Research Article119861119888rarr 119861119875 119861119881Decays with the QCD Factorization Approach

Junfeng Sun1 Na Wang12 Qin Chang1 and Yueling Yang1

1 Institute of Particle and Nuclear Physics Henan Normal University Xinxiang 453007 China2Institute of Particle and Key Laboratory of Quark and Lepton Physics Central China Normal University Wuhan 430079 China

Correspondence should be addressed to Qin Chang changqinhtucn

Received 7 January 2015 Accepted 17 March 2015

Academic Editor Michal Kreps

Copyright copy 2015 Junfeng Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We studied the nonleptonic 119861119888rarr 119861119875 119861119881 decays with the QCD factorization approach It is found that the Cabibbo favored

processes of 119861119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 are the promising decay channels with branching ratio larger than 1 which should be observed

earlier by the LHCb collaboration

1 Introduction

The119861119888meson is the ground pseudoscalarmeson of the 119887119888 sys-

tem [1] Compared with the heavy unflavored charmonium119888119888 and bottomonium 119887119887 the 119861

119888meson is unique in some

respects (1)Heavy quarkonia could be created in the parton-parton process 119894119895 rarr 119876119876 at the order of 1205722

119904(where 119894119895 = 119892119892

or 119902119902 119876 = 119887 119888) while the production probability for the119861119888meson is at least at the order of 1205724

119904via 119894119895 rarr 119861

(lowast)+

119888+ 119887119888

where the gluon-gluon fusion mechanism is dominant atTevatron and LHC [2] The 119861

119888meson is difficult to produce

experimentally but it was observed for the first time via thesemileptonic decaymode119861

119888rarr 119869120595ℓ] in119901119901 collisions by the

CDF collaboration in 1998 [3 4] which showed the realisticpossibility of experimental study of the 119861

119888meson One of

the best measurements on the mass and lifetime of the 119861119888

meson is reported recently by the LHCb collaboration119898119861119888

=

627628plusmn144plusmn036MeV [5] and 120591119861119888

= 5134plusmn110plusmn57 fs [6]With the running of the LHC the 119861

119888meson has a promising

prospect It is estimated that one could expect some 1010 119861119888

events at the high-luminosity LHC experiments per year [78] The studies on the 119861

119888meson have entered a new precision

era (2) For charmonium and bottomonium the strong andelectromagnetic interactions are mainly responsible for anni-hilation of the 119876119876 quark pair into final states The 119861

119888meson

carrying nonzero flavor number 119861 = 119862 = plusmn1 and lying below

the 119861119863 meson pair threshold can decay only via the weakinteraction which offers an ideal sample to investigate theweak decay mechanism of heavy flavors that is inaccessibleto both charmonium and bottomonium The 119861

119888weak decay

provides great opportunities to investigate the perturbativeand nonperturbative QCD final state interactions and soforth

With respect to the heavy-light 119861119906119889119904

mesons the doublyheavy 119861

119888meson has rich decay channels because of its

relatively large mass and that both 119887 and 119888 quarks can decayindividually The decay processes of the 119861

119888meson can be

divided into the following three classes [2 9ndash11] (1) the 119888quark decays with the 119887 quark as a spectator (2) the 119887 quarkdecays with the 119888 quark as a spectator (3) the 119887 and 119888 quarksannihilate into a virtual 119882 boson with the ratios of sim7020 and 10 respectively [2] Up to now the experimentalevidences of pure annihilation decay mode [class (3)] are stillnothing The 119887 rarr 119888 transition belonging to the class (2)offers a well-constructed experimental structure of charmo-nium at the Tevatron and LHC Although the detection ofthe 119888 quark decay is very challenging to experimentalists theclear signal of the 119861

119888rarr 119861

119904120587 decay is presented by the LHCb

group using the 119861119904rarr 119863

119904120587 and 119861

119904rarr 119869120595120601 channels with

statistical significance of 77120590 and 61120590 respectively [12]Anticipating the forthcoming accurate measurements on

the 119861119888meson at hadron colliders and the lionrsquos share of the 119861

119888

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 104378 10 pageshttpdxdoiorg1011552015104378

2 Advances in High Energy Physics

decay width from the 119888 quark decay [31ndash33] many theoreticalpapers were devoted to the study of the 119861

119888rarr 119861119875 119861119881 decays

(where 119875 and 119881 denote the 119878119880(3) ground pseudoscalar andvectormesons resp) such as [17 34ndash37]with the BSWmodel[38 39] or IGSWmodel [40] [18 19 41] based on the Bethe-Salpeter (BS) equation [20ndash24 42] with potential models[25] with constituent quark model [26ndash28] with QCD sumrules [43] with the quark diagram scheme [29 30] with theperturbativeQCD approach (pQCD) [44ndash49] and so onTheprevious predictions on the branching ratios for the 119861

119888rarr

119861119875 119861119881 decays are collected in Table 3 The discrepancies ofprevious investigations arise mainly from the different modelassumptions Recently several phenomenological methodshave been fully developed to cope with the hadronic matrixelements and successfully applied to the nonleptonic 119861 decaysuch as the pQCD approach [44ndash49] based on the 119896

119879

factorization scheme the soft-collinear effective theory [50ndash57] and the QCD-improved factorization (QCDF) approach[58ndash63] based on the collinear approximation and powercountering rules in the heavy quark limits In this paper wewill study the119861

119888rarr 119861119875 119861119881 decays with the QCDF approach

to provide a ready reference to the existing and upcomingexperiments

This paper is organized as follows In Section 2 we willpresent the theoretical framework and the amplitudes forthe 119861

119888rarr 119861119875 119861119881 decays within the QCDF framework

Section 3 is devoted to numerical results and discussionFinally Section 4 is our summation

2 Theoretical Framework

21The EffectiveHamiltonian The low energy effectiveHam-iltonian responsible for the nonleptonic bottom-conserving119861119888rarr 119861119875 119861119881 decays constructed by means of the

operator product expansion and the renormalization group(RG) method is usually written in terms of the four-quarkinteractions [64 65] Consider

119867eff

=

119866119865

radic2

119881119906119887119881lowast

119888119887[119862

119886

1(120583)119876

119886

1(120583) + 119862

119886

2(120583)119876

119886

2(120583)]

+ sum

11990211199022=119889119904

1198811199061199021

119881lowast

1198881199022

[1198621(120583)119876

1(120583) + 119862

2(120583)119876

2(120583)]

+ sum

1199023=119889119904

1198811199061199023

119881lowast

1198881199023

10

sum

119896=3

119862119896(120583)119876

119896(120583)

+ hc

(1)

where the Fermi coupling constant 119866119865≃ 1166 times 10

minus5 GeVminus2

[1]11987612119876119886

12 and119876

310are the relevant local tree annihila-

tion and penguin four-quark operators respectively whichgovern the decays in question The Cabibbo-Kobayashi-Maskawa (CKM) factor 119881

119906119902119894

119881lowast

119888119902119895

and Wilson coefficients 119862119894

describe the coupling strength for a given operator

Using the unitarity of the CKM matrix there is a largecancellation of the CKM factors

119881119906119889119881lowast

119888119889+ 119881

119906119904119881lowast

119888119904= minus119881

119906119887119881lowast

119888119887sim O (120582

5

) (2)

where the Wolfenstein parameter 120582 = sin 120579119888= 0225 37(61)

[1] and 120579119888is theCabibbo angleHence comparedwith the tree

contributions the contributions of annihilation and penguinoperators are strongly suppressed by the CKM factor If the119862119875-violating asymmetries that are expected to be very smalldue to the small weak phase difference for 119888 quark decay areprescinded from the present consideration then the penguinand annihilation contributions could be safely neglectedThelocal tree operators 119876

12in (1) are expressed as follows

1198761= [119902

2120572120574120583(1 minus 120574

5) 119888120572] [119906

120573120574120583

(1 minus 1205745) 119902

1120573]

1198762= [119902

2120572120574120583(1 minus 120574

5) 119888120573] [119906

120573120574120583

(1 minus 1205745) 119902

1120572]

(3)

where 120572 and 120573 are the 119878119880(3) color indicesThe Wilson coefficients 119862

119894(120583) summarize the physics

contributions from scales higher than 120583 They are calculablewith the RG improved perturbation theory and have properlybeen evaluated to the next-to-leading order (NLO) [64 65]They can be evolved from a higher scale 120583 sim O(119898

119882) down to

a characteristic scale 120583 sim O(119898119888) with the functions including

the flavor thresholds [64 65]

(120583) = 1198804(120583119898

119887)119872 (119898

119887) 119880

5(119898

119887 119898

119882) (119898

119882) (4)

where 119880119891(120583119891 120583

119894) is the RG evolution matrix converting

coefficients from the scale 120583119894to 120583

119891 and 119872(120583) is the quark

thresholdmatchingmatrixThe expressions of119880119891(120583119891 120583

119894) and

119872(120583) can be found in [64 65] The numerical values ofLO and NLO 119862

12with the naive dimensional regularization

scheme are listed in Table 1 The values of NLO Wilsoncoefficients in Table 1 are consistent with those given by [6465] where a trick with ldquoeffectiverdquo number of active flavors119891 =415 rather than formula (4) is used

To obtain the decay amplitudes the remaining work ishow to accurately evaluate the hadronic matrix elements⟨119861119872|119876

119894(120583)|119861

119888⟩ which summarize the physics contributions

from scales lower than 120583 Since the hadronic matrix elementsinvolve long distance contributions one is forced to useeither nonperturbative methods such as lattice calculationsand QCD sum rules or phenomenological models relyingon some assumptions Consequently it is very unfortunatethat hadronic matrix elements cannot be reliably calculatedat present and that the most intricate part and the dominanttheoretical uncertainties in the decay amplitudes reside in thehadronic matrix elements

22 Hadronic Matrix Elements Phenomenologically basedon the power counting rules in the heavy quark limit Benekeet al proposed that the hadronic matrix elements could bewritten as the convolution integrals of hard scattering kernelsand the light cone distribution amplitudes with the QCDFmaster formula [58ndash63] The QCDF approach is widelyapplied to nonleptonic 119861 decays and it works well [66ndash76]

Advances in High Energy Physics 3

Table 1 The numerical values of the Wilson coefficients and the effective coefficients for 119861119888rarr 119861120587 decay where119898

119888= 1275 plusmn 0025GeV [1]

120583

LO NLO QCDF1198621

1198622

1198621

1198622

Re (1198861) Im (119886

1) Re (119886

2) Im (119886

2)

08119898119888

1334 minus0587 1274 minus0503 1270 0096 minus0450 minus0218119898119888

1275 minus0503 1222 minus0424 1216 0068 minus0361 minus017312119898

1198881239 minus0449 1189 minus0373 1184 0054 minus0306 minus0148

which encourage us to apply the QCDF approach to thestudy of 119861

119888rarr 119861119875 119861119881 decays Since the spectator is the

heavy 119887 quark who is almost always on shell the virtuality ofthe gluon linked with the spectator should be sim O(Λ2QCD)The dominant behavior of the 119861

119888rarr 119861 transition form

factors and the contributions of hard spectator scatteringinteractions are governed by soft processes According to thebasic idea of the QCDF approach [69 70] the hard and softcontributions to the form factors entangle with each otherand cannot be identified reasonably so the physical formfactors are used as the inputs The hard spectator scatteringcontributions are power suppressed in the heavy quark limitFinally the hadronic matrix elements can be written as

⟨119861119872100381610038161003816100381611987612

1003816100381610038161003816119861119888⟩ = sum

119894

119865119861119888rarr119861

119894int119889119909119867

119894(119909)Φ

119872(119909)

prop sum

119894

119865119861119888rarr119861

1198941198911198721 + 120572

1199041199031+ sdot sdot sdot

(5)

where 119865119861119888rarr119861

119894is the transition form factor and Φ

119872(119909) is the

light-cone distribution amplitudes of the emitted meson 119872with the decay constant119891

119872The hard scattering kernels119867

119894(119909)

are computable order by order with the perturbation theoryin principle At the leading order 1205720

119904 119867

119894(119909) = 1 that is

the convolution integral of (5) results in the meson decayconstant The hadronic matrix elements are parameterizedby the product of form factors and decay constants whichare real and renormalization scale independent One goesback to the simple ldquonaive factorizationrdquo (NF) scenario Atthe order 120572

119904and higher orders the information of strong

phases and the renormalization scale dependence of hadronicmatrix elements could be partly recuperated Combined thenonfactorizable contributions with the Wilson coefficientsthe scale independent effective coefficients at the order 120572

119904can

be obtained [58ndash63] as follows

1198861= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO2119881

1198862= 119862

NLO2

+

1

119873119888

119862NLO1

+

120572119904

4120587

119862119865

119873119888

119862LO1119881

(6)

where the expressions of vertex corrections are [58ndash63]

119881 = 6 log(1198982

119888

1205832) minus 18 minus (

1

2

+ 1198943120587) 119886119872

0

+ (

11

2

minus 1198943120587) 119886119872

1minus

21

20

119886119872

2+ sdot sdot sdot

(7)

with the twist-2 quark-antiquark distribution amplitudes ofpseudoscalar 119875 and longitudinally polarized vector 119881mesonin terms of Gegenbauer polynomials [14ndash16] One has

120601119872(119909) = 6119909119909

infin

sum

119899=0

119886119872

11989911986232

119899(119909 minus 119909) (8)

where 119909 = 1 minus 119909 119886119872119899

is the Gegenbauer moment and 1198861198720equiv 1

From the numbers in Table 1 it is found that (1) for thecoefficient 119886

1the nonfactorizable contributions accompanied

by the Wilson coefficient 1198622can provide ge10 enhancement

compared with the NFrsquos result and a relatively small strongphase le 5

∘ (2) for the coefficient 1198862 the nonfactorizable

contributions assisted with the largeWilson coefficient1198621are

significant In addition a relatively large strong phasesim minus155∘is obtained (3) the QCDFrsquos values of 119886

12agree basically with

the real coefficients 1198861≃ 120 and 119886

2≃ minus0317 which are

used by previous studies on the 119861119888rarr 119861119875 119861119881 decays in

[17 18 20ndash28 34ndash37 42] but with more information on thestrong phases

23 Decay Amplitudes Within the QCDF framework theamplitudes for 119861

119888rarr 119861119872 decays are expressed as

A (119861119888997888rarr 119861119872) = ⟨119861119872

1003816100381610038161003816Heff

1003816100381610038161003816119861119888⟩

=

119866119865

radic2

1198811199061199021

119881lowast

1198881199022

119886119894⟨119872100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119861

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816119861119888⟩

(9)

The matrix elements of current operators are defined as

⟨119875 (119901)10038161003816100381610038161199021120574120583

(1 minus 1205745) 119902

2

10038161003816100381610038160⟩ = minus119894119891

119875119901120583

⟨119881 (120598 119901)10038161003816100381610038161199021120574120583

(1 minus 1205745) 119902

2

10038161003816100381610038160⟩ = 119891

119881119898119881120598120583

(10)

where 119891119875and 119891

119881are the decay constants of pseudoscalar 119875

and vector119881mesons respectively119898119881and 120598 denote the mass

and polarization of vector meson respectivelyFor the mixing of physical pseudoscalar 120578 and 1205781015840 meson

we adopt the quark-flavor basis description proposed in [13]and neglect the contributions from possible gluonium and 119888119888compositions that is

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (11)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393plusmn10)∘ [13]We assume that the vectormesons are ideally

mixed that is 120596 = (119906119906 + 119889119889)radic2 and 120601 = 119904119904

4 Advances in High Energy Physics

Table 2 The numerical values of input parameters

Wolfenstein parameters120582 = 022537 plusmn 000061 [1] 119860 = 0814

+0023

minus0024[1]

120588 = 0117 plusmn 0021 [1] 120578 = 0353 plusmn 0013 [1]Decay constant of mesons

119891120587= 13041 plusmn 020MeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119891120578119902

= (107 plusmn 002)119891120587[13] 119891

120578119904

= (134 plusmn 006)119891120587[13]

119891120588= 216 plusmn 3MeV [14ndash16] 119891

120596= 187 plusmn 5MeV [14ndash16]

119891119870lowast = 220 plusmn 5MeV [14ndash16]

Gegenbauer moments at the scale 120583 = 1 GeV119886120587

1= 119886

120578119902

1= 119886

120578119904

1= 0 [14ndash16] 119886

120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [14ndash16]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [14ndash16] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [14ndash16]

119886

1120588= 119886

1120596= 0 [14ndash16] 119886

2120588= 119886

2120596= 015 plusmn 007 [14ndash16]

119886

1119870

lowast = minus119886

1119870lowast = 003 plusmn 002 [14ndash16] 119886

2119870lowast = 119886

2119870

lowast = 011 plusmn 009 [14ndash16]

The transition form factors are defined as [38 39]

⟨119861 (119896)1003816100381610038161003816119902120574

120583

(1 minus 1205745) 1198881003816100381610038161003816119861119888(119901)⟩

= [119901 + 119896 minus

1198982

119861119888

minus 1198982

119861

1199022

119902]

120583

119865119861119888rarr119861

1(119902

2

)

+

1198982

119861119888

minus 1198982

119861

1199022

119902120583

119865119861119888rarr119861

0(119902

2

)

(12)

where 119902 = 119901 minus 119896 and the condition of 119865119861119888rarr119861

0(0) = 119865

119861119888rarr119861

1(0)

is required compulsorily to cancel the singularity at the pole1199022

= 0For the119861

119888rarr 119861 transition form factors since the velocity

of the recoiled 119861 meson is very low in the rest frame of the119861119888meson the wave functions of 119861 and 119861

119888mesons overlap

strongly It is believed that the form factors 119865119861119888rarr119861

01should

be close to the result using the nonrelativistic harmonicoscillator wave functions with the BSWmodel [17] Consider

119865119861119888rarr119861

01≃ (

2119898119861119888

119898119861

1198982

119861119888

+ 1198982

119861

)

12

≃ 099 (13)

The flavor symmetry breaking effects on the form factors areneglectable in (13) For simplification we take 119865119861119888rarr119861

119906119889119904

01= 10

in our numerical calculation to give a rough estimation

3 Numerical Results and Discussions

The branching ratios of nonleptonic two-body 119861119888decays in

the rest frame of the 119861119888meson can be written as

B119903 (119861119888997888rarr 119861119872) =

120591119861119888

8120587

119901

1198982

119861119888

1003816100381610038161003816A (119861

119888997888rarr 119861119872)

1003816100381610038161003816

2

(14)

where the lifetime of the 119861119888meson 120591

119861119888

= 5134 plusmn 110 plusmn 57 fs[6] and 119901 is the common momentum of final particles Thedecay amplitudesA(119861

119888rarr 119861119872) are listed in the Appendix

The input parameters in our calculation including theCKM Wolfenstein parameters decay constants and Gegen-bauer moments of distribution amplitudes in (8) are col-lected in Table 2 If not specified explicitly we will take theircentral values as the default inputs Our numerical results onthe 119862119875-averaged branching ratios are presented in Table 3where theoretical uncertainties of the ldquoQCDFrdquo column comefrom the CKM parameters the renormalization scale 120583 =(1 plusmn 02)119898

119888 decay constants and Gegenbauer moments

respectively For comparison previous results calculated withthe fixed coefficients 119886

1≃ 122 and 119886

2≃ minus04 are also listed

There are some comments on the branching ratios(1) From the numbers in Table 3 it is seen that different

branching ratios for the 119861119888rarr 119861119875 119861119881 decays were obtained

with different approaches in previous works although thesame coefficients 119886

12are used Much of the discrepancy

comes from the different values of the transition formfactors If the same value of the form factor is used thenthe disparities on branching ratios for the 119886

1-dominated 119861

119888

decays will be highly alleviated For example all previouspredictions on B119903(119861

119888rarr 119861

119904120587) will be about 10 with the

same form factor119865119861119888rarr119861119904

0≃ 10 which is generally in linewith

the QCDF estimation within uncertainties and also agreeswith the recent LHCb measurement [12](2)There is a hierarchical structure between the QCDFrsquos

results on branching ratios for the 119861119888rarr 119861119875 and 119861119881 decays

with the same final 119861119902meson for example

B119903 (119861119888997888rarr 119861

119902120587) gtB119903 (119861

119888997888rarr 119861

119902120588)

B119903 (119861119888997888rarr 119861

119902119870) ≳ 5B119903 (119861

119888997888rarr 119861

119902119870lowast

)

(15)

which differs from the previous resultsThere are two decisivefactors One is the kinematic factor The phase space for the119861119888rarr 119861119881 decays is more compressed than that for the 119861

119888rarr

119861119875 decays because the mass of the light pseudoscalar meson(except for the exotic 1205781015840meson) is generally less than themassof the corresponding vector meson with the same valencequark components The other is the dynamical factor The

Advances in High Energy Physics 5

Table3Th

e119862119875-averagedbranchingratio

sfor

the119861

119888rarr119861119875119861119881decays

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr1198610 119904120587+

1-a10times10minus1

68times10minus2

18times10minus2

29times10minus2

40times10minus2

43times10minus2

(43times10minus2

)13times10minus1

46times10minus2

19times10minus1

88times10minus2

(113+000+011+001

minus000minus006minus001)times10minus1

119861119888rarr1198610 119904119870+

1-b76times10minus3

49times10minus3

20times10minus3

24times10minus3

33times10minus3

33times10minus3

(33times10minus3

)85times10minus3

34times10minus3

12times10minus2

52times10minus3

(741+004+070+009

minus004minus039minus009)times10minus3

119861119888rarr1198610 119904120588+

1-a63times10minus2

52times10minus2

46times10minus2

16times10minus2

27times10minus2

30times10minus2

(27times10minus2

)11times10minus1

27times10minus2

84times10minus2

32times10minus2

(444+000+041+013

minus000minus023minus013)times10minus2

119861119888rarr1198610 119904119870lowast+

1-b32times10minus4

12times10minus3

35times10minus5

15times10minus4

80times10minus5

(71times10minus5

)40times10minus4

13times10minus4

97times10minus5

(125+001+012+006

minus001minus007minus006)times10minus4

119861119888rarr1198610 1198891205870

1-b74times10minus3

38times10minus3

12times10minus3

12times10minus3

13times10minus3

18times10minus3

(15times10minus3

)84times10minus3

24times10minus3

12times10minus2

69times10minus3

(783+004+073+004

minus004minus041minus004)times10minus3

119861119888rarr1198610 119889119870+

1-c30times10minus4

12times10minus4

10times10minus4

11times10minus4

15times10minus4

(12times10minus4

)59times10minus4

18times10minus4

81times10minus4

44times10minus4

(529+006+050+007

minus006minus028minus007)times10minus4

119861119888rarr1198610 119889120588+

1-b83times10minus3

69times10minus3

33times10minus3

15times10minus3

16times10minus3

22times10minus3

(17times10minus3

)14times10minus2

24times10minus3

11times10minus2

43times10minus3

(532+003+049+015

minus003minus028minus015)times10minus3

119861119888rarr1198610 119889119870lowast+

1-c21times10minus4

15times10minus4

46times10minus5

44times10minus5

49times10minus5

(37times10minus5

)35times10minus4

57times10minus5

17times10minus4

83times10minus5

(106+001+010+005

minus001minus006minus005)times10minus4

119861119888rarr119861+ 119906119870

0

2-a

21times10minus2

12times10minus2

49times10minus3

42times10minus3

44times10minus3

60times10minus3

(49times10minus3

)23times10minus2

68times10minus3

36times10minus2

22times10minus3

(197+000+111+005

minus000minus054minus005)times10minus2

119861119888rarr119861+ 1199061198700

2-c

16times10minus5

(13times10minus5

)63times10minus6

(571+006+320+015

minus006minus158minus014)times10minus5

119861119888rarr119861+ 119906119870

lowast0

2-a

78times10minus3

85times10minus3

58times10minus3

16times10minus3

16times10minus3

19times10minus3

(14times10minus3

)13times10minus2

21times10minus3

80times10minus3

20times10minus4

(372+000+209+021

minus000minus103minus020)times10minus3

119861119888rarr119861+ 119906119870lowast0

2-c

50times10minus6

(37times10minus6

)56times10minus7

(107+001+060+006

minus001minus030minus006)times10minus5

119861119888rarr119861+ 1199061205870

2-b

40times10minus4

21times10minus4

64times10minus5

62times10minus5

67times10minus5

97times10minus5

(81times10minus5

)45times10minus4

13times10minus4

66times10minus4

52times10minus5

(423+002+237+007

minus002minus117minus007)times10minus4

119861119888rarr119861+ 1199061205880

2-b

44times10minus4

37times10minus4

17times10minus4

87times10minus5

89times10minus5

12times10minus4

(94times10minus5

)74times10minus4

13times10minus4

55times10minus4

18times10minus5

(286+001+160+010

minus001minus079minus010)times10minus4

119861119888rarr119861+ 119906120596

2-b

41times10minus4

90times10minus5

(70times10minus5

)13times10minus5

(205+001+115+012

minus001minus057minus012)times10minus4

6 Advances in High Energy Physics

Table3Con

tinued

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr119861+ 119906120578

50times10minus4

(41times10minus4

)14times10minus4

(146+001+082+013

minus001minus040minus012)times10minus3

119861119888rarr119861+ 1199061205781015840

67times10minus6

(56times10minus6

)42times10minus6

(728+004+409+166

minus004minus201minus147)times10minus5

a Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=0925119865

119861119888rarr119861

0=091andparameter120596=1GeV

[17]

basedon

theB

SWmod

elb Itise

stimated

with

theinstantaneous

nonrela

tivisticapproxim

ationandthep

otentia

lmod

elbasedon

theB

Sequatio

nc Itise

stimated

inar

elativ

isticmod

elwith

aone-gluon

interactionplus

ascalarc

onfin

ementp

otentia

lbased

ontheB

Sequatio

nd Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=05119865

119861119888rarr119861

0=039usingaq

uasip

otentia

linther

elativisticqu

arkmod

el[20]

e Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=058119865

119861119888rarr119861

0=039in

then

onrelativisticconstituent

quarkmod

el[2122]

f Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=0573(0571)119865119861119888rarr119861

0=0467(0426)inthelight-fr

ontq

uark

mod

elbasedon

theC

oulombplus

linear(harm

onicoscillator)po

tentialtogether

with

theh

yperfin

einteraction[23]

g Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=103119865

119861119888rarr119861

0=101in

ther

elativisticindepend

entq

uark

mod

el[24]

h Itise

stimated

with

inar

elativisticconstituent

quarkmod

el[25]

i Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=13119865

119861119888rarr119861

0=127in

theQ

CDsum

rules[26ndash28]

j Itisestim

ated

with

thep

erturbativeQ

CDapproach

basedon

the119896

119879factorizationscheme[2930]

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Advances in High Energy Physics

decay width from the 119888 quark decay [31ndash33] many theoreticalpapers were devoted to the study of the 119861

119888rarr 119861119875 119861119881 decays

(where 119875 and 119881 denote the 119878119880(3) ground pseudoscalar andvectormesons resp) such as [17 34ndash37]with the BSWmodel[38 39] or IGSWmodel [40] [18 19 41] based on the Bethe-Salpeter (BS) equation [20ndash24 42] with potential models[25] with constituent quark model [26ndash28] with QCD sumrules [43] with the quark diagram scheme [29 30] with theperturbativeQCD approach (pQCD) [44ndash49] and so onTheprevious predictions on the branching ratios for the 119861

119888rarr

119861119875 119861119881 decays are collected in Table 3 The discrepancies ofprevious investigations arise mainly from the different modelassumptions Recently several phenomenological methodshave been fully developed to cope with the hadronic matrixelements and successfully applied to the nonleptonic 119861 decaysuch as the pQCD approach [44ndash49] based on the 119896

119879

factorization scheme the soft-collinear effective theory [50ndash57] and the QCD-improved factorization (QCDF) approach[58ndash63] based on the collinear approximation and powercountering rules in the heavy quark limits In this paper wewill study the119861

119888rarr 119861119875 119861119881 decays with the QCDF approach

to provide a ready reference to the existing and upcomingexperiments

This paper is organized as follows In Section 2 we willpresent the theoretical framework and the amplitudes forthe 119861

119888rarr 119861119875 119861119881 decays within the QCDF framework

Section 3 is devoted to numerical results and discussionFinally Section 4 is our summation

2 Theoretical Framework

21The EffectiveHamiltonian The low energy effectiveHam-iltonian responsible for the nonleptonic bottom-conserving119861119888rarr 119861119875 119861119881 decays constructed by means of the

operator product expansion and the renormalization group(RG) method is usually written in terms of the four-quarkinteractions [64 65] Consider

119867eff

=

119866119865

radic2

119881119906119887119881lowast

119888119887[119862

119886

1(120583)119876

119886

1(120583) + 119862

119886

2(120583)119876

119886

2(120583)]

+ sum

11990211199022=119889119904

1198811199061199021

119881lowast

1198881199022

[1198621(120583)119876

1(120583) + 119862

2(120583)119876

2(120583)]

+ sum

1199023=119889119904

1198811199061199023

119881lowast

1198881199023

10

sum

119896=3

119862119896(120583)119876

119896(120583)

+ hc

(1)

where the Fermi coupling constant 119866119865≃ 1166 times 10

minus5 GeVminus2

[1]11987612119876119886

12 and119876

310are the relevant local tree annihila-

tion and penguin four-quark operators respectively whichgovern the decays in question The Cabibbo-Kobayashi-Maskawa (CKM) factor 119881

119906119902119894

119881lowast

119888119902119895

and Wilson coefficients 119862119894

describe the coupling strength for a given operator

Using the unitarity of the CKM matrix there is a largecancellation of the CKM factors

119881119906119889119881lowast

119888119889+ 119881

119906119904119881lowast

119888119904= minus119881

119906119887119881lowast

119888119887sim O (120582

5

) (2)

where the Wolfenstein parameter 120582 = sin 120579119888= 0225 37(61)

[1] and 120579119888is theCabibbo angleHence comparedwith the tree

contributions the contributions of annihilation and penguinoperators are strongly suppressed by the CKM factor If the119862119875-violating asymmetries that are expected to be very smalldue to the small weak phase difference for 119888 quark decay areprescinded from the present consideration then the penguinand annihilation contributions could be safely neglectedThelocal tree operators 119876

12in (1) are expressed as follows

1198761= [119902

2120572120574120583(1 minus 120574

5) 119888120572] [119906

120573120574120583

(1 minus 1205745) 119902

1120573]

1198762= [119902

2120572120574120583(1 minus 120574

5) 119888120573] [119906

120573120574120583

(1 minus 1205745) 119902

1120572]

(3)

where 120572 and 120573 are the 119878119880(3) color indicesThe Wilson coefficients 119862

119894(120583) summarize the physics

contributions from scales higher than 120583 They are calculablewith the RG improved perturbation theory and have properlybeen evaluated to the next-to-leading order (NLO) [64 65]They can be evolved from a higher scale 120583 sim O(119898

119882) down to

a characteristic scale 120583 sim O(119898119888) with the functions including

the flavor thresholds [64 65]

(120583) = 1198804(120583119898

119887)119872 (119898

119887) 119880

5(119898

119887 119898

119882) (119898

119882) (4)

where 119880119891(120583119891 120583

119894) is the RG evolution matrix converting

coefficients from the scale 120583119894to 120583

119891 and 119872(120583) is the quark

thresholdmatchingmatrixThe expressions of119880119891(120583119891 120583

119894) and

119872(120583) can be found in [64 65] The numerical values ofLO and NLO 119862

12with the naive dimensional regularization

scheme are listed in Table 1 The values of NLO Wilsoncoefficients in Table 1 are consistent with those given by [6465] where a trick with ldquoeffectiverdquo number of active flavors119891 =415 rather than formula (4) is used

To obtain the decay amplitudes the remaining work ishow to accurately evaluate the hadronic matrix elements⟨119861119872|119876

119894(120583)|119861

119888⟩ which summarize the physics contributions

from scales lower than 120583 Since the hadronic matrix elementsinvolve long distance contributions one is forced to useeither nonperturbative methods such as lattice calculationsand QCD sum rules or phenomenological models relyingon some assumptions Consequently it is very unfortunatethat hadronic matrix elements cannot be reliably calculatedat present and that the most intricate part and the dominanttheoretical uncertainties in the decay amplitudes reside in thehadronic matrix elements

22 Hadronic Matrix Elements Phenomenologically basedon the power counting rules in the heavy quark limit Benekeet al proposed that the hadronic matrix elements could bewritten as the convolution integrals of hard scattering kernelsand the light cone distribution amplitudes with the QCDFmaster formula [58ndash63] The QCDF approach is widelyapplied to nonleptonic 119861 decays and it works well [66ndash76]

Advances in High Energy Physics 3

Table 1 The numerical values of the Wilson coefficients and the effective coefficients for 119861119888rarr 119861120587 decay where119898

119888= 1275 plusmn 0025GeV [1]

120583

LO NLO QCDF1198621

1198622

1198621

1198622

Re (1198861) Im (119886

1) Re (119886

2) Im (119886

2)

08119898119888

1334 minus0587 1274 minus0503 1270 0096 minus0450 minus0218119898119888

1275 minus0503 1222 minus0424 1216 0068 minus0361 minus017312119898

1198881239 minus0449 1189 minus0373 1184 0054 minus0306 minus0148

which encourage us to apply the QCDF approach to thestudy of 119861

119888rarr 119861119875 119861119881 decays Since the spectator is the

heavy 119887 quark who is almost always on shell the virtuality ofthe gluon linked with the spectator should be sim O(Λ2QCD)The dominant behavior of the 119861

119888rarr 119861 transition form

factors and the contributions of hard spectator scatteringinteractions are governed by soft processes According to thebasic idea of the QCDF approach [69 70] the hard and softcontributions to the form factors entangle with each otherand cannot be identified reasonably so the physical formfactors are used as the inputs The hard spectator scatteringcontributions are power suppressed in the heavy quark limitFinally the hadronic matrix elements can be written as

⟨119861119872100381610038161003816100381611987612

1003816100381610038161003816119861119888⟩ = sum

119894

119865119861119888rarr119861

119894int119889119909119867

119894(119909)Φ

119872(119909)

prop sum

119894

119865119861119888rarr119861

1198941198911198721 + 120572

1199041199031+ sdot sdot sdot

(5)

where 119865119861119888rarr119861

119894is the transition form factor and Φ

119872(119909) is the

light-cone distribution amplitudes of the emitted meson 119872with the decay constant119891

119872The hard scattering kernels119867

119894(119909)

are computable order by order with the perturbation theoryin principle At the leading order 1205720

119904 119867

119894(119909) = 1 that is

the convolution integral of (5) results in the meson decayconstant The hadronic matrix elements are parameterizedby the product of form factors and decay constants whichare real and renormalization scale independent One goesback to the simple ldquonaive factorizationrdquo (NF) scenario Atthe order 120572

119904and higher orders the information of strong

phases and the renormalization scale dependence of hadronicmatrix elements could be partly recuperated Combined thenonfactorizable contributions with the Wilson coefficientsthe scale independent effective coefficients at the order 120572

119904can

be obtained [58ndash63] as follows

1198861= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO2119881

1198862= 119862

NLO2

+

1

119873119888

119862NLO1

+

120572119904

4120587

119862119865

119873119888

119862LO1119881

(6)

where the expressions of vertex corrections are [58ndash63]

119881 = 6 log(1198982

119888

1205832) minus 18 minus (

1

2

+ 1198943120587) 119886119872

0

+ (

11

2

minus 1198943120587) 119886119872

1minus

21

20

119886119872

2+ sdot sdot sdot

(7)

with the twist-2 quark-antiquark distribution amplitudes ofpseudoscalar 119875 and longitudinally polarized vector 119881mesonin terms of Gegenbauer polynomials [14ndash16] One has

120601119872(119909) = 6119909119909

infin

sum

119899=0

119886119872

11989911986232

119899(119909 minus 119909) (8)

where 119909 = 1 minus 119909 119886119872119899

is the Gegenbauer moment and 1198861198720equiv 1

From the numbers in Table 1 it is found that (1) for thecoefficient 119886

1the nonfactorizable contributions accompanied

by the Wilson coefficient 1198622can provide ge10 enhancement

compared with the NFrsquos result and a relatively small strongphase le 5

∘ (2) for the coefficient 1198862 the nonfactorizable

contributions assisted with the largeWilson coefficient1198621are

significant In addition a relatively large strong phasesim minus155∘is obtained (3) the QCDFrsquos values of 119886

12agree basically with

the real coefficients 1198861≃ 120 and 119886

2≃ minus0317 which are

used by previous studies on the 119861119888rarr 119861119875 119861119881 decays in

[17 18 20ndash28 34ndash37 42] but with more information on thestrong phases

23 Decay Amplitudes Within the QCDF framework theamplitudes for 119861

119888rarr 119861119872 decays are expressed as

A (119861119888997888rarr 119861119872) = ⟨119861119872

1003816100381610038161003816Heff

1003816100381610038161003816119861119888⟩

=

119866119865

radic2

1198811199061199021

119881lowast

1198881199022

119886119894⟨119872100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119861

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816119861119888⟩

(9)

The matrix elements of current operators are defined as

⟨119875 (119901)10038161003816100381610038161199021120574120583

(1 minus 1205745) 119902

2

10038161003816100381610038160⟩ = minus119894119891

119875119901120583

⟨119881 (120598 119901)10038161003816100381610038161199021120574120583

(1 minus 1205745) 119902

2

10038161003816100381610038160⟩ = 119891

119881119898119881120598120583

(10)

where 119891119875and 119891

119881are the decay constants of pseudoscalar 119875

and vector119881mesons respectively119898119881and 120598 denote the mass

and polarization of vector meson respectivelyFor the mixing of physical pseudoscalar 120578 and 1205781015840 meson

we adopt the quark-flavor basis description proposed in [13]and neglect the contributions from possible gluonium and 119888119888compositions that is

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (11)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393plusmn10)∘ [13]We assume that the vectormesons are ideally

mixed that is 120596 = (119906119906 + 119889119889)radic2 and 120601 = 119904119904

4 Advances in High Energy Physics

Table 2 The numerical values of input parameters

Wolfenstein parameters120582 = 022537 plusmn 000061 [1] 119860 = 0814

+0023

minus0024[1]

120588 = 0117 plusmn 0021 [1] 120578 = 0353 plusmn 0013 [1]Decay constant of mesons

119891120587= 13041 plusmn 020MeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119891120578119902

= (107 plusmn 002)119891120587[13] 119891

120578119904

= (134 plusmn 006)119891120587[13]

119891120588= 216 plusmn 3MeV [14ndash16] 119891

120596= 187 plusmn 5MeV [14ndash16]

119891119870lowast = 220 plusmn 5MeV [14ndash16]

Gegenbauer moments at the scale 120583 = 1 GeV119886120587

1= 119886

120578119902

1= 119886

120578119904

1= 0 [14ndash16] 119886

120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [14ndash16]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [14ndash16] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [14ndash16]

119886

1120588= 119886

1120596= 0 [14ndash16] 119886

2120588= 119886

2120596= 015 plusmn 007 [14ndash16]

119886

1119870

lowast = minus119886

1119870lowast = 003 plusmn 002 [14ndash16] 119886

2119870lowast = 119886

2119870

lowast = 011 plusmn 009 [14ndash16]

The transition form factors are defined as [38 39]

⟨119861 (119896)1003816100381610038161003816119902120574

120583

(1 minus 1205745) 1198881003816100381610038161003816119861119888(119901)⟩

= [119901 + 119896 minus

1198982

119861119888

minus 1198982

119861

1199022

119902]

120583

119865119861119888rarr119861

1(119902

2

)

+

1198982

119861119888

minus 1198982

119861

1199022

119902120583

119865119861119888rarr119861

0(119902

2

)

(12)

where 119902 = 119901 minus 119896 and the condition of 119865119861119888rarr119861

0(0) = 119865

119861119888rarr119861

1(0)

is required compulsorily to cancel the singularity at the pole1199022

= 0For the119861

119888rarr 119861 transition form factors since the velocity

of the recoiled 119861 meson is very low in the rest frame of the119861119888meson the wave functions of 119861 and 119861

119888mesons overlap

strongly It is believed that the form factors 119865119861119888rarr119861

01should

be close to the result using the nonrelativistic harmonicoscillator wave functions with the BSWmodel [17] Consider

119865119861119888rarr119861

01≃ (

2119898119861119888

119898119861

1198982

119861119888

+ 1198982

119861

)

12

≃ 099 (13)

The flavor symmetry breaking effects on the form factors areneglectable in (13) For simplification we take 119865119861119888rarr119861

119906119889119904

01= 10

in our numerical calculation to give a rough estimation

3 Numerical Results and Discussions

The branching ratios of nonleptonic two-body 119861119888decays in

the rest frame of the 119861119888meson can be written as

B119903 (119861119888997888rarr 119861119872) =

120591119861119888

8120587

119901

1198982

119861119888

1003816100381610038161003816A (119861

119888997888rarr 119861119872)

1003816100381610038161003816

2

(14)

where the lifetime of the 119861119888meson 120591

119861119888

= 5134 plusmn 110 plusmn 57 fs[6] and 119901 is the common momentum of final particles Thedecay amplitudesA(119861

119888rarr 119861119872) are listed in the Appendix

The input parameters in our calculation including theCKM Wolfenstein parameters decay constants and Gegen-bauer moments of distribution amplitudes in (8) are col-lected in Table 2 If not specified explicitly we will take theircentral values as the default inputs Our numerical results onthe 119862119875-averaged branching ratios are presented in Table 3where theoretical uncertainties of the ldquoQCDFrdquo column comefrom the CKM parameters the renormalization scale 120583 =(1 plusmn 02)119898

119888 decay constants and Gegenbauer moments

respectively For comparison previous results calculated withthe fixed coefficients 119886

1≃ 122 and 119886

2≃ minus04 are also listed

There are some comments on the branching ratios(1) From the numbers in Table 3 it is seen that different

branching ratios for the 119861119888rarr 119861119875 119861119881 decays were obtained

with different approaches in previous works although thesame coefficients 119886

12are used Much of the discrepancy

comes from the different values of the transition formfactors If the same value of the form factor is used thenthe disparities on branching ratios for the 119886

1-dominated 119861

119888

decays will be highly alleviated For example all previouspredictions on B119903(119861

119888rarr 119861

119904120587) will be about 10 with the

same form factor119865119861119888rarr119861119904

0≃ 10 which is generally in linewith

the QCDF estimation within uncertainties and also agreeswith the recent LHCb measurement [12](2)There is a hierarchical structure between the QCDFrsquos

results on branching ratios for the 119861119888rarr 119861119875 and 119861119881 decays

with the same final 119861119902meson for example

B119903 (119861119888997888rarr 119861

119902120587) gtB119903 (119861

119888997888rarr 119861

119902120588)

B119903 (119861119888997888rarr 119861

119902119870) ≳ 5B119903 (119861

119888997888rarr 119861

119902119870lowast

)

(15)

which differs from the previous resultsThere are two decisivefactors One is the kinematic factor The phase space for the119861119888rarr 119861119881 decays is more compressed than that for the 119861

119888rarr

119861119875 decays because the mass of the light pseudoscalar meson(except for the exotic 1205781015840meson) is generally less than themassof the corresponding vector meson with the same valencequark components The other is the dynamical factor The

Advances in High Energy Physics 5

Table3Th

e119862119875-averagedbranchingratio

sfor

the119861

119888rarr119861119875119861119881decays

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr1198610 119904120587+

1-a10times10minus1

68times10minus2

18times10minus2

29times10minus2

40times10minus2

43times10minus2

(43times10minus2

)13times10minus1

46times10minus2

19times10minus1

88times10minus2

(113+000+011+001

minus000minus006minus001)times10minus1

119861119888rarr1198610 119904119870+

1-b76times10minus3

49times10minus3

20times10minus3

24times10minus3

33times10minus3

33times10minus3

(33times10minus3

)85times10minus3

34times10minus3

12times10minus2

52times10minus3

(741+004+070+009

minus004minus039minus009)times10minus3

119861119888rarr1198610 119904120588+

1-a63times10minus2

52times10minus2

46times10minus2

16times10minus2

27times10minus2

30times10minus2

(27times10minus2

)11times10minus1

27times10minus2

84times10minus2

32times10minus2

(444+000+041+013

minus000minus023minus013)times10minus2

119861119888rarr1198610 119904119870lowast+

1-b32times10minus4

12times10minus3

35times10minus5

15times10minus4

80times10minus5

(71times10minus5

)40times10minus4

13times10minus4

97times10minus5

(125+001+012+006

minus001minus007minus006)times10minus4

119861119888rarr1198610 1198891205870

1-b74times10minus3

38times10minus3

12times10minus3

12times10minus3

13times10minus3

18times10minus3

(15times10minus3

)84times10minus3

24times10minus3

12times10minus2

69times10minus3

(783+004+073+004

minus004minus041minus004)times10minus3

119861119888rarr1198610 119889119870+

1-c30times10minus4

12times10minus4

10times10minus4

11times10minus4

15times10minus4

(12times10minus4

)59times10minus4

18times10minus4

81times10minus4

44times10minus4

(529+006+050+007

minus006minus028minus007)times10minus4

119861119888rarr1198610 119889120588+

1-b83times10minus3

69times10minus3

33times10minus3

15times10minus3

16times10minus3

22times10minus3

(17times10minus3

)14times10minus2

24times10minus3

11times10minus2

43times10minus3

(532+003+049+015

minus003minus028minus015)times10minus3

119861119888rarr1198610 119889119870lowast+

1-c21times10minus4

15times10minus4

46times10minus5

44times10minus5

49times10minus5

(37times10minus5

)35times10minus4

57times10minus5

17times10minus4

83times10minus5

(106+001+010+005

minus001minus006minus005)times10minus4

119861119888rarr119861+ 119906119870

0

2-a

21times10minus2

12times10minus2

49times10minus3

42times10minus3

44times10minus3

60times10minus3

(49times10minus3

)23times10minus2

68times10minus3

36times10minus2

22times10minus3

(197+000+111+005

minus000minus054minus005)times10minus2

119861119888rarr119861+ 1199061198700

2-c

16times10minus5

(13times10minus5

)63times10minus6

(571+006+320+015

minus006minus158minus014)times10minus5

119861119888rarr119861+ 119906119870

lowast0

2-a

78times10minus3

85times10minus3

58times10minus3

16times10minus3

16times10minus3

19times10minus3

(14times10minus3

)13times10minus2

21times10minus3

80times10minus3

20times10minus4

(372+000+209+021

minus000minus103minus020)times10minus3

119861119888rarr119861+ 119906119870lowast0

2-c

50times10minus6

(37times10minus6

)56times10minus7

(107+001+060+006

minus001minus030minus006)times10minus5

119861119888rarr119861+ 1199061205870

2-b

40times10minus4

21times10minus4

64times10minus5

62times10minus5

67times10minus5

97times10minus5

(81times10minus5

)45times10minus4

13times10minus4

66times10minus4

52times10minus5

(423+002+237+007

minus002minus117minus007)times10minus4

119861119888rarr119861+ 1199061205880

2-b

44times10minus4

37times10minus4

17times10minus4

87times10minus5

89times10minus5

12times10minus4

(94times10minus5

)74times10minus4

13times10minus4

55times10minus4

18times10minus5

(286+001+160+010

minus001minus079minus010)times10minus4

119861119888rarr119861+ 119906120596

2-b

41times10minus4

90times10minus5

(70times10minus5

)13times10minus5

(205+001+115+012

minus001minus057minus012)times10minus4

6 Advances in High Energy Physics

Table3Con

tinued

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr119861+ 119906120578

50times10minus4

(41times10minus4

)14times10minus4

(146+001+082+013

minus001minus040minus012)times10minus3

119861119888rarr119861+ 1199061205781015840

67times10minus6

(56times10minus6

)42times10minus6

(728+004+409+166

minus004minus201minus147)times10minus5

a Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=0925119865

119861119888rarr119861

0=091andparameter120596=1GeV

[17]

basedon

theB

SWmod

elb Itise

stimated

with

theinstantaneous

nonrela

tivisticapproxim

ationandthep

otentia

lmod

elbasedon

theB

Sequatio

nc Itise

stimated

inar

elativ

isticmod

elwith

aone-gluon

interactionplus

ascalarc

onfin

ementp

otentia

lbased

ontheB

Sequatio

nd Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=05119865

119861119888rarr119861

0=039usingaq

uasip

otentia

linther

elativisticqu

arkmod

el[20]

e Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=058119865

119861119888rarr119861

0=039in

then

onrelativisticconstituent

quarkmod

el[2122]

f Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=0573(0571)119865119861119888rarr119861

0=0467(0426)inthelight-fr

ontq

uark

mod

elbasedon

theC

oulombplus

linear(harm

onicoscillator)po

tentialtogether

with

theh

yperfin

einteraction[23]

g Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=103119865

119861119888rarr119861

0=101in

ther

elativisticindepend

entq

uark

mod

el[24]

h Itise

stimated

with

inar

elativisticconstituent

quarkmod

el[25]

i Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=13119865

119861119888rarr119861

0=127in

theQ

CDsum

rules[26ndash28]

j Itisestim

ated

with

thep

erturbativeQ

CDapproach

basedon

the119896

119879factorizationscheme[2930]

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 3

Table 1 The numerical values of the Wilson coefficients and the effective coefficients for 119861119888rarr 119861120587 decay where119898

119888= 1275 plusmn 0025GeV [1]

120583

LO NLO QCDF1198621

1198622

1198621

1198622

Re (1198861) Im (119886

1) Re (119886

2) Im (119886

2)

08119898119888

1334 minus0587 1274 minus0503 1270 0096 minus0450 minus0218119898119888

1275 minus0503 1222 minus0424 1216 0068 minus0361 minus017312119898

1198881239 minus0449 1189 minus0373 1184 0054 minus0306 minus0148

which encourage us to apply the QCDF approach to thestudy of 119861

119888rarr 119861119875 119861119881 decays Since the spectator is the

heavy 119887 quark who is almost always on shell the virtuality ofthe gluon linked with the spectator should be sim O(Λ2QCD)The dominant behavior of the 119861

119888rarr 119861 transition form

factors and the contributions of hard spectator scatteringinteractions are governed by soft processes According to thebasic idea of the QCDF approach [69 70] the hard and softcontributions to the form factors entangle with each otherand cannot be identified reasonably so the physical formfactors are used as the inputs The hard spectator scatteringcontributions are power suppressed in the heavy quark limitFinally the hadronic matrix elements can be written as

⟨119861119872100381610038161003816100381611987612

1003816100381610038161003816119861119888⟩ = sum

119894

119865119861119888rarr119861

119894int119889119909119867

119894(119909)Φ

119872(119909)

prop sum

119894

119865119861119888rarr119861

1198941198911198721 + 120572

1199041199031+ sdot sdot sdot

(5)

where 119865119861119888rarr119861

119894is the transition form factor and Φ

119872(119909) is the

light-cone distribution amplitudes of the emitted meson 119872with the decay constant119891

119872The hard scattering kernels119867

119894(119909)

are computable order by order with the perturbation theoryin principle At the leading order 1205720

119904 119867

119894(119909) = 1 that is

the convolution integral of (5) results in the meson decayconstant The hadronic matrix elements are parameterizedby the product of form factors and decay constants whichare real and renormalization scale independent One goesback to the simple ldquonaive factorizationrdquo (NF) scenario Atthe order 120572

119904and higher orders the information of strong

phases and the renormalization scale dependence of hadronicmatrix elements could be partly recuperated Combined thenonfactorizable contributions with the Wilson coefficientsthe scale independent effective coefficients at the order 120572

119904can

be obtained [58ndash63] as follows

1198861= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO2119881

1198862= 119862

NLO2

+

1

119873119888

119862NLO1

+

120572119904

4120587

119862119865

119873119888

119862LO1119881

(6)

where the expressions of vertex corrections are [58ndash63]

119881 = 6 log(1198982

119888

1205832) minus 18 minus (

1

2

+ 1198943120587) 119886119872

0

+ (

11

2

minus 1198943120587) 119886119872

1minus

21

20

119886119872

2+ sdot sdot sdot

(7)

with the twist-2 quark-antiquark distribution amplitudes ofpseudoscalar 119875 and longitudinally polarized vector 119881mesonin terms of Gegenbauer polynomials [14ndash16] One has

120601119872(119909) = 6119909119909

infin

sum

119899=0

119886119872

11989911986232

119899(119909 minus 119909) (8)

where 119909 = 1 minus 119909 119886119872119899

is the Gegenbauer moment and 1198861198720equiv 1

From the numbers in Table 1 it is found that (1) for thecoefficient 119886

1the nonfactorizable contributions accompanied

by the Wilson coefficient 1198622can provide ge10 enhancement

compared with the NFrsquos result and a relatively small strongphase le 5

∘ (2) for the coefficient 1198862 the nonfactorizable

contributions assisted with the largeWilson coefficient1198621are

significant In addition a relatively large strong phasesim minus155∘is obtained (3) the QCDFrsquos values of 119886

12agree basically with

the real coefficients 1198861≃ 120 and 119886

2≃ minus0317 which are

used by previous studies on the 119861119888rarr 119861119875 119861119881 decays in

[17 18 20ndash28 34ndash37 42] but with more information on thestrong phases

23 Decay Amplitudes Within the QCDF framework theamplitudes for 119861

119888rarr 119861119872 decays are expressed as

A (119861119888997888rarr 119861119872) = ⟨119861119872

1003816100381610038161003816Heff

1003816100381610038161003816119861119888⟩

=

119866119865

radic2

1198811199061199021

119881lowast

1198881199022

119886119894⟨119872100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119861

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816119861119888⟩

(9)

The matrix elements of current operators are defined as

⟨119875 (119901)10038161003816100381610038161199021120574120583

(1 minus 1205745) 119902

2

10038161003816100381610038160⟩ = minus119894119891

119875119901120583

⟨119881 (120598 119901)10038161003816100381610038161199021120574120583

(1 minus 1205745) 119902

2

10038161003816100381610038160⟩ = 119891

119881119898119881120598120583

(10)

where 119891119875and 119891

119881are the decay constants of pseudoscalar 119875

and vector119881mesons respectively119898119881and 120598 denote the mass

and polarization of vector meson respectivelyFor the mixing of physical pseudoscalar 120578 and 1205781015840 meson

we adopt the quark-flavor basis description proposed in [13]and neglect the contributions from possible gluonium and 119888119888compositions that is

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (11)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393plusmn10)∘ [13]We assume that the vectormesons are ideally

mixed that is 120596 = (119906119906 + 119889119889)radic2 and 120601 = 119904119904

4 Advances in High Energy Physics

Table 2 The numerical values of input parameters

Wolfenstein parameters120582 = 022537 plusmn 000061 [1] 119860 = 0814

+0023

minus0024[1]

120588 = 0117 plusmn 0021 [1] 120578 = 0353 plusmn 0013 [1]Decay constant of mesons

119891120587= 13041 plusmn 020MeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119891120578119902

= (107 plusmn 002)119891120587[13] 119891

120578119904

= (134 plusmn 006)119891120587[13]

119891120588= 216 plusmn 3MeV [14ndash16] 119891

120596= 187 plusmn 5MeV [14ndash16]

119891119870lowast = 220 plusmn 5MeV [14ndash16]

Gegenbauer moments at the scale 120583 = 1 GeV119886120587

1= 119886

120578119902

1= 119886

120578119904

1= 0 [14ndash16] 119886

120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [14ndash16]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [14ndash16] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [14ndash16]

119886

1120588= 119886

1120596= 0 [14ndash16] 119886

2120588= 119886

2120596= 015 plusmn 007 [14ndash16]

119886

1119870

lowast = minus119886

1119870lowast = 003 plusmn 002 [14ndash16] 119886

2119870lowast = 119886

2119870

lowast = 011 plusmn 009 [14ndash16]

The transition form factors are defined as [38 39]

⟨119861 (119896)1003816100381610038161003816119902120574

120583

(1 minus 1205745) 1198881003816100381610038161003816119861119888(119901)⟩

= [119901 + 119896 minus

1198982

119861119888

minus 1198982

119861

1199022

119902]

120583

119865119861119888rarr119861

1(119902

2

)

+

1198982

119861119888

minus 1198982

119861

1199022

119902120583

119865119861119888rarr119861

0(119902

2

)

(12)

where 119902 = 119901 minus 119896 and the condition of 119865119861119888rarr119861

0(0) = 119865

119861119888rarr119861

1(0)

is required compulsorily to cancel the singularity at the pole1199022

= 0For the119861

119888rarr 119861 transition form factors since the velocity

of the recoiled 119861 meson is very low in the rest frame of the119861119888meson the wave functions of 119861 and 119861

119888mesons overlap

strongly It is believed that the form factors 119865119861119888rarr119861

01should

be close to the result using the nonrelativistic harmonicoscillator wave functions with the BSWmodel [17] Consider

119865119861119888rarr119861

01≃ (

2119898119861119888

119898119861

1198982

119861119888

+ 1198982

119861

)

12

≃ 099 (13)

The flavor symmetry breaking effects on the form factors areneglectable in (13) For simplification we take 119865119861119888rarr119861

119906119889119904

01= 10

in our numerical calculation to give a rough estimation

3 Numerical Results and Discussions

The branching ratios of nonleptonic two-body 119861119888decays in

the rest frame of the 119861119888meson can be written as

B119903 (119861119888997888rarr 119861119872) =

120591119861119888

8120587

119901

1198982

119861119888

1003816100381610038161003816A (119861

119888997888rarr 119861119872)

1003816100381610038161003816

2

(14)

where the lifetime of the 119861119888meson 120591

119861119888

= 5134 plusmn 110 plusmn 57 fs[6] and 119901 is the common momentum of final particles Thedecay amplitudesA(119861

119888rarr 119861119872) are listed in the Appendix

The input parameters in our calculation including theCKM Wolfenstein parameters decay constants and Gegen-bauer moments of distribution amplitudes in (8) are col-lected in Table 2 If not specified explicitly we will take theircentral values as the default inputs Our numerical results onthe 119862119875-averaged branching ratios are presented in Table 3where theoretical uncertainties of the ldquoQCDFrdquo column comefrom the CKM parameters the renormalization scale 120583 =(1 plusmn 02)119898

119888 decay constants and Gegenbauer moments

respectively For comparison previous results calculated withthe fixed coefficients 119886

1≃ 122 and 119886

2≃ minus04 are also listed

There are some comments on the branching ratios(1) From the numbers in Table 3 it is seen that different

branching ratios for the 119861119888rarr 119861119875 119861119881 decays were obtained

with different approaches in previous works although thesame coefficients 119886

12are used Much of the discrepancy

comes from the different values of the transition formfactors If the same value of the form factor is used thenthe disparities on branching ratios for the 119886

1-dominated 119861

119888

decays will be highly alleviated For example all previouspredictions on B119903(119861

119888rarr 119861

119904120587) will be about 10 with the

same form factor119865119861119888rarr119861119904

0≃ 10 which is generally in linewith

the QCDF estimation within uncertainties and also agreeswith the recent LHCb measurement [12](2)There is a hierarchical structure between the QCDFrsquos

results on branching ratios for the 119861119888rarr 119861119875 and 119861119881 decays

with the same final 119861119902meson for example

B119903 (119861119888997888rarr 119861

119902120587) gtB119903 (119861

119888997888rarr 119861

119902120588)

B119903 (119861119888997888rarr 119861

119902119870) ≳ 5B119903 (119861

119888997888rarr 119861

119902119870lowast

)

(15)

which differs from the previous resultsThere are two decisivefactors One is the kinematic factor The phase space for the119861119888rarr 119861119881 decays is more compressed than that for the 119861

119888rarr

119861119875 decays because the mass of the light pseudoscalar meson(except for the exotic 1205781015840meson) is generally less than themassof the corresponding vector meson with the same valencequark components The other is the dynamical factor The

Advances in High Energy Physics 5

Table3Th

e119862119875-averagedbranchingratio

sfor

the119861

119888rarr119861119875119861119881decays

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr1198610 119904120587+

1-a10times10minus1

68times10minus2

18times10minus2

29times10minus2

40times10minus2

43times10minus2

(43times10minus2

)13times10minus1

46times10minus2

19times10minus1

88times10minus2

(113+000+011+001

minus000minus006minus001)times10minus1

119861119888rarr1198610 119904119870+

1-b76times10minus3

49times10minus3

20times10minus3

24times10minus3

33times10minus3

33times10minus3

(33times10minus3

)85times10minus3

34times10minus3

12times10minus2

52times10minus3

(741+004+070+009

minus004minus039minus009)times10minus3

119861119888rarr1198610 119904120588+

1-a63times10minus2

52times10minus2

46times10minus2

16times10minus2

27times10minus2

30times10minus2

(27times10minus2

)11times10minus1

27times10minus2

84times10minus2

32times10minus2

(444+000+041+013

minus000minus023minus013)times10minus2

119861119888rarr1198610 119904119870lowast+

1-b32times10minus4

12times10minus3

35times10minus5

15times10minus4

80times10minus5

(71times10minus5

)40times10minus4

13times10minus4

97times10minus5

(125+001+012+006

minus001minus007minus006)times10minus4

119861119888rarr1198610 1198891205870

1-b74times10minus3

38times10minus3

12times10minus3

12times10minus3

13times10minus3

18times10minus3

(15times10minus3

)84times10minus3

24times10minus3

12times10minus2

69times10minus3

(783+004+073+004

minus004minus041minus004)times10minus3

119861119888rarr1198610 119889119870+

1-c30times10minus4

12times10minus4

10times10minus4

11times10minus4

15times10minus4

(12times10minus4

)59times10minus4

18times10minus4

81times10minus4

44times10minus4

(529+006+050+007

minus006minus028minus007)times10minus4

119861119888rarr1198610 119889120588+

1-b83times10minus3

69times10minus3

33times10minus3

15times10minus3

16times10minus3

22times10minus3

(17times10minus3

)14times10minus2

24times10minus3

11times10minus2

43times10minus3

(532+003+049+015

minus003minus028minus015)times10minus3

119861119888rarr1198610 119889119870lowast+

1-c21times10minus4

15times10minus4

46times10minus5

44times10minus5

49times10minus5

(37times10minus5

)35times10minus4

57times10minus5

17times10minus4

83times10minus5

(106+001+010+005

minus001minus006minus005)times10minus4

119861119888rarr119861+ 119906119870

0

2-a

21times10minus2

12times10minus2

49times10minus3

42times10minus3

44times10minus3

60times10minus3

(49times10minus3

)23times10minus2

68times10minus3

36times10minus2

22times10minus3

(197+000+111+005

minus000minus054minus005)times10minus2

119861119888rarr119861+ 1199061198700

2-c

16times10minus5

(13times10minus5

)63times10minus6

(571+006+320+015

minus006minus158minus014)times10minus5

119861119888rarr119861+ 119906119870

lowast0

2-a

78times10minus3

85times10minus3

58times10minus3

16times10minus3

16times10minus3

19times10minus3

(14times10minus3

)13times10minus2

21times10minus3

80times10minus3

20times10minus4

(372+000+209+021

minus000minus103minus020)times10minus3

119861119888rarr119861+ 119906119870lowast0

2-c

50times10minus6

(37times10minus6

)56times10minus7

(107+001+060+006

minus001minus030minus006)times10minus5

119861119888rarr119861+ 1199061205870

2-b

40times10minus4

21times10minus4

64times10minus5

62times10minus5

67times10minus5

97times10minus5

(81times10minus5

)45times10minus4

13times10minus4

66times10minus4

52times10minus5

(423+002+237+007

minus002minus117minus007)times10minus4

119861119888rarr119861+ 1199061205880

2-b

44times10minus4

37times10minus4

17times10minus4

87times10minus5

89times10minus5

12times10minus4

(94times10minus5

)74times10minus4

13times10minus4

55times10minus4

18times10minus5

(286+001+160+010

minus001minus079minus010)times10minus4

119861119888rarr119861+ 119906120596

2-b

41times10minus4

90times10minus5

(70times10minus5

)13times10minus5

(205+001+115+012

minus001minus057minus012)times10minus4

6 Advances in High Energy Physics

Table3Con

tinued

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr119861+ 119906120578

50times10minus4

(41times10minus4

)14times10minus4

(146+001+082+013

minus001minus040minus012)times10minus3

119861119888rarr119861+ 1199061205781015840

67times10minus6

(56times10minus6

)42times10minus6

(728+004+409+166

minus004minus201minus147)times10minus5

a Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=0925119865

119861119888rarr119861

0=091andparameter120596=1GeV

[17]

basedon

theB

SWmod

elb Itise

stimated

with

theinstantaneous

nonrela

tivisticapproxim

ationandthep

otentia

lmod

elbasedon

theB

Sequatio

nc Itise

stimated

inar

elativ

isticmod

elwith

aone-gluon

interactionplus

ascalarc

onfin

ementp

otentia

lbased

ontheB

Sequatio

nd Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=05119865

119861119888rarr119861

0=039usingaq

uasip

otentia

linther

elativisticqu

arkmod

el[20]

e Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=058119865

119861119888rarr119861

0=039in

then

onrelativisticconstituent

quarkmod

el[2122]

f Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=0573(0571)119865119861119888rarr119861

0=0467(0426)inthelight-fr

ontq

uark

mod

elbasedon

theC

oulombplus

linear(harm

onicoscillator)po

tentialtogether

with

theh

yperfin

einteraction[23]

g Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=103119865

119861119888rarr119861

0=101in

ther

elativisticindepend

entq

uark

mod

el[24]

h Itise

stimated

with

inar

elativisticconstituent

quarkmod

el[25]

i Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=13119865

119861119888rarr119861

0=127in

theQ

CDsum

rules[26ndash28]

j Itisestim

ated

with

thep

erturbativeQ

CDapproach

basedon

the119896

119879factorizationscheme[2930]

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

Table 2 The numerical values of input parameters

Wolfenstein parameters120582 = 022537 plusmn 000061 [1] 119860 = 0814

+0023

minus0024[1]

120588 = 0117 plusmn 0021 [1] 120578 = 0353 plusmn 0013 [1]Decay constant of mesons

119891120587= 13041 plusmn 020MeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119891120578119902

= (107 plusmn 002)119891120587[13] 119891

120578119904

= (134 plusmn 006)119891120587[13]

119891120588= 216 plusmn 3MeV [14ndash16] 119891

120596= 187 plusmn 5MeV [14ndash16]

119891119870lowast = 220 plusmn 5MeV [14ndash16]

Gegenbauer moments at the scale 120583 = 1 GeV119886120587

1= 119886

120578119902

1= 119886

120578119904

1= 0 [14ndash16] 119886

120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [14ndash16]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [14ndash16] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [14ndash16]

119886

1120588= 119886

1120596= 0 [14ndash16] 119886

2120588= 119886

2120596= 015 plusmn 007 [14ndash16]

119886

1119870

lowast = minus119886

1119870lowast = 003 plusmn 002 [14ndash16] 119886

2119870lowast = 119886

2119870

lowast = 011 plusmn 009 [14ndash16]

The transition form factors are defined as [38 39]

⟨119861 (119896)1003816100381610038161003816119902120574

120583

(1 minus 1205745) 1198881003816100381610038161003816119861119888(119901)⟩

= [119901 + 119896 minus

1198982

119861119888

minus 1198982

119861

1199022

119902]

120583

119865119861119888rarr119861

1(119902

2

)

+

1198982

119861119888

minus 1198982

119861

1199022

119902120583

119865119861119888rarr119861

0(119902

2

)

(12)

where 119902 = 119901 minus 119896 and the condition of 119865119861119888rarr119861

0(0) = 119865

119861119888rarr119861

1(0)

is required compulsorily to cancel the singularity at the pole1199022

= 0For the119861

119888rarr 119861 transition form factors since the velocity

of the recoiled 119861 meson is very low in the rest frame of the119861119888meson the wave functions of 119861 and 119861

119888mesons overlap

strongly It is believed that the form factors 119865119861119888rarr119861

01should

be close to the result using the nonrelativistic harmonicoscillator wave functions with the BSWmodel [17] Consider

119865119861119888rarr119861

01≃ (

2119898119861119888

119898119861

1198982

119861119888

+ 1198982

119861

)

12

≃ 099 (13)

The flavor symmetry breaking effects on the form factors areneglectable in (13) For simplification we take 119865119861119888rarr119861

119906119889119904

01= 10

in our numerical calculation to give a rough estimation

3 Numerical Results and Discussions

The branching ratios of nonleptonic two-body 119861119888decays in

the rest frame of the 119861119888meson can be written as

B119903 (119861119888997888rarr 119861119872) =

120591119861119888

8120587

119901

1198982

119861119888

1003816100381610038161003816A (119861

119888997888rarr 119861119872)

1003816100381610038161003816

2

(14)

where the lifetime of the 119861119888meson 120591

119861119888

= 5134 plusmn 110 plusmn 57 fs[6] and 119901 is the common momentum of final particles Thedecay amplitudesA(119861

119888rarr 119861119872) are listed in the Appendix

The input parameters in our calculation including theCKM Wolfenstein parameters decay constants and Gegen-bauer moments of distribution amplitudes in (8) are col-lected in Table 2 If not specified explicitly we will take theircentral values as the default inputs Our numerical results onthe 119862119875-averaged branching ratios are presented in Table 3where theoretical uncertainties of the ldquoQCDFrdquo column comefrom the CKM parameters the renormalization scale 120583 =(1 plusmn 02)119898

119888 decay constants and Gegenbauer moments

respectively For comparison previous results calculated withthe fixed coefficients 119886

1≃ 122 and 119886

2≃ minus04 are also listed

There are some comments on the branching ratios(1) From the numbers in Table 3 it is seen that different

branching ratios for the 119861119888rarr 119861119875 119861119881 decays were obtained

with different approaches in previous works although thesame coefficients 119886

12are used Much of the discrepancy

comes from the different values of the transition formfactors If the same value of the form factor is used thenthe disparities on branching ratios for the 119886

1-dominated 119861

119888

decays will be highly alleviated For example all previouspredictions on B119903(119861

119888rarr 119861

119904120587) will be about 10 with the

same form factor119865119861119888rarr119861119904

0≃ 10 which is generally in linewith

the QCDF estimation within uncertainties and also agreeswith the recent LHCb measurement [12](2)There is a hierarchical structure between the QCDFrsquos

results on branching ratios for the 119861119888rarr 119861119875 and 119861119881 decays

with the same final 119861119902meson for example

B119903 (119861119888997888rarr 119861

119902120587) gtB119903 (119861

119888997888rarr 119861

119902120588)

B119903 (119861119888997888rarr 119861

119902119870) ≳ 5B119903 (119861

119888997888rarr 119861

119902119870lowast

)

(15)

which differs from the previous resultsThere are two decisivefactors One is the kinematic factor The phase space for the119861119888rarr 119861119881 decays is more compressed than that for the 119861

119888rarr

119861119875 decays because the mass of the light pseudoscalar meson(except for the exotic 1205781015840meson) is generally less than themassof the corresponding vector meson with the same valencequark components The other is the dynamical factor The

Advances in High Energy Physics 5

Table3Th

e119862119875-averagedbranchingratio

sfor

the119861

119888rarr119861119875119861119881decays

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr1198610 119904120587+

1-a10times10minus1

68times10minus2

18times10minus2

29times10minus2

40times10minus2

43times10minus2

(43times10minus2

)13times10minus1

46times10minus2

19times10minus1

88times10minus2

(113+000+011+001

minus000minus006minus001)times10minus1

119861119888rarr1198610 119904119870+

1-b76times10minus3

49times10minus3

20times10minus3

24times10minus3

33times10minus3

33times10minus3

(33times10minus3

)85times10minus3

34times10minus3

12times10minus2

52times10minus3

(741+004+070+009

minus004minus039minus009)times10minus3

119861119888rarr1198610 119904120588+

1-a63times10minus2

52times10minus2

46times10minus2

16times10minus2

27times10minus2

30times10minus2

(27times10minus2

)11times10minus1

27times10minus2

84times10minus2

32times10minus2

(444+000+041+013

minus000minus023minus013)times10minus2

119861119888rarr1198610 119904119870lowast+

1-b32times10minus4

12times10minus3

35times10minus5

15times10minus4

80times10minus5

(71times10minus5

)40times10minus4

13times10minus4

97times10minus5

(125+001+012+006

minus001minus007minus006)times10minus4

119861119888rarr1198610 1198891205870

1-b74times10minus3

38times10minus3

12times10minus3

12times10minus3

13times10minus3

18times10minus3

(15times10minus3

)84times10minus3

24times10minus3

12times10minus2

69times10minus3

(783+004+073+004

minus004minus041minus004)times10minus3

119861119888rarr1198610 119889119870+

1-c30times10minus4

12times10minus4

10times10minus4

11times10minus4

15times10minus4

(12times10minus4

)59times10minus4

18times10minus4

81times10minus4

44times10minus4

(529+006+050+007

minus006minus028minus007)times10minus4

119861119888rarr1198610 119889120588+

1-b83times10minus3

69times10minus3

33times10minus3

15times10minus3

16times10minus3

22times10minus3

(17times10minus3

)14times10minus2

24times10minus3

11times10minus2

43times10minus3

(532+003+049+015

minus003minus028minus015)times10minus3

119861119888rarr1198610 119889119870lowast+

1-c21times10minus4

15times10minus4

46times10minus5

44times10minus5

49times10minus5

(37times10minus5

)35times10minus4

57times10minus5

17times10minus4

83times10minus5

(106+001+010+005

minus001minus006minus005)times10minus4

119861119888rarr119861+ 119906119870

0

2-a

21times10minus2

12times10minus2

49times10minus3

42times10minus3

44times10minus3

60times10minus3

(49times10minus3

)23times10minus2

68times10minus3

36times10minus2

22times10minus3

(197+000+111+005

minus000minus054minus005)times10minus2

119861119888rarr119861+ 1199061198700

2-c

16times10minus5

(13times10minus5

)63times10minus6

(571+006+320+015

minus006minus158minus014)times10minus5

119861119888rarr119861+ 119906119870

lowast0

2-a

78times10minus3

85times10minus3

58times10minus3

16times10minus3

16times10minus3

19times10minus3

(14times10minus3

)13times10minus2

21times10minus3

80times10minus3

20times10minus4

(372+000+209+021

minus000minus103minus020)times10minus3

119861119888rarr119861+ 119906119870lowast0

2-c

50times10minus6

(37times10minus6

)56times10minus7

(107+001+060+006

minus001minus030minus006)times10minus5

119861119888rarr119861+ 1199061205870

2-b

40times10minus4

21times10minus4

64times10minus5

62times10minus5

67times10minus5

97times10minus5

(81times10minus5

)45times10minus4

13times10minus4

66times10minus4

52times10minus5

(423+002+237+007

minus002minus117minus007)times10minus4

119861119888rarr119861+ 1199061205880

2-b

44times10minus4

37times10minus4

17times10minus4

87times10minus5

89times10minus5

12times10minus4

(94times10minus5

)74times10minus4

13times10minus4

55times10minus4

18times10minus5

(286+001+160+010

minus001minus079minus010)times10minus4

119861119888rarr119861+ 119906120596

2-b

41times10minus4

90times10minus5

(70times10minus5

)13times10minus5

(205+001+115+012

minus001minus057minus012)times10minus4

6 Advances in High Energy Physics

Table3Con

tinued

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr119861+ 119906120578

50times10minus4

(41times10minus4

)14times10minus4

(146+001+082+013

minus001minus040minus012)times10minus3

119861119888rarr119861+ 1199061205781015840

67times10minus6

(56times10minus6

)42times10minus6

(728+004+409+166

minus004minus201minus147)times10minus5

a Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=0925119865

119861119888rarr119861

0=091andparameter120596=1GeV

[17]

basedon

theB

SWmod

elb Itise

stimated

with

theinstantaneous

nonrela

tivisticapproxim

ationandthep

otentia

lmod

elbasedon

theB

Sequatio

nc Itise

stimated

inar

elativ

isticmod

elwith

aone-gluon

interactionplus

ascalarc

onfin

ementp

otentia

lbased

ontheB

Sequatio

nd Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=05119865

119861119888rarr119861

0=039usingaq

uasip

otentia

linther

elativisticqu

arkmod

el[20]

e Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=058119865

119861119888rarr119861

0=039in

then

onrelativisticconstituent

quarkmod

el[2122]

f Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=0573(0571)119865119861119888rarr119861

0=0467(0426)inthelight-fr

ontq

uark

mod

elbasedon

theC

oulombplus

linear(harm

onicoscillator)po

tentialtogether

with

theh

yperfin

einteraction[23]

g Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=103119865

119861119888rarr119861

0=101in

ther

elativisticindepend

entq

uark

mod

el[24]

h Itise

stimated

with

inar

elativisticconstituent

quarkmod

el[25]

i Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=13119865

119861119888rarr119861

0=127in

theQ

CDsum

rules[26ndash28]

j Itisestim

ated

with

thep

erturbativeQ

CDapproach

basedon

the119896

119879factorizationscheme[2930]

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

Table3Th

e119862119875-averagedbranchingratio

sfor

the119861

119888rarr119861119875119861119881decays

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr1198610 119904120587+

1-a10times10minus1

68times10minus2

18times10minus2

29times10minus2

40times10minus2

43times10minus2

(43times10minus2

)13times10minus1

46times10minus2

19times10minus1

88times10minus2

(113+000+011+001

minus000minus006minus001)times10minus1

119861119888rarr1198610 119904119870+

1-b76times10minus3

49times10minus3

20times10minus3

24times10minus3

33times10minus3

33times10minus3

(33times10minus3

)85times10minus3

34times10minus3

12times10minus2

52times10minus3

(741+004+070+009

minus004minus039minus009)times10minus3

119861119888rarr1198610 119904120588+

1-a63times10minus2

52times10minus2

46times10minus2

16times10minus2

27times10minus2

30times10minus2

(27times10minus2

)11times10minus1

27times10minus2

84times10minus2

32times10minus2

(444+000+041+013

minus000minus023minus013)times10minus2

119861119888rarr1198610 119904119870lowast+

1-b32times10minus4

12times10minus3

35times10minus5

15times10minus4

80times10minus5

(71times10minus5

)40times10minus4

13times10minus4

97times10minus5

(125+001+012+006

minus001minus007minus006)times10minus4

119861119888rarr1198610 1198891205870

1-b74times10minus3

38times10minus3

12times10minus3

12times10minus3

13times10minus3

18times10minus3

(15times10minus3

)84times10minus3

24times10minus3

12times10minus2

69times10minus3

(783+004+073+004

minus004minus041minus004)times10minus3

119861119888rarr1198610 119889119870+

1-c30times10minus4

12times10minus4

10times10minus4

11times10minus4

15times10minus4

(12times10minus4

)59times10minus4

18times10minus4

81times10minus4

44times10minus4

(529+006+050+007

minus006minus028minus007)times10minus4

119861119888rarr1198610 119889120588+

1-b83times10minus3

69times10minus3

33times10minus3

15times10minus3

16times10minus3

22times10minus3

(17times10minus3

)14times10minus2

24times10minus3

11times10minus2

43times10minus3

(532+003+049+015

minus003minus028minus015)times10minus3

119861119888rarr1198610 119889119870lowast+

1-c21times10minus4

15times10minus4

46times10minus5

44times10minus5

49times10minus5

(37times10minus5

)35times10minus4

57times10minus5

17times10minus4

83times10minus5

(106+001+010+005

minus001minus006minus005)times10minus4

119861119888rarr119861+ 119906119870

0

2-a

21times10minus2

12times10minus2

49times10minus3

42times10minus3

44times10minus3

60times10minus3

(49times10minus3

)23times10minus2

68times10minus3

36times10minus2

22times10minus3

(197+000+111+005

minus000minus054minus005)times10minus2

119861119888rarr119861+ 1199061198700

2-c

16times10minus5

(13times10minus5

)63times10minus6

(571+006+320+015

minus006minus158minus014)times10minus5

119861119888rarr119861+ 119906119870

lowast0

2-a

78times10minus3

85times10minus3

58times10minus3

16times10minus3

16times10minus3

19times10minus3

(14times10minus3

)13times10minus2

21times10minus3

80times10minus3

20times10minus4

(372+000+209+021

minus000minus103minus020)times10minus3

119861119888rarr119861+ 119906119870lowast0

2-c

50times10minus6

(37times10minus6

)56times10minus7

(107+001+060+006

minus001minus030minus006)times10minus5

119861119888rarr119861+ 1199061205870

2-b

40times10minus4

21times10minus4

64times10minus5

62times10minus5

67times10minus5

97times10minus5

(81times10minus5

)45times10minus4

13times10minus4

66times10minus4

52times10minus5

(423+002+237+007

minus002minus117minus007)times10minus4

119861119888rarr119861+ 1199061205880

2-b

44times10minus4

37times10minus4

17times10minus4

87times10minus5

89times10minus5

12times10minus4

(94times10minus5

)74times10minus4

13times10minus4

55times10minus4

18times10minus5

(286+001+160+010

minus001minus079minus010)times10minus4

119861119888rarr119861+ 119906120596

2-b

41times10minus4

90times10minus5

(70times10minus5

)13times10minus5

(205+001+115+012

minus001minus057minus012)times10minus4

6 Advances in High Energy Physics

Table3Con

tinued

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr119861+ 119906120578

50times10minus4

(41times10minus4

)14times10minus4

(146+001+082+013

minus001minus040minus012)times10minus3

119861119888rarr119861+ 1199061205781015840

67times10minus6

(56times10minus6

)42times10minus6

(728+004+409+166

minus004minus201minus147)times10minus5

a Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=0925119865

119861119888rarr119861

0=091andparameter120596=1GeV

[17]

basedon

theB

SWmod

elb Itise

stimated

with

theinstantaneous

nonrela

tivisticapproxim

ationandthep

otentia

lmod

elbasedon

theB

Sequatio

nc Itise

stimated

inar

elativ

isticmod

elwith

aone-gluon

interactionplus

ascalarc

onfin

ementp

otentia

lbased

ontheB

Sequatio

nd Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=05119865

119861119888rarr119861

0=039usingaq

uasip

otentia

linther

elativisticqu

arkmod

el[20]

e Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=058119865

119861119888rarr119861

0=039in

then

onrelativisticconstituent

quarkmod

el[2122]

f Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=0573(0571)119865119861119888rarr119861

0=0467(0426)inthelight-fr

ontq

uark

mod

elbasedon

theC

oulombplus

linear(harm

onicoscillator)po

tentialtogether

with

theh

yperfin

einteraction[23]

g Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=103119865

119861119888rarr119861

0=101in

ther

elativisticindepend

entq

uark

mod

el[24]

h Itise

stimated

with

inar

elativisticconstituent

quarkmod

el[25]

i Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=13119865

119861119888rarr119861

0=127in

theQ

CDsum

rules[26ndash28]

j Itisestim

ated

with

thep

erturbativeQ

CDapproach

basedon

the119896

119879factorizationscheme[2930]

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

Table3Con

tinued

Decay

mod

eCa

seRe

ference

[17]

aRe

ference

[18]

bRe

ference

[19]c

Reference

[20]

dRe

ference

[2122]e

Reference

[23]

fRe

ference

[24]

gRe

ference

[25]

hRe

ference

[26ndash

28]i

Reference

[2930]j

QCD

F

119861119888rarr119861+ 119906120578

50times10minus4

(41times10minus4

)14times10minus4

(146+001+082+013

minus001minus040minus012)times10minus3

119861119888rarr119861+ 1199061205781015840

67times10minus6

(56times10minus6

)42times10minus6

(728+004+409+166

minus004minus201minus147)times10minus5

a Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=0925119865

119861119888rarr119861

0=091andparameter120596=1GeV

[17]

basedon

theB

SWmod

elb Itise

stimated

with

theinstantaneous

nonrela

tivisticapproxim

ationandthep

otentia

lmod

elbasedon

theB

Sequatio

nc Itise

stimated

inar

elativ

isticmod

elwith

aone-gluon

interactionplus

ascalarc

onfin

ementp

otentia

lbased

ontheB

Sequatio

nd Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=05119865

119861119888rarr119861

0=039usingaq

uasip

otentia

linther

elativisticqu

arkmod

el[20]

e Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=058119865

119861119888rarr119861

0=039in

then

onrelativisticconstituent

quarkmod

el[2122]

f Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=0573(0571)119865119861119888rarr119861

0=0467(0426)inthelight-fr

ontq

uark

mod

elbasedon

theC

oulombplus

linear(harm

onicoscillator)po

tentialtogether

with

theh

yperfin

einteraction[23]

g Itise

stimated

with

theform

factors119865

119861119888rarr119861119904

0=103119865

119861119888rarr119861

0=101in

ther

elativisticindepend

entq

uark

mod

el[24]

h Itise

stimated

with

inar

elativisticconstituent

quarkmod

el[25]

i Itisestim

ated

with

theform

factors119865

119861119888rarr119861119904

0=13119865

119861119888rarr119861

0=127in

theQ

CDsum

rules[26ndash28]

j Itisestim

ated

with

thep

erturbativeQ

CDapproach

basedon

the119896

119879factorizationscheme[2930]

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

Table 4 Hierarchy of amplitudes among the QCDFrsquos branching ratios for 119861119888decay

Case Coefficient CKM factor Branching ratio Decay modes1 a 119886

1|119881119906119889119881lowast

119888119904| sim 1 ≳10minus2 119861

119904120587 119861

119904120588

1 b 1198861

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus3 119861

119904119870 119861

119889120587 119861

119889120588

1 c 1198861

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus5 119861119889119870 119861

119889119870lowast

2 a 1198862

|119881119906119889119881lowast

119888119904| sim 1 ≳10minus3 119861

+

119906119870

0 119861+119906119870

lowast0

2 b 1198862

|119881119906119889119881lowast

119888119889| |119881

119906119904119881lowast

119888119904| sim 120582 ≳10minus4 119861

119906120587 119861

119906120588 119861

119906120596

2 c 1198862

|119881119906119904119881lowast

119888119889| sim 120582

2

≳10minus6 119861+

1199061198700 119861+

119906119870lowast0

orbital angular momentum for the 119861119875 final states is ℓ119861119875= 0

while the orbital angular momentum is ℓ119861119881= 1 for the 119861119881

final states(3) According to the CKM factors and the coefficients

11988612 there is another hierarchy of amplitudes among the

QCDFrsquos branching ratios for the 119861119888decays which could

be subdivided into different cases (see Table 4) The CKM-favored 119886

1-dominated 119861

119888rarr 119861

119904120587 decays are expected to

have the largest branching ratio sim10 within the QCDFframework In addition the branching ratios for the Cabibbofavored 119861+

119888rarr 119861

0

119904120588+ 119861+

119906119870

0 decays are also larger than 1which might be promisingly detected at experiments(4)There are many uncertainties on the QCDFrsquos results

The first uncertainty from the CKM factors is small due tothe high precision on Wolfenstein parameter 120582 with only03 relative errors [1] Large uncertainty comes from therenormalization scale especially for the 119886

2dominated 119861

119888rarr

119861119906119875 119861

119906119881 decays In principle the second uncertainty could

be reduced by the inclusion of higher order 120572119904corrections to

hadronic matrix elements It has been showed [77 78] thattree amplitudes incorporatingwith theNNLOcorrections arerelatively less sensitive to the choice of scale than the NLOamplitudes As aforementioned large uncertainty mainlycomes from hadron parameters such as the transition formfactors which is expected to be cancelled from the rate ofbranching ratios For example

B119903 (119861119888rarr 119861

119904119870)

B119903 (119861119888rarr 119861

119904120587)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870

1198912

120587

asymp

B119903 (119861119888rarr 119861

119889119870)

B119903 (119861119888rarr 119861

119889120587)

(16)

B119903 (119861119888rarr 119861

119904119870lowast

)

B119903 (119861119888rarr 119861

119904120588)

asymp1003816100381610038161003816119881119906119904

1003816100381610038161003816

2 1198912

119870lowast

1198912

120588

asymp

B119903 (119861119888rarr 119861

119889119870lowast

)

B119903 (119861119888rarr 119861

119889120588)

(17)

B119903 (119861119888rarr 119861

119906120587)

B119903 (119861119888rarr 119861

119889120587)

asymp

1

2

10038161003816100381610038161198862

1003816100381610038161003816

2

10038161003816100381610038161198861

1003816100381610038161003816

2asymp

B119903 (119861119888rarr 119861

119906120588)

B119903 (119861119888rarr 119861

119889120588)

(18)

Particularly the relation of (18) might be used to givesome information on the coefficients 119886

12and to provide an

interesting feasibility research on the validity of the QCDFapproach for the charm quark decay Finally we would like topoint out that many uncertainties from other factors such asthe final state interactions which deserve the dedicated studyare not considered here So one should not be too seriousabout the numbers inTable 3Despite this our resultswill still

provide some useful information to experimental physiciststhat is the Cabibbo favored119861

119888rarr 119861

119904120587119861

119904120588119861

119906119870 decays have

large branching ratios ≳ 1 which could be detected earlier

4 Summary

In prospects of the potential 119861119888meson at the LHCb experi-

ments accurate and thorough studies of the 119861119888decays will be

accessible very soonThe carefully theoretical study on the 119861119888

decays is urgently desiderated In this paper we concentratedon the nonfactorizable contributions to hadronic matrixelements within the QCDF framework while the transitionform factors are taken as nonperturbative inputs which isdifferent from previous studies It is found that the branchingratios for the Cabibbo favored 119861

119888rarr 119861

119904120587 119861

119904120588 119861

119906119870 decays

are very large and could be measured earlier by the runningLHCb experiment in the forthcoming years

Appendix

Decay Amplitudes

A (119861+

119888997888rarr 119861

0

119904120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891120587(119898

2

119861119888

minus 1198982

119861119904

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119904

0119891119870(119898

2

119861119888

minus 1198982

119861119904

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904120588+

)

= radic2119866119865119865119861119888rarr119861119904

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119904119870lowast+

)

= radic2119866119865119865119861119888rarr119861119904

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881199041198861

A (119861+

119888997888rarr 119861

0

119889120587+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891120587(119898

2

119861119888

minus 1198982

119861119889

)119881119906119889119881lowast

1198881198891198861

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

A (119861+

119888997888rarr 119861

0

119889119870+

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119889

0119891119870(119898

2

119861119888

minus 1198982

119861119889

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889120588+

)

= radic2119866119865119865119861119888rarr119861119889

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

0

119889119870lowast+

)

= radic2119866119865119865119861119888rarr119861119889

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198861

A (119861+

119888997888rarr 119861

+

119906119870

0

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

1199061198700

)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891119870(119898

2

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906119870

lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119889119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906119870lowast0

)

= radic2119866119865119865119861119888rarr119861119906

1119891119870lowast119898

119870lowast (120598

119870lowast sdot 119901

119861119888

)119881119906119904119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205870

)

= +119894

119866119865

2

119865119861119888rarr119861119906

0119891120587(119898

2

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

1199061205880

)

= minus119866119865119865119861119888rarr119861119906

1119891120588119898120588(120598120588sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120596)

= +119866119865119865119861119888rarr119861119906

1119891120596119898120596(120598120596sdot 119901

119861119888

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119902)

= minus119894

119866119865

2

119865119861119888rarr119861119906

0119891120578119902

(1198982

119861119888

minus 1198982

119861119906

)119881119906119889119881lowast

1198881198891198862

A (119861+

119888997888rarr 119861

+

119906120578119904)

= minus119894

119866119865

radic2

119865119861119888rarr119861119906

0119891120578119904

(1198982

119861119888

minus 1198982

119861119906

)119881119906119904119881lowast

1198881199041198862

A (119861+

119888997888rarr 119861

+

119906120578)

= cos120601A (119861+119888997888rarr 119861

+

119906120578119902) minus sin120601A (119861+

119888997888rarr 119861

+

119906120578119904)

A (119861+

119888997888rarr 119861

+

1199061205781015840

)

= sin120601A (119861+119888997888rarr 119861

+

119906120578119902) + cos120601A (119861+

119888997888rarr 119861

+

119906120578119904)

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant nos 11475055 11275057 andU1232101) N Wang thanks the support from CCNU-QLPLInnovation Fund (QLPL201411) Q Chang is also supportedby the Foundation for the Author of National ExcellentDoctoral Dissertation of China (Grant no 201317) and theProgram for Science and Technology Innovation Talents inUniversities of Henan Province (Grant no 14HASTIT036)

References

[1] K Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 38 no 9 Article ID 0900012014

[2] N Brambilla M Kramer R Mussa et al ldquoHeavy quarkoniumphysicsrdquo httparxivorgabshep-ph0412158

[3] F Abe H Akimoto A Akopian et al ldquoObservation of Bcmesons in 119901119901 collisions at radic119904 = 18TeVrdquo Physical Review Dvol 58 no 11 Article ID 112004 29 pages 1998

[4] F Abe H Akimoto A Akopian et al ldquoObservation of the 119861119888

meson in 119901119901 collisions atradic119904 = 18TeVrdquo Physical Review Lettersvol 81 p 2432 1998

[5] R Aaij C A Beteta B Adeva et al ldquoObservation of 119861+119888rarr

119869120595119863+

119904and 119861+

119888rarr 119869120595119863

lowast+

119904decaysrdquo Physical Review D vol 87

Article ID 112012 2013[6] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of the

lifetime of the 119861+119888meson using the 119861+

119888rarr 119869120595120587

+ decay moderdquoPhysics Letters B vol 742 pp 29ndash37 2015

[7] I P Gouz V V Kiselev A K Likhoded V I Romanovskyand O P Yushchenko ldquoProspects for the Bc studies at LHCbrdquoPhysics of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004

[8] A Likhoded and A Luchinsky ldquoLight hadron production in119861119888rarr 119861

119904

(lowast)

+ 119883 decaysrdquo Physical Review D vol 82 Article ID014012 2010

[9] M Lusignoli and M Masetti ldquoBc decaysrdquo Zeitschrift fur PhysikC Particles and Fields vol 51 no 4 pp 549ndash555 1991

[10] C Chang and Y Chen ldquoDecays of the 119861119888mesonrdquo Physical

Review D vol 49 no 7 pp 3399ndash3411 1994[11] S S Gershtein V V Kiselev A K Likhoded and A V

Tkabladze ldquoReviews of topical problems physics of Bc-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[12] R Aaij B Adeva M Adinolfi et al ldquoObservation of the decay119861+

119888rarr 119861

0

119904120587+rdquo Physical Review Letters vol 111 Article ID 181801

2013[13] T Feldmann P Kroll andB Stech ldquoMixing and decay constants

of pseudoscalar mesonsrdquo Physical Review D vol 58 Article ID114006 1998

[14] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 10 1999

[15] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 4 2006

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

[16] P Ball and G Jones ldquoTwist-3 distribution amplitudes of119870lowast and120601mesonsrdquo Journal ofHigh Energy Physics vol 2007 no 3 article069 2007

[17] D Du and Z Wang ldquoPredictions of the standard model for 119861plusmn119888

weak decaysrdquo Physical Review D vol 39 no 5 pp 1342ndash13481989

[18] C-H Chang andY-Q Chen ldquoDecays of theB119888mesonrdquoPhysical

Review D vol 49 no 7 Article ID 3399 1994[19] A El-Hady J Munoz and J Vary ldquoSemileptonic and nonlep-

tonic 119861119888decaysrdquo Physical Review D vol 62 Article ID 014019

2000[20] D Ebert R N Faustov and V O Galkin ldquoWeak decays of the

119861119888meson to 119861

119904and B mesons in the relativistic quark modelrdquo

European Physical Journal C vol 32 no 1 pp 29ndash43 2003[21] E Hernandez J Nieves and J Verde-Velasco ldquoStudy of exclu-

sive semileptonic andnonleptonic decays of119861119888in a nonrelativis-

tic quark modelrdquo Physical Review D vol 74 Article ID 0740082006

[22] E Hernandez J Nieves and J M Verde-Velasco ldquoStudyof semileptonic and nonleptonic decays of the Bc-MesonrdquoEuropean Physical Journal A vol 31 no 4 pp 714ndash717 2007

[23] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861119888

meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D vol 80 Article ID 1140032009

[24] S Naimuddin S Kar M Priyadarsini N Barik and P C DashldquoNonleptonic two-body 119861

119888-meson decaysrdquo Physical Review D

vol 86 Article ID 094028 2012[25] M Ivanov J Korner and P Santorelli ldquoExclusive semileptonic

and nonleptonic decays of the 119861119888mesonrdquo Physical Review D

vol 73 Article ID 054024 2006[26] V V Kiselev A E Kovalsky andA K Likhoded ldquoB

119888decays and

lifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[27] V V Kiselev A E Kovalsky and A K Likhoded ldquoBc-Mesondecays and lifetime within QCD sum rulesrdquo Physics of AtomicNuclei vol 64 no 10 pp 1860ndash1875 2001

[28] I P Gouz V V Kiselev A K Likhoded V I Romanovsky andO P Yushchenko ldquoProspects for the119861

119888studies at LHCbrdquoPhysics

of Atomic Nuclei vol 67 no 8 pp 1559ndash1570 2004[29] J Sun Y Yang Q Chang and G Lu ldquoPhenomenological study

of the 119861119888rarr 119861119875 BV decays with perturbative QCD approachrdquo

Physical Review D vol 89 no 11 Article ID 114019 17 pages2014

[30] J Sun Y Yang and G Lu ldquoStudy of the 119861119888rarr 119861

119904120587 decay

with the perturbative QCD approachrdquo Science China PhysicsMechanics amp Astronomy vol 57 no 10 pp 1891ndash1897 2014

[31] M Beneke and G Buchalla ldquo119861119888meson lifetimerdquo Physical

Review D vol 53 article 4991 1996[32] C Chang S Chen T Feng and X Li ldquoLifetime of the 119861

119888meson

and some relevant problemsrdquo Physical ReviewD vol 64 ArticleID 014003 2001

[33] C-H Chang S-L Chen T-F Feng and X-Q Li ldquoStudy of theB119888-Meson LifetimerdquoCommunications inTheoretical Physics vol

35 no 1 p 57 2001[34] M Lusignoil and M Masetti ldquo119861

119888decaysrdquo Zeitschrift fur Physik

C Particles and Fields vol 51 no 4 pp 549ndash555 1991[35] A Y Anisimov P Y Kulikov I M Narodetskii and K A Ter-

Martirosyan ldquoExclusive and inclusive decays of the Bc mesonin the light-front ISGW modelrdquo Physics of Atomic Nuclei vol62 no 10 pp 1739ndash1753 1999

[36] R Dhir N Sharma and R Verma ldquoFlavor dependence of 119861119888

meson form factors and 119861119888rarr 119875119875 decaysrdquo Journal of Physics

G Nuclear and Particle Physics vol 35 no 8 Article ID 0850022008

[37] R Dhir and R Verma ldquoB119888meson form factors and 119861

119888rarr

119875119881 decays involving flavor dependence of transverse quarkmomentumrdquo Physical Review D vol 79 Article ID 0340042009

[38] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C Particles andFields vol 29 no 4 pp 637ndash642 1985

[39] M Bauer B Stech and M Wirbel ldquoExclusive non-leptonicdecays of D- D

119904- and B-mesonsrdquo Zeitschrift fur Physik C

Particles and Fields vol 34 no 1 pp 103ndash115 1987[40] N Isgur D Scora B Grinstein and M B Wise ldquoSemileptonic

B and D decays in the quark modelrdquo Physical Review D vol 39no 3 pp 799ndash818 1989

[41] J Liu and K Chao ldquo119861119888meson weak decays and CP violationrdquo

Physical Review D vol 56 no 7 pp 4133ndash4145 1997[42] P Colangelo and F de Fazio ldquoUsing heavy quark spin symmetry

in semileptonic B119888decaysrdquo Physical Review D vol 61 no 3

Article ID 034012 11 pages 2000[43] R C Verma and A Sharma ldquoQuark diagram analysis of weak

hadronic decays of the 119861+119888mesonrdquo Physical Review D vol 65

Article ID 114007 2002[44] C Chang and H Li ldquoThree-scale factorization theorem and

effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 p 5577 1997

[45] T-W Yeh and H-N Li ldquoFactorization theorems effective fieldtheory and nonleptonic heavy meson decaysrdquo Physical ReviewD vol 56 pp 1615ndash1631 1997

[46] Y Keum H Li and A Sanda ldquoFat penguins and imaginarypenguins in perturbative QCDrdquo Physics Letters B vol 504 no1-2 pp 6ndash14 2001

[47] Y-Y Keum and H-N Li ldquoNonleptonic charmless B decaysfactorization versus perturbative QCDrdquo Physical Review D vol63 Article ID 074006 2001

[48] C Lu K Ukai andM Yang ldquoBranching ratio and CP violationof 120587120587 decays in the perturbative QCD approachrdquo PhysicalReview D vol 63 Article ID 074009 2001

[49] C-D Lu and M-Z Yang ldquo119861 rarr 120587120588 120587120596 decays in perturbativeQCD approachrdquoThe European Physical Journal C vol 23 no 2pp 275ndash287 2002

[50] C Bauer S Fleming and M Luke ldquoSumming Sudakov loga-rithms in 119861 rarr 119883

119904120574 in effective field theoryrdquo Physical Review

D vol 63 Article ID 014006 2001[51] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

effective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 Article ID 114020 2001

[52] C W Bauer and I W Stewart ldquoInvariant operators in collineareffective theoryrdquo Physics Letters B vol 516 no 1-2 pp 134ndash1422001

[53] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 Article ID 054022 2002

[54] C Bauer S Fleming D Pirjol I Z Rothstein and I WStewart ldquoHard scattering factorization from effective fieldtheoryrdquo Physical Review D vol 66 no 1 Article ID 014017 23pages 2002

[55] M Beneke A P Chapovsky M Diehl and T Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyond

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

leading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[56] M Beneke and T Feldmann ldquoMultipole-expanded soft-collinear effective theory with non-Abelian gauge symmetryrdquoPhysics Letters B vol 553 no 3-4 pp 267ndash276 2003

[57] M Beneke and T Feldmann ldquoFactorization of heavy-to-lightform factors in soft-collinear effective theoryrdquo Nuclear PhysicsB vol 685 no 1ndash3 pp 249ndash296 2004

[58] M Beneke G BuchallaMNeubert andC T Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[59] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization for exclusive non-leptonicB-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[60] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD factorization in 119861 rarr 120587119870 120587120587 decays and extraction ofWolfenstein parametersrdquoNuclear Physics B vol 606 no 1-2 pp245ndash321 2001

[61] D Du D Yang and G Zhu ldquoInfrared divergence and twist-3 distribution amplitudes in QCD factorization for Brarr PPrdquoPhysics Letters B vol 509 no 3-4 pp 263ndash272 2001

[62] D Du D Yang and G Zhu ldquoAnalysis of the decays 119861 rarr

120587120587 and 120587K with QCD factorization in the heavy quark limitrdquoPhysics Letters B vol 488 no 1 pp 46ndash54 2000

[63] D Du D Yang and G Zhu ldquoQCD factorization for PPrdquoPhysical Review D vol 64 Article ID 014036 2001

[64] G Buchalla A Buras and M Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68p 1125 1996

[65] A J Buras ldquoWeak hamiltonian CP violation and rare decaysrdquohttparxivorgabshep-ph9806471

[66] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomenolog-ical analysis of PP decays with QCD factorizationrdquo PhysicalReview D vol 65 Article ID 074001 2002

[67] D Du H Gong J Sun D Yang and G Zhu ldquoPhenomeno-logical analysis of charmless decays 119861 rarr PV with QCDfactorizationrdquo Physical Review D vol 65 Article ID 0940252002 Erratum in Physical Review D vol 66 Article ID 0799042002

[68] J Sun G Zhu and D Du ldquoPhenomenological analysis ofcharmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 Article ID 054003 2003[69] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

and 119861 rarr 119875119881 decaysrdquo Nuclear Physics B vol 675 no 1-2 pp333ndash415 2003

[70] M Beneke J Rohrer and D Yang ldquoBranching fractionspolarisation and asymmetries of B rarr VV decaysrdquo NuclearPhysics B vol 774 no 1ndash3 pp 64ndash101 2007

[71] X Li and Y Yang ldquoReexamining charmless 119861 rarr 119875119881 decaysin the QCD factorization approachrdquo Physical Review D vol 73no 11 Article ID 114027 24 pages 2006

[72] H-Y Cheng andK-C Yang ldquoBranching ratios and polarizationin 119861 rarr 119881119881VAAA decaysrdquo Physical Review D vol 78 ArticleID 094001 2008 Erratum in Physical Review D vol 79 ArticleID 039903 2009

[73] H-Y Cheng andC-K Chua ldquoResolving119861minus119862119875 puzzles inQCDfactorizationrdquo Physical Review D vol 80 Article ID 0740312009

[74] H Cheng and C Chua ldquoRevisiting charmless hadronic B119906119889

decays in QCD factorizationrdquo Physical Review D vol 80 no11 Article ID 114008 33 pages 2009

[75] H Cheng and C Chua ldquoQCD factorization for charmlesshadronic B

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 25 pages 2009[76] H Cheng and C Chua ldquoCharmless hadronic B decays into a

tensor mesonrdquo Physical Review D vol 83 Article ID 0340012011

[77] G Bell ldquoNNLO vertex corrections in charmless hadronic Bdecays real partrdquo Nuclear Physics B vol 822 no 1-2 pp 172ndash200 2009

[78] M Beneke T Huber and X-Q Li ldquoNNLO vertex correctionsto non-leptonic B decays tree amplitudesrdquo Nuclear Physics Bvol 832 no 1-2 pp 109ndash151 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of