Advances in High Energy Physics

Volume 2015 (2015), Article ID 104378, 10 pages

http://dx.doi.org/10.1155/2015/104378

## Decays with the QCD Factorization Approach

^{1}Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China^{2}Institute of Particle and Key Laboratory of Quark and Lepton Physics, Central China Normal University, Wuhan 430079, China

Received 7 January 2015; Accepted 17 March 2015

Academic Editor: Michal Kreps

Copyright © 2015 Junfeng Sun et al. 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 publication of this article was funded by SCOAP^{3}.

#### Abstract

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.

#### 1. Introduction

The meson is the ground pseudoscalar meson of the system [1]. 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 [2]. 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 [5] and fs [6]. 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 [2]. 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 [12].

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 [40], [18, 19, 41] based on the Bethe-Salpeter (BS) equation, [20–24, 42] with potential models, [25] with constituent quark model, [26–28] with QCD sum rules, [43] 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 [1]; , , 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 [1] 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.