study of nonleptonic weak decays, which involve much more decay channels, is also essential. In this paper, we will focus on the (, and ) decays, which are tree-dominated and thus principally have relative large branching fractions. In the meson system, in order to evaluate the QCD corrections, some calculation approaches, such as the QCD factorization (QCDF) [21, 22], the perturbative QCD [23, 24], and the soft-collinear effective theory [25–28], are presented. Then, the nonleptonic decays provide another testing ground for these approaches. In our following calculation, the QCD factorization approach is employed.
Our paper is organized as follows. In Section 2, the basic theoretical framework and the amplitudes of decays calculated through the QCDF approach are presented. Section 3 is devoted to the numerical results and discussion. Finally, a short summary is given.
2. Theoretical Framework
The low energy effective Hamiltonian responsible for the decays could be written as [31, 32]where is the Fermi coupling and is product of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The Wilson coefficients summarize the physical contributions above scale of and are calculable with the perturbation theory. Their values at scales of in naive dimensional regularization scheme are listed in Table 2. and are local tree four-quark operators and defined aswhere and are color indices and the sum over repeated indices is understood. One can refer to [31, 32] for details of this part. To obtain the decay amplitudes, the remaining works are to accurately calculate the hadronic matrix elements of local operators.
The simplest way to deal with the hadronic matrix elements is the naive factorization (NF) scheme [33, 34] based on the color transparency mechanism [35–37]. Within the NF scheme, the hadronic matrix element is approximated by the product of two current matrix elements, which are further parameterized by decay constants and transition form factors. Explicitly, for the () decays, the hadronic matrix element could be written as For the case that is a light pseudoscalar (), evaluating (3), we getwhere is the polarization four-vector of meson, is the momentum of , and their production could be simplified by replacingFor the case that is a light vector (), corresponding to the different helicity amplitudes, (3) could be written asIn the evaluations, the definition of decay constantsthe form factors (the expression of parameterization of the hadronic matrix , such as (8), could be obtained through taking the Hermitian conjugate of ; for the latter of which, we take the same conventions as .)and the relationare used. Then, one can easily get the amplitudes of decays within NF, which are proportional to or .
However, in the NF framework, the amplitudes are renormalization scale dependence, and the nonfactorizable contributions dominated by the hard gluon exchange are lost. As a result, the amplitude is unphysical, and the strong phase, which is essential for evaluating CP asymmetry, cannot be calculated. In order to remedy these deficiencies and take into account the nonfactorizable contribution, the QCDF approach is proposed by BBNS [21, 22] and has been widely used to deal with the hadronic matrix elements (e.g., [39–49]). Within the framework of QCDF, in the heavy-quark limit (), the hadronic matrix elements are expressed by the factorization formula [21, 22]:Here, is form factor; is hard-scattering function, which is perturbatively calculable; is the light-cone distribution amplitude for the quark-antiquark Fock state of meson . The leading-twist distribution amplitudes of pseudoscalars ( and ) and longitudinal polarized vectors ( and ) are conventionally expanded in Gegenbauer polynomials [29, 30, 50]:where is the Gegenbauer moment. In factorization formula (10), the spectator scattering contribution does not appear due to the fact that it is not only -suppressed but also power-suppressed by the factor of relative to the LO contribution for case of heavy-light final states, while the vertex correction included in (10) is only -suppressed. In fact, after a detailed analysis for decays, the authors of  have concluded that the spectator interaction does not contribute to heavy-light final states at leading power in the heavy-quark expansion .
Applying the QCDF formula, the matrix elements of the effective weak Hamiltonian for () decays could be written aswhere the products of matrix elements of two current, , have been given explicitly by (4) and (6). The effective coefficient in the amplitude, including nonfactorizable contributions from QCD radiative vertex corrections, is defined as [22, 51]Obviously, the NF result is recovered if the QCD-loop correction (the third term in (13)) is neglected. With the modified minimal subtraction () scheme, for the pseudoscalar and the longitudinally polarized vector meson, the function is written aswhere the loop function iswith the definitions ,,, , and . For the transversely polarized vector meson, the leading-twist contribution to is zero. The result of (see (14)) is exactly the same as the result in decays, where is a light meson, given in .
With the amplitudes given in (12), we can evaluate the observables of decays. In the rest frame of meson, the spin-averaged branching fractions could be written aswhere is the total decay width of . Besides of the branching fraction, the polarization fractions of decays are also important observables. They are defined aswhere and are parallel and perpendicular amplitudes and could be easily obtained through .
3. Numerical Results and Discussions
With the theoretical framework given in Section 2, we then present our numerical results and discussions. Firstly, we would like to clarify the input parameters in our numerical evaluation. The input values of Wolfenstein parameters, masses of quarks, decay constants, and Gegenbauer moments are summarized in Table 1. Our numerical results of the Wilson coefficients and at different scales are listed in Table 2. Besides that, to evaluate the branching fractions of decays, the total decay widths are essential. Unfortunately, there are no available experimental or theoretical results until now. In our numerical evaluation, the approximation is taken because of the known fact that the radiative process dominates meson decays. Theoretically, the predictions on have been widely evaluated in various models, such as relativistic quark model [52, 53], QCD sum rules , light-cone QCD sum rules , light front quark model , heavy-quark effective theory with vector meson dominance hypothesis , or covariant model . In this paper, we employ the most recent results [56, 58]:which are consistent with the results in the other models.
In addition, the values of transition form factors are also unknown. In this paper, the Bauer-Stech-Wirbel (BSW) model  is employed to evaluate the values of , , and , which could be written as the overlap integrals of wave functions of mesons . With the meson wave function as solution of a relativistic scalar harmonic oscillator potential and = 0.4 GeV which determines the average transverse quark momentum, we getIn our numerical evaluation, these numbers and of them are treated as default inputs and uncertainties, respectively.
Using the given values of input parameters and the theoretical formula, we then present QCDF predictions of the -averaged branching ratios of () decays in Table 3, in which the three theoretical uncertainties are induced by the CKM parameters, hadronic parameters (decay constants and form factors), and total decay widths, respectively. In comparison, the NF results are also listed in Table 3. The followings are some analyses and discussions:(1)In Table 2, the values of effective coefficient within NF and QCDF are summarized. It could be found that information of strong phases is obtained by considering gluon radiative corrections to vertex, which plays an important role in exploring the direct violation. However, due to lack of interference, the direct CP asymmetries of () decays are zero. In Figure 1, the dependence of tree coefficient on the renormalization scale is shown. As Figure 1(b) shows, the imaginary part , which is zero at LO (NF result), arises after taking into account the NLO corrections. For the real part , as Figure 1(a) shows, the scale dependence has been reduced partly at low scales when the NLO corrections are taken into account. To further clarify such partial reduction, we define the quantity , which is equal to zero if is totally scale-independent. It is found that the value of at NLO is a little bit smaller than the one at LO; for instance, (LO), (NLO), and (LO), (NLO), as found from Figure 1(a). However, one also should note that the reduction of scale dependence is not very obvious as one expected, which could be attributed to the fact that NLO correction is color-suppressed , while the scale dependence reduction effect becomes very significant when the NNLO correction, which is no longer color-suppressed, is taken into account as found in .(2)From Table 3, one may find a clear hierarchy of branching fractions, . It is mainly induced by the following two reasons: (i) The CKM element responsible for decays is suppressed by factor of compared with the one for decays and (ii) the decays are suppressed relatively by the orbital angular momentum compared with the corresponding decays. In addition, one also may find that the decay is always about two times larger than the corresponding decay; for instance, . It is mainly induced by the theoretical prediction (see (19)) and the assumption . Explicitly, such relation could be expressed as which is a useful observable for measuring experimentally and further testing the theoretical predictions of . From Table 3, it could be found that decays have the largest branching fractions, about , and thus are hopefully to be well measured by Belle-II experiment in the near future. In addition, the processes that decays into two light mesons, such as and final states, generally have much more interesting phenomena. However, they are generally CKM- and/or loop-suppressed and therefore hard to be observed soon.(3)Besides the branching ratio, the polarization fractions are also important observables. In the decays, the hierarchy pattern of helicity amplitudes is expected [61–63], especially for the tree-dominated decays. For the decays, such hierarchical relation is also naively expected due to the following: taking decay ( transition) as an example for convenience of discussion, in the longitudinal transition, the quark and antiquark in each meson have opposite helicities, in which the case is favored by interaction. Relative to , for to occur the quark has to flip its helicity, which results in the so-called “helicity-flip” suppression. For , in addition to the “helicity-flip” suppression, a further chirality suppression appears since the quark in the interaction has a “wrong” helicity at this moment. Exactly, from (6), it could be found that the transverse amplitudes are suppressed by a factor relative to . In addition, the axial-vector and vector contribution to cancel in the heavy-quark limit. As a result, the hierarchy pattern of helicity amplitudes (see (22)) is still fulfilled by the decays within NF framework. Further, considering that the QCD NLO correction in (see (13)) is much smaller than the LO one, the very large longitudinal polarization fractions of decays are generally expected in both NF and QCDF frameworks. Numerically, within the QCDF, using the default values of input parameters and taking , we get(4)In order to explore the relation between and decays, we define the ratio which is independent of the decay constants and the coefficient and close to . Further, evaluating the branching fractions, we get in which the prefactor of corresponds to the factor in (16) caused by averaging over the initial spin. With the values of masses given by PDG , the ratio of masses in (25) is equal to for and . Moreover, the ratio of form factors in (25) is generally close to ; for instance, for within WSB model. So, the relation is expected. Numerically, with the assumption and the values of given by (19), we get for and for , which could be tested experimentally. For the and decays, the relation between their polarization fractions is much interesting. It could be found that the relation is generally expected, because (i) the expressions of their helicity amplitudes are very similar to each other, except for the replacements and everywhere in (6), and (ii) different from branching ratio (see (26)) the polarization fraction is sensitive to the relative strengths of form factors rather than the absolute ones. In order to test the relation, we take and decays for example. One may find that our prediction numerically agrees well with predicted in , which is consistent with experimental results .