Research Article | Open Access
Decays with the QCD Factorization Approach
We studied the nonleptonic , decays with the QCD factorization approach. It is found that the Cabibbo favored processes of , , are the promising decay channels with branching ratio larger than 1%, which should be observed earlier by the LHCb collaboration.
The meson is the ground pseudoscalar meson of the system . Compared with the heavy unflavored charmonium and bottomonium , the meson is unique in some respects. Heavy quarkonia could be created in the parton-parton process at the order of (where or , , ), while the production probability for the meson is at least at the order of via , where the gluon-gluon fusion mechanism is dominant at Tevatron and LHC . The meson is difficult to produce experimentally, but it was observed for the first time via the semileptonic decay mode in collisions by the CDF collaboration in 1998 [3, 4], which showed the realistic possibility of experimental study of the meson. One of the best measurements on the mass and lifetime of the meson is reported recently by the LHCb collaboration, MeV  and fs . With the running of the LHC, the meson has a promising prospect. It is estimated that one could expect some events at the high-luminosity LHC experiments per year [7, 8]. The studies on the meson have entered a new precision era. For charmonium and bottomonium, the strong and electromagnetic interactions are mainly responsible for annihilation of the quark pair into final states. The meson, carrying nonzero flavor number and lying below the meson pair threshold, can decay only via the weak interaction, which offers an ideal sample to investigate the weak decay mechanism of heavy flavors that is inaccessible to both charmonium and bottomonium. The weak decay provides great opportunities to investigate the perturbative and nonperturbative QCD, final state interactions, and so forth.
With respect to the heavy-light mesons, the doubly heavy meson has rich decay channels because of its relatively large mass and that both and quarks can decay individually. The decay processes of the meson can be divided into the following three classes [2, 9–11]: the quark decays with the quark as a spectator; the quark decays with the quark as a spectator; the and quarks annihilate into a virtual boson, with the ratios of 70%, 20% and 10%, respectively . Up to now, the experimental evidences of pure annihilation decay mode [class ] are still nothing. The transition, belonging to the class , offers a well-constructed experimental structure of charmonium at the Tevatron and LHC. Although the detection of the quark decay is very challenging to experimentalists, the clear signal of the decay is presented by the LHCb group using the and channels with statistical significance of and , respectively .
Anticipating the forthcoming accurate measurements on the meson at hadron colliders and the lion’s share of the decay width from the quark decay [31–33], many theoretical papers were devoted to the study of the , decays (where and denote the ground pseudoscalar and vector mesons, resp.), such as [17, 34–37] with the BSW model [38, 39] or IGSW model , [18, 19, 41] based on the Bethe-Salpeter (BS) equation, [20–24, 42] with potential models,  with constituent quark model, [26–28] with QCD sum rules,  with the quark diagram scheme, [29, 30] with the perturbative QCD approach (pQCD) [44–49], and so on. The previous predictions on the branching ratios for the , decays are collected in Table 3. The discrepancies of previous investigations arise mainly from the different model assumptions. Recently, several phenomenological methods have been fully developed to cope with the hadronic matrix elements and successfully applied to the nonleptonic decay, such as the pQCD approach [44–49] based on the factorization scheme, the soft-collinear effective theory [50–57] and the QCD-improved factorization (QCDF) approach [58–63] based on the collinear approximation and power countering rules in the heavy quark limits. In this paper, we will study the , decays with the QCDF approach to provide a ready reference to the existing and upcoming experiments.
This paper is organized as follows. In Section 2, we will present the theoretical framework and the amplitudes for the , decays within the QCDF framework. Section 3 is devoted to numerical results and discussion. Finally, Section 4 is our summation.
2. Theoretical Framework
2.1. The Effective Hamiltonian
The low energy effective Hamiltonian responsible for the nonleptonic bottom-conserving , decays constructed by means of the operator product expansion and the renormalization group (RG) method is usually written in terms of the four-quark interactions [64, 65]. Considerwhere the Fermi coupling constant ; , , and are the relevant local tree, annihilation, and penguin four-quark operators, respectively, which govern the decays in question. The Cabibbo-Kobayashi-Maskawa (CKM) factor and Wilson coefficients describe the coupling strength for a given operator.
Using the unitarity of the CKM matrix, there is a large cancellation of the CKM factors where the Wolfenstein parameter  and is the Cabibbo angle. Hence, compared with the tree contributions, the contributions of annihilation and penguin operators are strongly suppressed by the CKM factor. If the -violating asymmetries that are expected to be very small due to the small weak phase difference for quark decay are prescinded from the present consideration, then the penguin and annihilation contributions could be safely neglected. The local tree operators in (1) are expressed as follows: where and are the color indices.
The Wilson coefficients summarize the physics contributions from scales higher than . They are calculable with the RG improved perturbation theory and have properly been evaluated to the next-to-leading order (NLO) [64, 65]. They can be evolved from a higher scale down to a characteristic scale with the functions including the flavor thresholds [64, 65]where is the RG evolution matrix converting coefficients from the scale to , and is the quark threshold matching matrix. The expressions of and can be found in [64, 65]. The numerical values of LO and NLO with the naive dimensional regularization scheme are listed in Table 1. The values of NLO Wilson coefficients in Table 1 are consistent with those given by [64, 65] where a trick with “effective” number of active flavors 4.15 rather than formula (4) is used.
To obtain the decay amplitudes, the remaining work is how to accurately evaluate the hadronic matrix elements which summarize the physics contributions from scales lower than . Since the hadronic matrix elements involve long distance contributions, one is forced to use either nonperturbative methods such as lattice calculations and QCD sum rules or phenomenological models relying on some assumptions. Consequently, it is very unfortunate that hadronic matrix elements cannot be reliably calculated at present, and that the most intricate part and the dominant theoretical uncertainties in the decay amplitudes reside in the hadronic matrix elements.
2.2. Hadronic Matrix Elements
Phenomenologically, based on the power counting rules in the heavy quark limit, Beneke et al. proposed that the hadronic matrix elements could be written as the convolution integrals of hard scattering kernels and the light cone distribution amplitudes with the QCDF master formula [58–63]. The QCDF approach is widely applied to nonleptonic decays and it works well [66–76], which encourage us to apply the QCDF approach to the study of , decays. Since the spectator is the heavy quark who is almost always on shell, the virtuality of the gluon linked with the spectator should be . The dominant behavior of the transition form factors and the contributions of hard spectator scattering interactions are governed by soft processes. According to the basic idea of the QCDF approach [69, 70], the hard and soft contributions to the form factors entangle with each other and cannot be identified reasonably, so the physical form factors are used as the inputs. The hard spectator scattering contributions are power suppressed in the heavy quark limit. Finally, the hadronic matrix elements can be written as where is the transition form factor and is the light-cone distribution amplitudes of the emitted meson with the decay constant . The hard scattering kernels are computable order by order with the perturbation theory in principle. At the leading order , , that is, the convolution integral of (5) results in the meson decay constant. The hadronic matrix elements are parameterized by the product of form factors and decay constants, which are real and renormalization scale independent. One goes back to the simple “naive factorization” (NF) scenario. At the order and higher orders, the information of strong phases and the renormalization scale dependence of hadronic matrix elements could be partly recuperated. Combined the nonfactorizable contributions with the Wilson coefficients, the scale independent effective coefficients at the order can be obtained [58–63] as follows: where the expressions of vertex corrections are [58–63] with the twist-2 quark-antiquark distribution amplitudes of pseudoscalar and longitudinally polarized vector meson in terms of Gegenbauer polynomials [14–16]. One has where −; is the Gegenbauer moment and .
From the numbers in Table 1, it is found that for the coefficient the nonfactorizable contributions accompanied by the Wilson coefficient can provide 10% enhancement compared with the NF’s result, and a relatively small strong phase ; for the coefficient , the nonfactorizable contributions assisted with the large Wilson coefficient are significant. In addition, a relatively large strong phase is obtained; the QCDF’s values of agree basically with the real coefficients and which are used by previous studies on the , decays in [17, 18, 20–28, 34–37, 42], but with more information on the strong phases.
2.3. Decay Amplitudes
Within the QCDF framework, the amplitudes for decays are expressed as
The matrix elements of current operators are defined as where and are the decay constants of pseudoscalar and vector mesons, respectively; and denote the mass and polarization of vector meson, respectively.
For the mixing of physical pseudoscalar and meson, we adopt the quark-flavor basis description proposed in  and neglect the contributions from possible gluonium and compositions; that is, where and ; the mixing angle . We assume that the vector mesons are ideally mixed; that is, and .
For the transition form factors, since the velocity of the recoiled meson is very low in the rest frame of the meson, the wave functions of and mesons overlap strongly. It is believed that the form factors should be close to the result using the nonrelativistic harmonic oscillator wave functions with the BSW model . Consider The flavor symmetry breaking effects on the form factors are neglectable in (13). For simplification, we take 1.0 in our numerical calculation to give a rough estimation.
3. Numerical Results and Discussions
The branching ratios of nonleptonic two-body decays in the rest frame of the meson can be written aswhere the lifetime of the meson fs  and is the common momentum of final particles. The decay amplitudes are listed in the Appendix.
The input parameters in our calculation, including the CKM Wolfenstein parameters, decay constants, and Gegenbauer moments of distribution amplitudes in (8), are collected in Table 2. If not specified explicitly, we will take their central values as the default inputs. Our numerical results on the -averaged branching ratios are presented in Table 3, where theoretical uncertainties of the “QCDF” column come from the CKM parameters, the renormalization scale , decay constants, and Gegenbauer moments, respectively. For comparison, previous results calculated with the fixed coefficients and are also listed. There are some comments on the branching ratios.