<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.458401pt" id="M1" height="16.0827pt" version="1.1" viewBox="0 -11.6243 73.2093 16.0827" width="73.2093pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.013,0,0,-0.013,10.659,4.308)"><path id="g50-100" d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"/></g><g transform="matrix(.018,0,0,-0.018,24.474,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><g transform="matrix(.018,0,0,-0.018,49.529,0)"><use xlink:href="#g113-67"/></g><g transform="matrix(.018,0,0,-0.018,59.739,0)"><path id="g113-81" d="M600 480C600 590 528 650 384 650H143L137 622C222 614 225 607 210 531L130 127C113 41 106 36 23 28L17 0H294L300 28C204 36 195 42 212 127L243 284L314 263C327 263 339 263 352 264C465 271 600 337 600 480ZM508 481C508 351 402 304 329 304C289 304 265 311 250 317L295 559C302 594 310 606 323 611C335 616 350 619 367 619C455 619 508 573 508 481Z"/></g><g transform="matrix(.018,0,0,-0.018,69.883,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g></svg> <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1968002pt" id="M2" height="11.8211pt" version="1.1" viewBox="0 -11.6243 23.2132 11.8211" width="23.2132pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><use xlink:href="#g113-67"/></g><g transform="matrix(.018,0,0,-0.018,10.749,0)"><path id="g113-87" d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"/></g></svg> Decays with the QCD Factorization Approach

Sun, Junfeng; Wang, Na; Chang, Qin; Yang, Yueling

doi:https://doi.org/10.1155/2015/104378

Advances in High Energy Physics

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Abstract Introduction Appendix Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 104378 | https://doi.org/10.1155/2015/104378

Decays with the QCD Factorization Approach

Junfeng Sun,¹Na Wang,^1,2Qin Chang,¹and Yueling Yang¹

Academic Editor: Michal Kreps

Received07 Jan 2015

Accepted17 Mar 2015

Published05 Apr 2015

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.

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 [13] and neglect the contributions from possible gluonium and compositions; that is, where and ; the mixing angle [13]. We assume that the vector mesons are ideally mixed; that is, and .

The transition form factors are defined as [38, 39] where −, and the condition of is required compulsorily to cancel the singularity at the pole .

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 [17]. 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 [6] 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.

From the numbers in Table 3, it is seen that different branching ratios for the , decays were obtained with different approaches in previous works, although the same coefficients are used. Much of the discrepancy comes from the different values of the transition form factors. If the same value of the form factor is used, then the disparities on branching ratios for the -dominated decays will be highly alleviated. For example, all previous predictions on will be about with the same form factor , which is generally in line with the QCDF estimation within uncertainties and also agrees with the recent LHCb measurement [12].

There is a hierarchical structure between the QCDF’s results on branching ratios for the and decays with the same final meson, for example, which differs from the previous results. There are two decisive factors. One is the kinematic factor. The phase space for the decays is more compressed than that for the decays, because the mass of the light pseudoscalar meson (except for the exotic meson) is generally less than the mass of the corresponding vector meson with the same valence quark components. The other is the dynamical factor. The orbital angular momentum for the final states is , while the orbital angular momentum is for the final states.

According to the CKM factors and the coefficients , there is another hierarchy of amplitudes among the QCDF’s branching ratios for the decays, which could be subdivided into different cases (see Table 4). The CKM-favored -dominated decays are expected to have the largest branching ratio, 10%, within the QCDF framework. In addition, the branching ratios for the Cabibbo favored , decays are also larger than 1%, which might be promisingly detected at experiments.

There are many uncertainties on the QCDF’s results. The first uncertainty from the CKM factors is small due to the high precision on Wolfenstein parameter with only 0.3% relative errors [1]. Large uncertainty comes from the renormalization scale, especially for the dominated , decays. In principle, the second uncertainty could be reduced by the inclusion of higher order corrections to hadronic matrix elements. It has been showed [77, 78] that tree amplitudes incorporating with the NNLO corrections are relatively less sensitive to the choice of scale than the NLO amplitudes. As aforementioned, large uncertainty mainly comes from hadron parameters, such as the transition form factors, which is expected to be cancelled from the rate of branching ratios. For example, Particularly, the relation of (18) might be used to give some information on the coefficients and to provide an interesting feasibility research on the validity of the QCDF approach for the charm quark decay. Finally, we would like to point out that many uncertainties from other factors, such as the final state interactions, which deserve the dedicated study, are not considered here. So one should not be too serious about the numbers in Table 3. Despite this, our results will still provide some useful information to experimental physicists; that is, the Cabibbo favored , , decays have large branching ratios 1%, which could be detected earlier.

4. Summary

In prospects of the potential meson at the LHCb experiments, accurate and thorough studies of the decays will be accessible very soon. The carefully theoretical study on the decays is urgently desiderated. In this paper, we concentrated on the nonfactorizable contributions to hadronic matrix elements within the QCDF framework, while the transition form factors are taken as nonperturbative inputs, which is different from previous studies. It is found that the branching ratios for the Cabibbo favored , , decays are very large and could be measured earlier by the running LHCb experiment in the forthcoming years.

Appendix

Decay Amplitudes

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant nos. 11475055, 11275057, and U1232101). N. Wang thanks the support from CCNU-QLPL Innovation Fund (QLPL201411). Q. Chang is also supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant no. 201317) and the Program for Science and Technology Innovation Talents in Universities of Henan Province (Grant no. 14HASTIT036).

References

K. Olive, K. Agashe, C. Amsler et al., “Review of particle physics,” Chinese Physics C, vol. 38, no. 9, Article ID 090001, 2014.
View at: Publisher Site | Google Scholar
N. Brambilla, M. Krämer, R. Mussa et al., “Heavy quarkonium physics,” http://arxiv.org/abs/hep-ph/0412158.
View at: Google Scholar
F. Abe, H. Akimoto, A. Akopian et al., “Observation of Bc mesons in $p \bar{p}$ collisions at $\sqrt{s} = 1.8$ TeV,” Physical Review D, vol. 58, no. 11, Article ID 112004, 29 pages, 1998.
View at: Publisher Site | Google Scholar
F. Abe, H. Akimoto, A. Akopian et al., “Observation of the $B_{c}$ meson in $p \bar{p}$ collisions at $\sqrt{s} = 1.8$ TeV,” Physical Review Letters, vol. 81, p. 2432, 1998.
View at: Publisher Site | Google Scholar
R. Aaij, C. A. Beteta, B. Adeva et al., “Observation of $B_{c}^{+} \to J / ψ D_{s}^{+}$ and $B_{c}^{+} \to J / ψ D_{s}^{* +}$ decays,” Physical Review D, vol. 87, Article ID 112012, 2013.
View at: Publisher Site | Google Scholar
R. Aaij, B. Adeva, M. Adinolfi et al., “Measurement of the lifetime of the $B_{c}^{+}$ meson using the $B_{c}^{+} \to J / ψ π^{+}$ decay mode,” Physics Letters B, vol. 742, pp. 29–37, 2015.
View at: Publisher Site | Google Scholar
I. P. Gouz, V. V. Kiselev, A. K. Likhoded, V. I. Romanovsky, and O. P. Yushchenko, “Prospects for the Bc studies at LHCb,” Physics of Atomic Nuclei, vol. 67, no. 8, pp. 1559–1570, 2004.
View at: Publisher Site | Google Scholar
A. Likhoded and A. Luchinsky, “Light hadron production in $B_{c} \to {B_{s}}^{(*)} + X$ decays,” Physical Review D, vol. 82, Article ID 014012, 2010.
View at: Publisher Site | Google Scholar
M. Lusignoli and M. Masetti, “B_c decays,” Zeitschrift für Physik C: Particles and Fields, vol. 51, no. 4, pp. 549–555, 1991.
View at: Publisher Site | Google Scholar
C. Chang and Y. Chen, “Decays of the $B_{c}$ meson,” Physical Review D, vol. 49, no. 7, pp. 3399–3411, 1994.
View at: Publisher Site | Google Scholar
S. S. Gershtein, V. V. Kiselev, A. K. Likhoded, and A. V. Tkabladze, “Reviews of topical problems: physics of Bc-mesons,” Physics-Uspekhi, vol. 38, no. 1, pp. 1–37, 1995.
View at: Publisher Site | Google Scholar
R. Aaij, B. Adeva, M. Adinolfi et al., “Observation of the decay $B_{c}^{+} \to B_{s}^{0} π^{+}$ ,” Physical Review Letters, vol. 111, Article ID 181801, 2013.
View at: Publisher Site | Google Scholar
T. Feldmann, P. Kroll, and B. Stech, “Mixing and decay constants of pseudoscalar mesons,” Physical Review D, vol. 58, Article ID 114006, 1998.
View at: Publisher Site | Google Scholar
P. Ball, “Theoretical update of pseudoscalar meson distribution amplitudes of higher twist: the nonsinglet case,” Journal of High Energy Physics, vol. 1999, no. 1, article 10, 1999.
View at: Google Scholar
P. Ball, V. M. Braun, and A. Lenz, “Higher-twist distribution amplitudes of the K meson in QCD,” Journal of High Energy Physics, vol. 2006, no. 5, article 4, 2006.
View at: Publisher Site | Google Scholar
P. Ball and G. Jones, “Twist-3 distribution amplitudes of $K^{*}$ and ϕ mesons,” Journal of High Energy Physics, vol. 2007, no. 3, article 069, 2007.
View at: Publisher Site | Google Scholar
D. Du and Z. Wang, “Predictions of the standard model for $B_{c}^{\pm}$ weak decays,” Physical Review D, vol. 39, no. 5, pp. 1342–1348, 1989.
View at: Publisher Site | Google Scholar
C.-H. Chang and Y.-Q. Chen, “Decays of the B_c meson,” Physical Review D, vol. 49, no. 7, Article ID 3399, 1994.
View at: Publisher Site | Google Scholar
A. El-Hady, J. Muñoz, and J. Vary, “Semileptonic and nonleptonic $B_{c}$ decays,” Physical Review D, vol. 62, Article ID 014019, 2000.
View at: Publisher Site | Google Scholar
D. Ebert, R. N. Faustov, and V. O. Galkin, “Weak decays of the $B_{c}$ meson to $B_{s}$ and B mesons in the relativistic quark model,” European Physical Journal C, vol. 32, no. 1, pp. 29–43, 2003.
View at: Publisher Site | Google Scholar
E. Hernández, J. Nieves, and J. Verde-Velasco, “Study of exclusive semileptonic and nonleptonic decays of $B_{\bar{c}}$ in a nonrelativistic quark model,” Physical Review D, vol. 74, Article ID 074008, 2006.
View at: Publisher Site | Google Scholar
E. Hernández, J. Nieves, and J. M. Verde-Velasco, “Study of semileptonic and nonleptonic decays of the Bc-Meson,” European Physical Journal A, vol. 31, no. 4, pp. 714–717, 2007.
View at: Publisher Site | Google Scholar
H. Choi and C. Ji, “Nonleptonic two-body decays of the $B_{c}$ meson in the light-front quark model and the QCD factorization approach,” Physical Review D, vol. 80, Article ID 114003, 2009.
View at: Publisher Site | Google Scholar
S. Naimuddin, S. Kar, M. Priyadarsini, N. Barik, and P. C. Dash, “Nonleptonic two-body $B_{c}$ -meson decays,” Physical Review D, vol. 86, Article ID 094028, 2012.
View at: Publisher Site | Google Scholar
M. Ivanov, J. Körner, and P. Santorelli, “Exclusive semileptonic and nonleptonic decays of the $B_{c}$ meson,” Physical Review D, vol. 73, Article ID 054024, 2006.
View at: Publisher Site | Google Scholar
V. V. Kiselev, A. E. Kovalsky, and A. K. Likhoded, “B_c decays and lifetime in QCD sum rules,” Nuclear Physics B, vol. 585, no. 1-2, pp. 353–382, 2000.
View at: Publisher Site | Google Scholar
V. V. Kiselev, A. E. Kovalsky, and A. K. Likhoded, “Bc-Meson decays and lifetime within QCD sum rules,” Physics of Atomic Nuclei, vol. 64, no. 10, pp. 1860–1875, 2001.
View at: Publisher Site | Google Scholar
I. P. Gouz, V. V. Kiselev, A. K. Likhoded, V. I. Romanovsky, and O. P. Yushchenko, “Prospects for the $B_{c}$ studies at LHCb,” Physics of Atomic Nuclei, vol. 67, no. 8, pp. 1559–1570, 2004.
View at: Publisher Site | Google Scholar
J. Sun, Y. Yang, Q. Chang, and G. Lu, “Phenomenological study of the $B_{c} \to B P$ , BV decays with perturbative QCD approach,” Physical Review D, vol. 89, no. 11, Article ID 114019, 17 pages, 2014.
View at: Publisher Site | Google Scholar
J. Sun, Y. Yang, and G. Lu, “Study of the $B_{c} \to B_{s}$ π decay with the perturbative QCD approach,” Science China Physics, Mechanics & Astronomy, vol. 57, no. 10, pp. 1891–1897, 2014.
View at: Publisher Site | Google Scholar
M. Beneke and G. Buchalla, “ $B_{c}$ meson lifetime,” Physical Review D, vol. 53, article 4991, 1996.
View at: Publisher Site | Google Scholar
C. Chang, S. Chen, T. Feng, and X. Li, “Lifetime of the $B_{c}$ meson and some relevant problems,” Physical Review D, vol. 64, Article ID 014003, 2001.
View at: Publisher Site | Google Scholar
C.-H. Chang, S.-L. Chen, T.-F. Feng, and X.-Q. Li, “Study of the B_c-Meson Lifetime,” Communications in Theoretical Physics, vol. 35, no. 1, p. 57, 2001.
View at: Publisher Site | Google Scholar
M. Lusignoil and M. Masetti, “ $B_{c}$ decays,” Zeitschrift für Physik C: Particles and Fields, vol. 51, no. 4, pp. 549–555, 1991.
View at: Publisher Site | Google Scholar
A. Y. Anisimov, P. Y. Kulikov, I. M. Narodetskii, and K. A. Ter-Martirosyan, “Exclusive and inclusive decays of the Bc meson in the light-front ISGW model,” Physics of Atomic Nuclei, vol. 62, no. 10, pp. 1739–1753, 1999.
View at: Google Scholar
R. Dhir, N. Sharma, and R. Verma, “Flavor dependence of $B_{c}$ meson form factors and $B_{c} \to P P$ decays,” Journal of Physics G: Nuclear and Particle Physics, vol. 35, no. 8, Article ID 085002, 2008.
View at: Publisher Site | Google Scholar
R. Dhir and R. Verma, “B_c meson form factors and $B_{c} \to P V$ decays involving flavor dependence of transverse quark momentum,” Physical Review D, vol. 79, Article ID 034004, 2009.
View at: Publisher Site | Google Scholar
M. Wirbel, B. Stech, and M. Bauer, “Exclusive semileptonic decays of heavy mesons,” Zeitschrift für Physik C Particles and Fields, vol. 29, no. 4, pp. 637–642, 1985.
View at: Publisher Site | Google Scholar
M. Bauer, B. Stech, and M. Wirbel, “Exclusive non-leptonic decays of D-, D_s- and B-mesons,” Zeitschrift für Physik C Particles and Fields, vol. 34, no. 1, pp. 103–115, 1987.
View at: Publisher Site | Google Scholar
N. Isgur, D. Scora, B. Grinstein, and M. B. Wise, “Semileptonic B and D decays in the quark model,” Physical Review D, vol. 39, no. 3, pp. 799–818, 1989.
View at: Publisher Site | Google Scholar
J. Liu and K. Chao, “ $B_{c}$ meson weak decays and CP violation,” Physical Review D, vol. 56, no. 7, pp. 4133–4145, 1997.
View at: Publisher Site | Google Scholar
P. Colangelo and F. de Fazio, “Using heavy quark spin symmetry in semileptonic B_c decays,” Physical Review D, vol. 61, no. 3, Article ID 034012, 11 pages, 2000.
View at: Publisher Site | Google Scholar
R. C. Verma and A. Sharma, “Quark diagram analysis of weak hadronic decays of the $B_{c}^{+}$ meson,” Physical Review D, vol. 65, Article ID 114007, 2002.
View at: Publisher Site | Google Scholar
C. Chang and H. Li, “Three-scale factorization theorem and effective field theory: analysis of nonleptonic heavy meson decays,” Physical Review D, vol. 55, p. 5577, 1997.
View at: Publisher Site | Google Scholar
T.-W. Yeh and H.-N. Li, “Factorization theorems, effective field theory, and nonleptonic heavy meson decays,” Physical Review D, vol. 56, pp. 1615–1631, 1997.
View at: Publisher Site | Google Scholar
Y. Keum, H. Li, and A. Sanda, “Fat penguins and imaginary penguins in perturbative QCD,” Physics Letters B, vol. 504, no. 1-2, pp. 6–14, 2001.
View at: Publisher Site | Google Scholar
Y.-Y. Keum and H.-N. Li, “Nonleptonic charmless B decays: factorization versus perturbative QCD,” Physical Review D, vol. 63, Article ID 074006, 2001.
View at: Publisher Site | Google Scholar
C. Lü, K. Ukai, and M. Yang, “Branching ratio and CP violation of $\vec{B} π π$ decays in the perturbative QCD approach,” Physical Review D, vol. 63, Article ID 074009, 2001.
View at: Publisher Site | Google Scholar
C.-D. Lü and M.-Z. Yang, “ $B \to π ρ$ , $π ω$ decays in perturbative QCD approach,” The European Physical Journal C, vol. 23, no. 2, pp. 275–287, 2002.
View at: Publisher Site | Google Scholar
C. Bauer, S. Fleming, and M. Luke, “Summing Sudakov logarithms in $B \to X_{s}$ γ in effective field theory,” Physical Review D, vol. 63, Article ID 014006, 2001.
View at: Google Scholar
C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, “An effective field theory for collinear and soft gluons: heavy to light decays,” Physical Review D, vol. 63, Article ID 114020, 2001.
View at: Publisher Site | Google Scholar
C. W. Bauer and I. W. Stewart, “Invariant operators in collinear effective theory,” Physics Letters B, vol. 516, no. 1-2, pp. 134–142, 2001.
View at: Publisher Site | Google Scholar
C. W. Bauer, D. Pirjol, and I. W. Stewart, “Soft-collinear factorization in effective field theory,” Physical Review D, vol. 65, Article ID 054022, 2002.
View at: Publisher Site | Google Scholar
C. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, “Hard scattering factorization from effective field theory,” Physical Review D, vol. 66, no. 1, Article ID 014017, 23 pages, 2002.
View at: Publisher Site | Google Scholar
M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, “Soft-collinear effective theory and heavy-to-light currents beyond leading power,” Nuclear Physics B, vol. 643, no. 1–3, pp. 431–476, 2002.
View at: Publisher Site | Google Scholar
M. Beneke and T. Feldmann, “Multipole-expanded soft-collinear effective theory with non-Abelian gauge symmetry,” Physics Letters B, vol. 553, no. 3-4, pp. 267–276, 2003.
View at: Publisher Site | Google Scholar
M. Beneke and T. Feldmann, “Factorization of heavy-to-light form factors in soft-collinear effective theory,” Nuclear Physics B, vol. 685, no. 1–3, pp. 249–296, 2004.
View at: Publisher Site | Google Scholar
M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, “QCD factorization for $B \to π π$ decays: strong phases and CP violation in the heavy quark limit,” Physical Review Letters, vol. 83, no. 10, pp. 1914–1917, 1999.
View at: Publisher Site | Google Scholar
M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, “QCD factorization for exclusive non-leptonic B-meson decays: general arguments and the case of heavy-light final states,” Nuclear Physics B, vol. 591, no. 1-2, pp. 313–418, 2000.
View at: Publisher Site | Google Scholar
M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, “QCD factorization in $B \to π K$ , $π π$ decays and extraction of Wolfenstein parameters,” Nuclear Physics B, vol. 606, no. 1-2, pp. 245–321, 2001.
View at: Publisher Site | Google Scholar
D. Du, D. Yang, and G. Zhu, “Infrared divergence and twist-3 distribution amplitudes in QCD factorization for B→PP,” Physics Letters B, vol. 509, no. 3-4, pp. 263–272, 2001.
View at: Publisher Site | Google Scholar
D. Du, D. Yang, and G. Zhu, “Analysis of the decays $B \to π π$ and πK with QCD factorization in the heavy quark limit,” Physics Letters B, vol. 488, no. 1, pp. 46–54, 2000.
View at: Publisher Site | Google Scholar
D. Du, D. Yang, and G. Zhu, “QCD factorization for $\vec{B} PP$ ,” Physical Review D, vol. 64, Article ID 014036, 2001.
View at: Publisher Site | Google Scholar
G. Buchalla, A. Buras, and M. Lautenbacher, “Weak decays beyond leading logarithms,” Reviews of Modern Physics, vol. 68, p. 1125, 1996.
View at: Publisher Site | Google Scholar
A. J. Buras, “Weak hamiltonian, CP violation and rare decays,” http://arxiv.org/abs/hep-ph/9806471.
View at: Google Scholar
D. Du, H. Gong, J. Sun, D. Yang, and G. Zhu, “Phenomenological analysis of $\vec{B}$ PP decays with QCD factorization,” Physical Review D, vol. 65, Article ID 074001, 2002.
View at: Publisher Site | Google Scholar
D. Du, H. Gong, J. Sun, D. Yang, and G. Zhu, “Phenomenological analysis of charmless decays $B \to PV$ with QCD factorization,” Physical Review D, vol. 65, Article ID 094025, 2002, Erratum in Physical Review D, vol. 66, Article ID 079904, 2002.
View at: Publisher Site | Google Scholar
J. Sun, G. Zhu, and D. Du, “Phenomenological analysis of charmless decays $B_{s} \to P P, P V$ with QCD factorization,” Physical Review D, vol. 68, Article ID 054003, 2003.
View at: Publisher Site | Google Scholar
M. Beneke and M. Neubert, “QCD factorization for $B \to P P$ and $B \to P V$ decays,” Nuclear Physics B, vol. 675, no. 1-2, pp. 333–415, 2003.
View at: Publisher Site | Google Scholar
M. Beneke, J. Rohrer, and D. Yang, “Branching fractions, polarisation and asymmetries of $B \to VV$ decays,” Nuclear Physics B, vol. 774, no. 1–3, pp. 64–101, 2007.
View at: Publisher Site | Google Scholar
X. Li and Y. Yang, “Reexamining charmless $B \to P V$ decays in the QCD factorization approach,” Physical Review D, vol. 73, no. 11, Article ID 114027, 24 pages, 2006.
View at: Publisher Site | Google Scholar
H.-Y. Cheng and K.-C. Yang, “Branching ratios and polarization in $B \to V V$ , VA, AA decays,” Physical Review D, vol. 78, Article ID 094001, 2008, Erratum in Physical Review D, vol. 79, Article ID 039903, 2009.
View at: Publisher Site | Google Scholar
H.-Y. Cheng and C.-K. Chua, “Resolving $B - C P$ puzzles in QCD factorization,” Physical Review D, vol. 80, Article ID 074031, 2009.
View at: Publisher Site | Google Scholar
H. Cheng and C. Chua, “Revisiting charmless hadronic B_u,d decays in QCD factorization,” Physical Review D, vol. 80, no. 11, Article ID 114008, 33 pages, 2009.
View at: Publisher Site | Google Scholar
H. Cheng and C. Chua, “QCD factorization for charmless hadronic B_s decays revisited,” Physical Review D, vol. 80, no. 11, Article ID 114026, 25 pages, 2009.
View at: Publisher Site | Google Scholar
H. Cheng and C. Chua, “Charmless hadronic B decays into a tensor meson,” Physical Review D, vol. 83, Article ID 034001, 2011.
View at: Publisher Site | Google Scholar
G. Bell, “NNLO vertex corrections in charmless hadronic B decays: real part,” Nuclear Physics B, vol. 822, no. 1-2, pp. 172–200, 2009.
View at: Publisher Site | Google Scholar
M. Beneke, T. Huber, and X.-Q. Li, “NNLO vertex corrections to non-leptonic B decays: tree amplitudes,” Nuclear Physics B, vol. 832, no. 1-2, pp. 109–151, 2010.
View at: Publisher Site | Google Scholar

Copyright

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 SCOAP³.

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