Research ArticleStudy of the 120595(1119878 2119878) and 120578

119888(1119878 2119878)Weak Decays into 119863119872

Junfeng Sun1 Yueling Yang1 Jinshu Huang2 Lili Chen1 and Qin Chang1

1 Institute of Particle and Nuclear Physics Henan Normal University Xinxiang 453007 China2College of Physics and Electronic Engineering Nanyang Normal University Nanyang 473061 China

Correspondence should be addressed to Yueling Yang yangyuelinghtueducn

Received 20 October 2015 Revised 19 January 2016 Accepted 20 January 2016

Academic Editor Sally Seidel

Copyright copy 2016 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

Inspired by the recent measurements on the 119869120595(1119878) rarr 119863119904120588 119863

119906119870

lowast weak decays at BESIII and the potential prospects ofcharmonium at high-luminosity heavy-flavor experiments we study120595(1119878 2119878) and 120578

119888(1119878 2119878)weak decays into final states including

one charmed meson plus one light meson considering QCD corrections to hadronic matrix elements with QCD factorizationapproach It is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863

minus

119904120588+119863minus

119904120587+1198630

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which

might be accessible at future experiments

1 Introduction

More than forty years after the discovery of the 119869120595(1119878)

meson the properties of charmonium (bound state of 119888119888)continue to be the subject of intensive theoretical and exper-imental study It is believed that charmonium resemblingbottomonium (bound state of 119887119887) plays the same role inexploring hadronic dynamics as positronium andor thehydrogen atom in understanding the atomic physics Char-monium and bottomonium are good objects to test the basicideas of QCD [1] There is a renewed interest in charmoniumdue to the plentiful dedicated investigation fromBES CLEO-c LHCb and the studies via decays of the 119861 mesons at 119861factories

The 120595(1119878 2119878) and 120578119888(1119878 2119878) mesons are 119878-wave char-

monium states below open-charm kinematic threshold andhave the well-established quantum numbers of 119868119866119869119875119862 =

0+

1minusminus and 0

+

0minus+ respectively They decay mainly through

the strong and electromagnetic interactions Because the 119866-parity conserving hadronic decays 120595(2119878) rarr 120587120587119869120595(1119878)120578119869120595(1119878) and 120578

119888(2119878) rarr 120587120587120578

119888(1119878) are suppressed by the

compact phase space of final states and because the decaysinto light hadrons are suppressed by the phenomenologicalOkubo-Zweig-Iizuka (OZI) rules [2ndash4] the total widths of120595(1119878 2119878) and 120578

119888(1119878 2119878) are narrow (see Table 1) which

might render the charmonium weak decay as a necessarysupplement Here we will concentrate on the 120595(1119878 2119878) and120578119888(1119878 2119878)weak decays into119863119872 final states where119872 denotes

the low-lying 119878119880(3) pseudoscalar and vector meson nonetOur motivation is listed as follows

From the experimental point of view (1) some10

9

120595(1119878 2119878) data samples have been collected by BESIIIsince 2009 [5] It is inspiringly expected to have about10 billion 119869120595(1119878) and 3 billion 120595(2119878) events at BESIIIexperiment per year of data taking with the designedluminosity [6] over 10

10

119869120595(1119878) at LHCb [7] ATLAS[8] and CMS [9] per fbminus1 data in 119901119901 collisions A largeamount of data sample offers a realistic possibility toexplore experimentally the charmonium weak decaysCorrespondingly theoretical study is very necessary toprovide a ready reference (2) Identification of the single119863 meson would provide an unambiguous signature ofthe charmonium weak decay into 119863119872 states With theimprovements of experimental instrumentation and particleidentification techniques accurate measurements on thenonleptonic charmonium weak decay might be feasibleRecently a search for the 119869120595(1119878) rarr 119863

119904120588 119863

119906119870

lowast decays hasbeen performed at BESIII although signals are unseen for themoment [10] Of course the branching ratios for the inclusivecharmonium weak decay are tiny within the standard model

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 5071671 11 pageshttpdxdoiorg10115520165071671

2 Advances in High Energy Physics

Table 1 The properties of 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons [12]

Meson 119868119866

119869119875119862 Mass (MeV) Width

120595(1119878) 0+

1minusminus

3096916 plusmn 0011 929 plusmn 28 keV120595(2119878) 0

+

1minusminus

3686109+0012

minus0014299 plusmn 8 keV

120578119888(1119878) 0

+

0minus+

29836 plusmn 07 322 plusmn 09MeV120578119888(2119878) 0

+

0minus+

36394 plusmn 13 113+32

minus29MeV

about 2(120591119863Γ120595) sim 10

minus8 and 2(120591119863Γ120578119888

) sim 10minus10 where 119863

denotes the neutral charmed meson [11] and Γ120595and Γ

120578119888

standfor the total widths of 120595(1119878 2119878) and 120578

119888(1119878 2119878) resonances

respectively Observation of an abnormally large productionrate of single charmed mesons in the final state would be ahint of new physics beyond the standard model [11]

From the theoretical point of view (1) the charm quarkweak decay is more favorable than the bottom quark weakdecay because the Cabibbo-Kobayashi-Maskawa (CKM)matrix elements obey |119881

119888119887| ≪ |119881

119888119904| [12] Penguin and

annihilation contributions to nonleptonic charm quark weakdecay being proportional to the CKM factor |119881

119888119887119881119906119887| sim

O(1205825) with the Wolfenstein parameter 120582 ≃ 022 [12]are highly suppressed and hence negligible relative to treecontributions Both 119888 and 119888 quarks in charmonium candecay individually which provides a good place to investigatethe dynamical mechanism of heavy-flavor weak decay andcrosscheck model parameters obtained from the charmedhadron weak decays (2) There are few works devoted tononleptonic 119869120595(1119878) weak decays in the past such as [13]with the covariant light-cone quark model [14] with QCDsum rules and [15ndash17] with the Wirbel-Stech-Bauer (WSB)model [18] Moreover previous works of [13ndash17] concernmainly the weak transition form factors between the 119869120595(1119878)and charmed mesons Fewer papers have been devoted tononleptonic120595(2119878) and 120578

119888(1119878 2119878)weak decays until now even

though a rough estimate of branching ratios is unavailableIn this paper we will estimate the branching ratios fornonleptonic two-body charmonium weak decay taking thenonfactorizable contributions to hadronic matrix elementsinto account with the attractive QCD factorization (QCDF)approach [19]

This paper is organized as follows In Section 2 we willpresent the theoretical framework and the amplitudes for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays Section 3 is devoted

to numerical results and discussion Finally Section 4 is oursummation

2 Theoretical Framework

21 The Effective Hamiltonian Phenomenologically theeffective Hamiltonian responsible for charmonium weakdecay into119863119872 final states can be written as follows [25]

Heff

=

119866119865

radic2

sum

11990211199022

119881lowast

1198881199021

1198811199061199022

1198621(120583)119876

1(120583) + 119862

2(120583)119876

2(120583)

+Hc

(1)

where 119866119865= 1166 times 10

minus5 GeVminus2 [12] is the Fermi couplingconstant 119881lowast

1198881199021

1198811199061199022

is the CKM factor with 11990212

= 119889 119904the Wilson coefficients 119862

12(120583) which are independent of

one particular process summarize the physical contributionsabove the scale of 120583 The expressions of the local tree four-quark operators are

1198761= [119902

1120572120574120583(1 minus 120574

5) 119888

120572] [119906

120573120574120583

(1 minus 1205745) 119902

2120573]

1198762= [119902

1120572120574120583(1 minus 120574

5) 119888

120573] [119906

120573120574120583

(1 minus 1205745) 119902

2120572]

(2)

where 120572 and 120573 are color indicesIt is well known that the Wilson coefficients 119862

119894could be

systematically calculated with perturbation theory and haveproperly been evaluated to the next-to-leading order (NLO)Their values at the scale of 120583 sim O(119898

119888) can be evaluated with

the renormalization group (RG) equation [25]

11986212(120583) = 119880

4(120583119898

119887) 119880

5(119898

119887 119898

119882) 119862

12(119898

119882) (3)

where 119880119891(120583

119891 120583

119894) is the RG evolution matrix which trans-

forms the Wilson coefficients from scale of 120583119894to 120583

119891 The

expression for119880119891(120583

119891 120583

119894) can be found in [25]The numerical

values of the leading-order (LO) and NLO 11986212

in the naivedimensional regularization scheme are listed in Table 2 Thevalues of coefficients 119862

12in Table 2 agree well with those

obtained with ldquoeffectiverdquo number of active flavors 119891 = 415

[25] rather than formula (3)To obtain the decay amplitudes and branching ratios

the remaining works are to evaluate accurately the hadronicmatrix elements (HME) where the local operators are sand-wiched between the charmonium and final states which isalso the most intricate work in dealing with the weak decayof heavy hadrons by now

22 Hadronic Matrix Elements Analogous to the exclusiveprocesses with perturbative QCD theory proposed by Lepageand Brodsky [26] the QCDF approach is developed byBeneke et al [19] to deal with HME based on the collinearfactorization approximation and power counting rules inthe heavy quark limit and has been extensively used for119861 meson decays Using the QCDF master formula HMEof nonleptonic decays could be written as the convolutionintegrals of the process-dependent hard scattering kernelsand universal light-cone distribution amplitudes (LCDA) ofparticipating hadrons

The spectator quark is the heavy-flavor charm quark forcharmonium weak decays into 119863119872 final states It is com-monly assumed that the virtuality of the gluon connectingto the heavy spectator is of order Λ2

QCD where ΛQCD isthe characteristic QCD scale Hence the transition formfactors between charmonium and 119863 mesons are assumed tobe dominated by the soft and nonperturbative contributionsand the amplitudes of the spectator rescattering subprocessare power-suppressed [19] Taking 120578

119888rarr 119863119872 decays for

example HME can be written as

⟨1198631198721003816100381610038161003816119876

12

1003816100381610038161003816120578119888⟩ = sum

119894

119865120578119888rarr119863

119894119891119872int119867

119894(119909)Φ

119872(119909) 119889119909 (4)

Advances in High Energy Physics 3

Table 2 Numerical values of theWilson coefficients11986212

and parameters 11988612

for 120578119888rarr 119863120587 decay with119898

119888= 1275GeV [12] where 119886

12in [20]

is used in the119863meson weak decay

120583

LO NLO QCDF Previous works119862

1119862

2119862

1119862

21198861

1198862

Ref 1198861

1198862

08119898119888

1335 minus0589 1275 minus0504 1275119890+1198944∘

0503119890minus119894154∘

[14 16 17] 126 minus051

119898119888

1276 minus0505 1222 minus0425 1219119890+1198943∘

0402119890minus119894154∘

[15] 13 plusmn 01 minus055 plusmn 010

12119898119888

1240 minus0450 1190 minus0374 1186119890+1198943∘

0342119890minus119894154∘

[20] 1274 minus0529

where 119865120578119888rarr119863

119894is the weak transition form factor and 119891

119872and

Φ119872(119909) are the decay constant and LCDA of the meson 119872

respectively The leading twist LCDA for the pseudoscalarand longitudinally polarized vector mesons can be expressedin terms of Gegenbauer polynomials [23 24]

Φ119872(119909) = 6119909119909

infin

sum

119899=0

119886119872

11989911986232

119899(119909 minus 119909) (5)

where 119909 = 1 minus 119909 11986232

119899(119911) is the Gegenbauer polynomial

11986232

0(119911) = 1

11986232

1(119911) = 3119911

11986232

2(119911) =

3

2

(51199112

minus 1)

(6)

119886119872

119899is the Gegenbauer moment corresponding to the Gegen-

bauer polynomials 11986232

119899(119911) 119886119872

0equiv 1 for the asymptotic form

and 119886119899= 0 for 119899 = 1 3 5 because of the 119866-parity invari-

ance of the 120587 120578(1015840) 120588 120596 120601 meson distribution amplitudesIn this paper to give a rough estimation the contributionsfrom higher-order 119899 ge 3 Gegenbauer polynomials are notconsidered for the moment

Hard scattering function 119867119894(119909) in (4) is in principle

calculable order by order with the perturbative QCD theoryAt the order of 1205720

119904 119867

119894(119909) = 1 This is the simplest scenario

and one goes back to the naive factorization where there is noinformation about the strong phases and the renormalizationscale hidden in the HME At the order of 120572

119904and higher

orders the renormalization scale dependence of hadronicmatrix elements could be recuperated to partly cancel the 120583-dependence of the Wilson coefficients In addition part ofthe strong phases could be reproduced from nonfactorizablecontributions

Within the QCDF framework amplitudes for 120578119888rarr 119863119872

decays can be expressed as

A (120578119888997888rarr 119863119872) = ⟨119863119872

1003816100381610038161003816Heff

1003816100381610038161003816120578119888⟩

=

119866119865

radic2

119881lowast

1198881199021

1198811199061199022

119886119894⟨119872

100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816120578119888⟩

(7)

In addition the HME for the 120595(1119878 2119878) rarr 119863119881 decays areconventionally expressed as the helicity amplitudes with thedecomposition [27 28]

H120582= ⟨119881

100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816120595⟩ = 120598

lowast120583

119881120598]120595119886119892

120583]

+

119887

119898120595119898

119881

(119901120595+ 119901

119863)

120583

119901]119881

+

119894119888

119898120595119898

119881

120598120583]120572120573119901

120572

119881(119901

120595+ 119901

119863)

120573

(8)

The relations among helicity amplitudes and invariant ampli-tudes 119886 119887 119888 are

H0= minus119886119909 minus 2119887 (119909

2

minus 1)

Hplusmn= 119886 plusmn 2119888radic119909

2minus 1

119909 =

119901120595sdot 119901

119881

119898120595119898

119881

=

1198982

120595minus 119898

2

119863+ 119898

2

119881

2119898120595119898

119881

(9)

where three scalar amplitudes 119886 119887 119888 describe 119904 119889 119901 wavecontributions respectively

The effective coefficient 119886119894at the order of 120572

119904can be

expressed as [19]

1198861= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO2V

1198862= 119862

NLO2

+

1

119873119888

119862NLO1

+

120572119904

4120587

119862119865

119873119888

119862LO1V

(10)

where the color factor 119862119865= 43 the color number 119873

119888= 3

For the transversely polarized light vector meson the factorV = 0 in the helicity H

plusmnamplitudes beyond the leading

twist contributions With the leading twist LCDA for thepseudoscalar and longitudinally polarized vectormesons thefactorV is written as [19]

V = 6 log(119898

2

119888

1205832) minus 18 minus (

1

2

+ 1198943120587)

+ (

11

2

minus 1198943120587) 119886119872

1minus

21

20

119886119872

2+ sdot sdot sdot

(11)

From the numbers in Table 2 it is found that (1) the valuesof coefficients 119886

12agree generally with those used in previous

works [14ndash17 20] (2) the strong phases appear by taking

4 Advances in High Energy Physics

nonfactorizable corrections into account which is necessaryfor119862119875 violation and (3) the strong phase of 119886

1is small due to

the suppression of 120572119904and 1119873

119888The strong phase of 119886

2is large

due to the enhancement from the large Wilson coefficients1198621

23 Form Factors The weak transition form factors betweencharmonium and a charmed meson are defined as follows[18]

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120578119888(119901

1)⟩

= (1199011+ 119901

2)120583minus

1198982

120578119888

minus 1198982

119863

1199022

119902120583119865

1(119902

2

)

+

1198982

120578119888

minus 1198982

119863

1199022

1199021205831198650(119902

2

)

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120595 (119901

1 120598)⟩

= minus120598120583]120572120573120598

]120595119902120572

(1199011+ 119901

2)120573

119881(1199022

)

119898120595+ 119898

119863

minus 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

0(119902

2

)

minus 119894120598120595120583

(119898120595+ 119898

119863)119860

1(119902

2

)

minus 119894

120598120595sdot 119902

119898120595+ 119898

119863

(1199011+ 119901

2)120583119860

2(119902

2

)

+ 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

3(119902

2

)

(12)

where 119902 = 1199011minus119901

2 120598

120595denotes the 120595rsquos polarization vectorThe

form factors 1198650(0) = 119865

1(0) and 119860

0(0) = 119860

3(0) are required

compulsorily to cancel singularities at the pole of 1199022 = 0There is a relation among these form factors

2119898120595119860

3(119902

2

) = (119898120595+ 119898

119863)119860

1(119902

2

)

+ (119898120595minus 119898

119863)119860

2(119902

2

)

(13)

There are four independent transition form factors1198650(0)

11986001(0) and119881(0) at the pole of 1199022 = 0 They could be written

as the overlap integrals of wave functions [18]

1198650(0) = intint

1

0

Φ120578119888

(119896perp 119909 0 0)

sdot Φ119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198600(0) = intint

1

0

Φ120595(119896perp 119909 1 0)

sdot 120590119911Φ

119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198601(0) =

119898119888+ 119898

119902

119898120595+ 119898

119863

119868

119881 (0) =

119898119888minus 119898

119902

119898120595minus 119898

119863

119868

119868 = radic2intint

1

0

Φ120595(119896perp 119909 1 minus1) 119894120590

119910Φ

119863(119896perp 119909 0 0)

sdot

1

119909

119889119909 119889119896perp

(14)

where 120590119910119911

is the Pauli matrix acting on the spin indices ofthe decaying charm quark 119909 and

119896perpdenote the fraction of

the longitudinal momentum and the transverse momentumof the nonspectator quark respectively

With the separation of the spin and spatial variables wavefunctions can be written as

Φ(119896perp 119909 119895 119895

119911) = 120601 (

119896perp 119909)

1003816100381610038161003816119904 119904

119911 119904

1 119904

2⟩ (15)

where the total angular momentum 119895 = + 1199041+ 119904

2= 119904

1+

1199042= 119904 because the orbital angular momentum between the

valence quarks in 120595(1119878 2119878) 120578119888(1119878 2119878)119863mesons in question

have = 0 11990412

denote the spins of valence quarks in meson119904 = 1 and 0 for the 120595 and 120578

119888mesons respectively

The charm quark in the charmonium state is nearlynonrelativistic with an average velocity V ≪ 1 basedon arguments of nonrelativistic quantum chromodynamics(NRQCD) [29ndash31] For the 119863 meson the valence quarks arealso nonrelativistic due to 119898

119863asymp 119898

119888+ 119898

119902 where the light

quark mass 119898119906

asymp 119898119889

asymp 310MeV and 119898119904asymp 510MeV

[32] Here we will take the solution of the Schrodingerequation with a scalar harmonic oscillator potential as thewave functions of the charmonium and119863mesons

1206011119878(119896) sim 119890

minus119896

2

21205722

1206012119878(119896) sim 119890

minus119896

2

21205722

(2119896

2

minus 31205722

)

(16)

where the parameter 120572 determines the average transversequark momentum ⟨120601

1119878|119896

2

perp|120601

1119878⟩ = 120572

2 With the NRQCDpower counting rules [29] | 119896

perp| sim 119898V sim 119898120572

119904for heavy

quarkonium Hence parameter 120572 is approximately taken as119898120572

119904in our calculationUsing the substitution ansatz [33]

119896

2

997888rarr

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

4119909119909

(17)

one can obtain

1206011119878(119896perp 119909) = 119860 exp

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

minus81205722119909119909

1206012119878(119896perp 119909) = 119861120601

1119878(119896perp 119909)

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

61205722119909119909

minus 1

(18)

Advances in High Energy Physics 5

Table 3 The numerical values of transition form factors at 1199022 = 0 where uncertainties of this work come from the charm quark mass

Transition Reference 1198650(0) 119860

0(0) 119860

1(0) 119881(0)

120578119888(1119878) 120595(1119878) rarr 119863

119906119889

This work 085 plusmn 001 085 plusmn 001 072 plusmn 001 176 plusmn 003

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 16 plusmn 01

[21]b sdot sdot sdot 027+002

minus003027

+003

minus002081

+012

minus008

[15]c sdot sdot sdot 040 (061) 044 (068) 117 (182)[17]d sdot sdot sdot 055 plusmn 002 077

+009

minus007214

+015

minus011

[17]e sdot sdot sdot 054 080 221

120578119888(1119878) 120595(1119878) rarr 119863

119904

This work 090 plusmn 001 090 plusmn 001 081 plusmn 001 155 plusmn 004

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 18

[21]b sdot sdot sdot 037 plusmn 002 038+002

minus001107

+005

minus002

[15]c sdot sdot sdot 047 (066) 055 (078) 125 (180)[17]d sdot sdot sdot 071

+004

minus002094 plusmn 007 230

+009

minus006

[17]e sdot sdot sdot 069 096 236120578119888(2119878) 120595(2119878) rarr 119863

119906119889This work 062 plusmn 001 061 plusmn 001 054 plusmn 001 100 plusmn 004

120578119888(2119878) 120595(2119878) rarr 119863

119904This work 065 plusmn 001 064 plusmn 001 059 plusmn 002 083 plusmn 004

aThe form factors are computed with the covariant light-front quark model where uncertainties come from the decay constant of charmed mesonbThe form factors are computed with QCD sum rules where uncertainties are from the Borel parameterscThe form factors are computed with parameter 120596 = 04 (05) GeV using the WSB modeldThe form factors are computed with flavor dependent parameter 120596 using the WSB modeleThe form factors are computed with parameter 120596 = 119898120572

119904using the WSB model

where the parameters 119860 and 119861 are the normalization coeffi-cients satisfying the normalization condition

intint

1

0

10038161003816100381610038161003816120601 (

119896perp 119909)

10038161003816100381610038161003816

2

119889119909 119889119896perp= 1 (19)

The numerical values of transition form factors at 1199022 = 0

are listed in Table 3 It is found that (1) themodel dependenceof form factors is large (2) isospin-breaking effects arenegligible and flavor breaking effects are small and (3) asstated in [18] 119865

0≃ 119860

0holds within collinear symmetry

3 Numerical Results and Discussion

In the charmonium center-of-mass frame the branchingratio for the charmonium weak decay can be written as

B119903 (120578119888997888rarr 119863119872) =

119901cm4120587119898

2

120578119888

Γ120578119888

1003816100381610038161003816A (120578

119888997888rarr 119863119872)

1003816100381610038161003816

2

B119903 (120595 997888rarr 119863119872) =

119901cm12120587119898

2

120595Γ120595

1003816100381610038161003816A (120595 997888rarr 119863119872)

1003816100381610038161003816

2

(20)

where the common momentum of final states is

119901cm

=

radic[1198982

120578119888120595minus (119898

119863+ 119898

119872)2

] [1198982

120578119888120595minus (119898

119863minus 119898

119872)2

]

2119898120578119888120595

(21)

The decay amplitudes for A(120595 rarr 119863119872) and A(120578119888rarr 119863119872)

are collected in Appendices A and B respectivelyIn our calculation we assume that the light vectormesons

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

the mixing of pseudoscalar 120578 and 1205781015840 meson we will adopt thequark-flavor basis description proposed in [22] and neglectthe contributions from possible gluonium compositions thatis

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (22)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393 plusmn 10)∘ [22] The mass relations are

1198982

120578119902

= 1198982

120578cos2120601 + 1198982

1205781015840sin2120601

minus

radic2119891120578119904

119891120578119902

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

1198982

120578119904

= 1198982

120578sin2120601 + 1198982

1205781015840cos2120601

minus

119891120578119902

radic2119891120578119904

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

(23)

The input parameters including the CKM Wolfensteinparameters decay constants and Gegenbauer moments arecollected in Table 4 If not specified explicitly we will taketheir central values as the default inputs Our numericalresults on branching ratios for the nonleptonic two-body120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 weak decays are displayed in

Tables 5 and 6 where the uncertainties of this work comefrom theCKMparameters the renormalization scale120583 = (1plusmn

02)119898119888 and hadronic parameters including decay constants

and Gegenbauer moments respectively For comparisonprevious results on 119869120595(1119878) weak decays [14 16 17] withparameters 119886

1= 126 and 119886

2= minus051 are also listed in Table 5

The following are some comments

6 Advances in High Energy Physics

Table 4 Numerical values of input parameters

120582 = 022537 plusmn 000061 [12] 119860 = 0814+0023

minus0024[12]

120588 = 0117 plusmn 0021 [12] 120578 = 0353 plusmn 0013 [12]119898

119888= 1275 plusmn 0025GeV [12] 119898

119863119906

= 186484 plusmn 007MeV [12]119898

119863119889

= 186961 plusmn 010MeV [12] 119898119863119904

= 196830 plusmn 011MeV [12]119891120587= 13041 plusmn 020MeV [12] 119891

119870= 1562 plusmn 07MeV [12]

119891120578119902

= (107 plusmn 002) 119891120587[22] 119891

120578119904

= (134 plusmn 006) 119891120587[22]

119891120588= 216 plusmn 3MeV [23] 119891

120596= 187 plusmn 5MeV [23]

119891120601= 215 plusmn 5MeV [23] 119891

119870lowast = 220 plusmn 5MeV [23]

119886120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [24] 119886

120588

2= 119886

120596

2= 015 plusmn 007 [23]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [24] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [24]

119886119870

lowast

1= minus119886

119870lowast

1= 003 plusmn 002 [23] 119886

119870lowast

2= 119886

119870

lowast

2= 011 plusmn 009 [23]

119886120587

1= 119886

120588

1= 119886

120596

1= 119886

120601

1= 0 119886

120601

2= 018 plusmn 008 [23]

Table 5 Branching ratios for the nonleptonic two-body 119869120595(1119878) weak decays where the uncertainties of this work come from the CKMparameters the renormalization scale 120583 = (1 plusmn 02)119898

119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively The results of [14 16 17] are calculated with 1198861= 126 and 119886

2= minus051 The results of [14] are based on QCD sum rules The

numbers in columns of ldquoArdquo ldquoBrdquo ldquoCrdquo and ldquoDrdquo are based on the WSB model with flavor dependent 120596 QCD inspired 120596 = 120572119904119898 and universal

120596 = 04GeV and 05GeV respectively

Final states Case Reference [14] Reference [17] Reference [16] This workA B C D

119863minus

119904120587+ 1-a 20 times 10

minus10

741 times 10minus10

713 times 10minus10

332 times 10minus10

874 times 10minus10

(109+001+010+001

minus001minus006minus001) times 10

minus9

119863minus

119904119870

+ 1-b 16 times 10minus11

53 times 10minus11

52 times 10minus11

24 times 10minus11

55 times 10minus11

(618+003+059+008

minus003minus033minus008) times 10

minus11

119863minus

119889120587+ 1-b 08 times 10

minus11

29 times 10minus11

28 times 10minus11

15 times 10minus11

55 times 10minus11

(637+003+060+003

minus003minus034minus003) times 10

minus11

119863minus

119889119870

+ 1-c sdot sdot sdot 23 times 10minus12

22 times 10minus12

12 times 10minus12

sdot sdot sdot (379+004+036+005

minus004minus020minus005) times 10

minus12

119863

0

1199061205870 2-b sdot sdot sdot 24 times 10

minus12

23 times 10minus12

12 times 10minus12

55 times 10minus12

(350+002+198+006

minus002minus097minus006) times 10

minus12

119863

0

119906119870

0 2-c sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

sdot sdot sdot (416+004+235+011

minus004minus115minus010) times 10

minus13

119863

0

119906119870

0 2-a 36 times 10minus11

139 times 10minus10

134 times 10minus10

72 times 10minus11

28 times 10minus10

(144+001+081+003

minus001minus040minus003) times 10

minus10

119863

0

119906120578 sdot sdot sdot 70 times 10

minus12

67 times 10minus12

36 times 10minus12

16 times 10minus12

(103+001+058+010

minus001minus028minus010) times 10

minus11

119863

0

1199061205781015840

sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

30 times 10minus13

(583+003+329+172

minus003minus161minus150) times 10

minus13

119863minus

119904120588+ 1-a 126 times 10

minus9

511 times 10minus9

532 times 10minus9

177 times 10minus9

363 times 10minus9

(382+001+036+011

minus001minus020minus011) times 10

minus9

119863minus

119904119870

lowast+ 1-b 082 times 10minus10

282 times 10minus10

296 times 10minus10

097 times 10minus10

212 times 10minus10

(200+001+019+010

minus001minus011minus009) times 10

minus10

119863minus

119889120588+ 1-b 042 times 10

minus10

216 times 10minus10

228 times 10minus10

072 times 10minus10

220 times 10minus10

(212+001+020+006

minus001minus011minus006) times 10

minus10

119863minus

119889119870

lowast+ 1-c sdot sdot sdot 13 times 10minus11

13 times 10minus11

42 times 10minus12

sdot sdot sdot (114+001+011+006

minus001minus006minus005) times 10

minus11

119863

0

1199061205880 2-b sdot sdot sdot 18 times 10

minus11

19 times 10minus11

60 times 10minus12

22 times 10minus11

(108+001+061+004

minus001minus030minus004) times 10

minus11

119863

0

119906120596 2-b sdot sdot sdot 16 times 10

minus11

17 times 10minus11

50 times 10minus12

18 times 10minus11

(810+004+456+050

minus004minus225minus048) times 10

minus12

119863

0

119906120601 2-b sdot sdot sdot 42 times 10

minus11

44 times 10minus11

14 times 10minus11

65 times 10minus11

(192+001+108+010

minus001minus053minus010) times 10

minus11

119863

0

119906119870

lowast0 2-c sdot sdot sdot 21 times 10minus12

22 times 10minus12

70 times 10minus13

sdot sdot sdot (119+001+067+007

minus001minus033minus007) times 10

minus12

119863

0

119906119870

lowast0 2-a 154 times 10minus10

761 times 10minus10

812 times 10minus10

251 times 10minus10

103 times 10minus9

(409+001+230+024

minus001minus114minus023) times 10

minus10

(1) There are some differences among the estimates ofbranching ratios for 119869120595(1119878) rarr 119863119872weak decays (seethe numbers in Table 5)These inconsistencies amongprevious works although the same values of param-eters 119886

12are used come principally from different

values of form factors Our results are generally in linewith the numbers in columns ldquoArdquo and ldquoBrdquo which arefavored by [17]

(2) Branching ratios for 119869120595(1119878) weak decay are abouttwo or more times as large as those for 120595(2119878) decayinto the same final states because the decay width of120595(2119878) is about three times as large as that of 119869120595(1119878)

(3) Due to the relatively small decay width and relativelylarge space phases for 120578

119888(2119878) decay branching ratios

for 120578119888(2119878) weak decay are some five (ten) or more

times as large as those for 120578119888(1119878) weak decay into the

same119863119875 (119863119881) final states

(4) Among 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons 120578

119888(1119878) has a

maximal decay width and a minimal mass resultingin a small phase space while 119869120595(1119878) has a minimaldecay width These facts lead to the smallest [or thelargest] branching ratio for 120578

119888(1119878) [or 119869120595(1119878)] weak

decay among 120595(1119878 2119878) 120578119888(1119878 2119878) weak decays into

the same final states

Advances in High Energy Physics 7

Table 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

Table 1 The properties of 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons [12]

Meson 119868119866

119869119875119862 Mass (MeV) Width

120595(1119878) 0+

1minusminus

3096916 plusmn 0011 929 plusmn 28 keV120595(2119878) 0

+

1minusminus

3686109+0012

minus0014299 plusmn 8 keV

120578119888(1119878) 0

+

0minus+

29836 plusmn 07 322 plusmn 09MeV120578119888(2119878) 0

+

0minus+

36394 plusmn 13 113+32

minus29MeV

about 2(120591119863Γ120595) sim 10

minus8 and 2(120591119863Γ120578119888

) sim 10minus10 where 119863

denotes the neutral charmed meson [11] and Γ120595and Γ

120578119888

standfor the total widths of 120595(1119878 2119878) and 120578

119888(1119878 2119878) resonances

respectively Observation of an abnormally large productionrate of single charmed mesons in the final state would be ahint of new physics beyond the standard model [11]

From the theoretical point of view (1) the charm quarkweak decay is more favorable than the bottom quark weakdecay because the Cabibbo-Kobayashi-Maskawa (CKM)matrix elements obey |119881

119888119887| ≪ |119881

119888119904| [12] Penguin and

annihilation contributions to nonleptonic charm quark weakdecay being proportional to the CKM factor |119881

119888119887119881119906119887| sim

O(1205825) with the Wolfenstein parameter 120582 ≃ 022 [12]are highly suppressed and hence negligible relative to treecontributions Both 119888 and 119888 quarks in charmonium candecay individually which provides a good place to investigatethe dynamical mechanism of heavy-flavor weak decay andcrosscheck model parameters obtained from the charmedhadron weak decays (2) There are few works devoted tononleptonic 119869120595(1119878) weak decays in the past such as [13]with the covariant light-cone quark model [14] with QCDsum rules and [15ndash17] with the Wirbel-Stech-Bauer (WSB)model [18] Moreover previous works of [13ndash17] concernmainly the weak transition form factors between the 119869120595(1119878)and charmed mesons Fewer papers have been devoted tononleptonic120595(2119878) and 120578

119888(1119878 2119878)weak decays until now even

though a rough estimate of branching ratios is unavailableIn this paper we will estimate the branching ratios fornonleptonic two-body charmonium weak decay taking thenonfactorizable contributions to hadronic matrix elementsinto account with the attractive QCD factorization (QCDF)approach [19]

This paper is organized as follows In Section 2 we willpresent the theoretical framework and the amplitudes for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays Section 3 is devoted

to numerical results and discussion Finally Section 4 is oursummation

2 Theoretical Framework

21 The Effective Hamiltonian Phenomenologically theeffective Hamiltonian responsible for charmonium weakdecay into119863119872 final states can be written as follows [25]

Heff

=

119866119865

radic2

sum

11990211199022

119881lowast

1198881199021

1198811199061199022

1198621(120583)119876

1(120583) + 119862

2(120583)119876

2(120583)

+Hc

(1)

where 119866119865= 1166 times 10

minus5 GeVminus2 [12] is the Fermi couplingconstant 119881lowast

1198881199021

1198811199061199022

is the CKM factor with 11990212

= 119889 119904the Wilson coefficients 119862

12(120583) which are independent of

one particular process summarize the physical contributionsabove the scale of 120583 The expressions of the local tree four-quark operators are

1198761= [119902

1120572120574120583(1 minus 120574

5) 119888

120572] [119906

120573120574120583

(1 minus 1205745) 119902

2120573]

1198762= [119902

1120572120574120583(1 minus 120574

5) 119888

120573] [119906

120573120574120583

(1 minus 1205745) 119902

2120572]

(2)

where 120572 and 120573 are color indicesIt is well known that the Wilson coefficients 119862

119894could be

systematically calculated with perturbation theory and haveproperly been evaluated to the next-to-leading order (NLO)Their values at the scale of 120583 sim O(119898

119888) can be evaluated with

the renormalization group (RG) equation [25]

11986212(120583) = 119880

4(120583119898

119887) 119880

5(119898

119887 119898

119882) 119862

12(119898

119882) (3)

where 119880119891(120583

119891 120583

119894) is the RG evolution matrix which trans-

forms the Wilson coefficients from scale of 120583119894to 120583

119891 The

expression for119880119891(120583

119891 120583

119894) can be found in [25]The numerical

values of the leading-order (LO) and NLO 11986212

in the naivedimensional regularization scheme are listed in Table 2 Thevalues of coefficients 119862

12in Table 2 agree well with those

obtained with ldquoeffectiverdquo number of active flavors 119891 = 415

[25] rather than formula (3)To obtain the decay amplitudes and branching ratios

the remaining works are to evaluate accurately the hadronicmatrix elements (HME) where the local operators are sand-wiched between the charmonium and final states which isalso the most intricate work in dealing with the weak decayof heavy hadrons by now

22 Hadronic Matrix Elements Analogous to the exclusiveprocesses with perturbative QCD theory proposed by Lepageand Brodsky [26] the QCDF approach is developed byBeneke et al [19] to deal with HME based on the collinearfactorization approximation and power counting rules inthe heavy quark limit and has been extensively used for119861 meson decays Using the QCDF master formula HMEof nonleptonic decays could be written as the convolutionintegrals of the process-dependent hard scattering kernelsand universal light-cone distribution amplitudes (LCDA) ofparticipating hadrons

The spectator quark is the heavy-flavor charm quark forcharmonium weak decays into 119863119872 final states It is com-monly assumed that the virtuality of the gluon connectingto the heavy spectator is of order Λ2

QCD where ΛQCD isthe characteristic QCD scale Hence the transition formfactors between charmonium and 119863 mesons are assumed tobe dominated by the soft and nonperturbative contributionsand the amplitudes of the spectator rescattering subprocessare power-suppressed [19] Taking 120578

119888rarr 119863119872 decays for

example HME can be written as

⟨1198631198721003816100381610038161003816119876

12

1003816100381610038161003816120578119888⟩ = sum

119894

119865120578119888rarr119863

119894119891119872int119867

119894(119909)Φ

119872(119909) 119889119909 (4)

Advances in High Energy Physics 3

Table 2 Numerical values of theWilson coefficients11986212

and parameters 11988612

for 120578119888rarr 119863120587 decay with119898

119888= 1275GeV [12] where 119886

12in [20]

is used in the119863meson weak decay

120583

LO NLO QCDF Previous works119862

1119862

2119862

1119862

21198861

1198862

Ref 1198861

1198862

08119898119888

1335 minus0589 1275 minus0504 1275119890+1198944∘

0503119890minus119894154∘

[14 16 17] 126 minus051

119898119888

1276 minus0505 1222 minus0425 1219119890+1198943∘

0402119890minus119894154∘

[15] 13 plusmn 01 minus055 plusmn 010

12119898119888

1240 minus0450 1190 minus0374 1186119890+1198943∘

0342119890minus119894154∘

[20] 1274 minus0529

where 119865120578119888rarr119863

119894is the weak transition form factor and 119891

119872and

Φ119872(119909) are the decay constant and LCDA of the meson 119872

respectively The leading twist LCDA for the pseudoscalarand longitudinally polarized vector mesons can be expressedin terms of Gegenbauer polynomials [23 24]

Φ119872(119909) = 6119909119909

infin

sum

119899=0

119886119872

11989911986232

119899(119909 minus 119909) (5)

where 119909 = 1 minus 119909 11986232

119899(119911) is the Gegenbauer polynomial

11986232

0(119911) = 1

11986232

1(119911) = 3119911

11986232

2(119911) =

3

2

(51199112

minus 1)

(6)

119886119872

119899is the Gegenbauer moment corresponding to the Gegen-

bauer polynomials 11986232

119899(119911) 119886119872

0equiv 1 for the asymptotic form

and 119886119899= 0 for 119899 = 1 3 5 because of the 119866-parity invari-

ance of the 120587 120578(1015840) 120588 120596 120601 meson distribution amplitudesIn this paper to give a rough estimation the contributionsfrom higher-order 119899 ge 3 Gegenbauer polynomials are notconsidered for the moment

Hard scattering function 119867119894(119909) in (4) is in principle

calculable order by order with the perturbative QCD theoryAt the order of 1205720

119904 119867

119894(119909) = 1 This is the simplest scenario

and one goes back to the naive factorization where there is noinformation about the strong phases and the renormalizationscale hidden in the HME At the order of 120572

119904and higher

orders the renormalization scale dependence of hadronicmatrix elements could be recuperated to partly cancel the 120583-dependence of the Wilson coefficients In addition part ofthe strong phases could be reproduced from nonfactorizablecontributions

Within the QCDF framework amplitudes for 120578119888rarr 119863119872

decays can be expressed as

A (120578119888997888rarr 119863119872) = ⟨119863119872

1003816100381610038161003816Heff

1003816100381610038161003816120578119888⟩

=

119866119865

radic2

119881lowast

1198881199021

1198811199061199022

119886119894⟨119872

100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816120578119888⟩

(7)

In addition the HME for the 120595(1119878 2119878) rarr 119863119881 decays areconventionally expressed as the helicity amplitudes with thedecomposition [27 28]

H120582= ⟨119881

100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816120595⟩ = 120598

lowast120583

119881120598]120595119886119892

120583]

+

119887

119898120595119898

119881

(119901120595+ 119901

119863)

120583

119901]119881

+

119894119888

119898120595119898

119881

120598120583]120572120573119901

120572

119881(119901

120595+ 119901

119863)

120573

(8)

The relations among helicity amplitudes and invariant ampli-tudes 119886 119887 119888 are

H0= minus119886119909 minus 2119887 (119909

2

minus 1)

Hplusmn= 119886 plusmn 2119888radic119909

2minus 1

119909 =

119901120595sdot 119901

119881

119898120595119898

119881

=

1198982

120595minus 119898

2

119863+ 119898

2

119881

2119898120595119898

119881

(9)

where three scalar amplitudes 119886 119887 119888 describe 119904 119889 119901 wavecontributions respectively

The effective coefficient 119886119894at the order of 120572

119904can be

expressed as [19]

1198861= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO2V

1198862= 119862

NLO2

+

1

119873119888

119862NLO1

+

120572119904

4120587

119862119865

119873119888

119862LO1V

(10)

where the color factor 119862119865= 43 the color number 119873

119888= 3

For the transversely polarized light vector meson the factorV = 0 in the helicity H

plusmnamplitudes beyond the leading

twist contributions With the leading twist LCDA for thepseudoscalar and longitudinally polarized vectormesons thefactorV is written as [19]

V = 6 log(119898

2

119888

1205832) minus 18 minus (

1

2

+ 1198943120587)

+ (

11

2

minus 1198943120587) 119886119872

1minus

21

20

119886119872

2+ sdot sdot sdot

(11)

From the numbers in Table 2 it is found that (1) the valuesof coefficients 119886

12agree generally with those used in previous

works [14ndash17 20] (2) the strong phases appear by taking

4 Advances in High Energy Physics

nonfactorizable corrections into account which is necessaryfor119862119875 violation and (3) the strong phase of 119886

1is small due to

the suppression of 120572119904and 1119873

119888The strong phase of 119886

2is large

due to the enhancement from the large Wilson coefficients1198621

23 Form Factors The weak transition form factors betweencharmonium and a charmed meson are defined as follows[18]

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120578119888(119901

1)⟩

= (1199011+ 119901

2)120583minus

1198982

120578119888

minus 1198982

119863

1199022

119902120583119865

1(119902

2

)

+

1198982

120578119888

minus 1198982

119863

1199022

1199021205831198650(119902

2

)

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120595 (119901

1 120598)⟩

= minus120598120583]120572120573120598

]120595119902120572

(1199011+ 119901

2)120573

119881(1199022

)

119898120595+ 119898

119863

minus 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

0(119902

2

)

minus 119894120598120595120583

(119898120595+ 119898

119863)119860

1(119902

2

)

minus 119894

120598120595sdot 119902

119898120595+ 119898

119863

(1199011+ 119901

2)120583119860

2(119902

2

)

+ 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

3(119902

2

)

(12)

where 119902 = 1199011minus119901

2 120598

120595denotes the 120595rsquos polarization vectorThe

form factors 1198650(0) = 119865

1(0) and 119860

0(0) = 119860

3(0) are required

compulsorily to cancel singularities at the pole of 1199022 = 0There is a relation among these form factors

2119898120595119860

3(119902

2

) = (119898120595+ 119898

119863)119860

1(119902

2

)

+ (119898120595minus 119898

119863)119860

2(119902

2

)

(13)

There are four independent transition form factors1198650(0)

11986001(0) and119881(0) at the pole of 1199022 = 0 They could be written

as the overlap integrals of wave functions [18]

1198650(0) = intint

1

0

Φ120578119888

(119896perp 119909 0 0)

sdot Φ119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198600(0) = intint

1

0

Φ120595(119896perp 119909 1 0)

sdot 120590119911Φ

119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198601(0) =

119898119888+ 119898

119902

119898120595+ 119898

119863

119868

119881 (0) =

119898119888minus 119898

119902

119898120595minus 119898

119863

119868

119868 = radic2intint

1

0

Φ120595(119896perp 119909 1 minus1) 119894120590

119910Φ

119863(119896perp 119909 0 0)

sdot

1

119909

119889119909 119889119896perp

(14)

where 120590119910119911

is the Pauli matrix acting on the spin indices ofthe decaying charm quark 119909 and

119896perpdenote the fraction of

the longitudinal momentum and the transverse momentumof the nonspectator quark respectively

With the separation of the spin and spatial variables wavefunctions can be written as

Φ(119896perp 119909 119895 119895

119911) = 120601 (

119896perp 119909)

1003816100381610038161003816119904 119904

119911 119904

1 119904

2⟩ (15)

where the total angular momentum 119895 = + 1199041+ 119904

2= 119904

1+

1199042= 119904 because the orbital angular momentum between the

valence quarks in 120595(1119878 2119878) 120578119888(1119878 2119878)119863mesons in question

have = 0 11990412

denote the spins of valence quarks in meson119904 = 1 and 0 for the 120595 and 120578

119888mesons respectively

The charm quark in the charmonium state is nearlynonrelativistic with an average velocity V ≪ 1 basedon arguments of nonrelativistic quantum chromodynamics(NRQCD) [29ndash31] For the 119863 meson the valence quarks arealso nonrelativistic due to 119898

119863asymp 119898

119888+ 119898

119902 where the light

quark mass 119898119906

asymp 119898119889

asymp 310MeV and 119898119904asymp 510MeV

[32] Here we will take the solution of the Schrodingerequation with a scalar harmonic oscillator potential as thewave functions of the charmonium and119863mesons

1206011119878(119896) sim 119890

minus119896

2

21205722

1206012119878(119896) sim 119890

minus119896

2

21205722

(2119896

2

minus 31205722

)

(16)

where the parameter 120572 determines the average transversequark momentum ⟨120601

1119878|119896

2

perp|120601

1119878⟩ = 120572

2 With the NRQCDpower counting rules [29] | 119896

perp| sim 119898V sim 119898120572

119904for heavy

quarkonium Hence parameter 120572 is approximately taken as119898120572

119904in our calculationUsing the substitution ansatz [33]

119896

2

997888rarr

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

4119909119909

(17)

one can obtain

1206011119878(119896perp 119909) = 119860 exp

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

minus81205722119909119909

1206012119878(119896perp 119909) = 119861120601

1119878(119896perp 119909)

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

61205722119909119909

minus 1

(18)

Advances in High Energy Physics 5

Table 3 The numerical values of transition form factors at 1199022 = 0 where uncertainties of this work come from the charm quark mass

Transition Reference 1198650(0) 119860

0(0) 119860

1(0) 119881(0)

120578119888(1119878) 120595(1119878) rarr 119863

119906119889

This work 085 plusmn 001 085 plusmn 001 072 plusmn 001 176 plusmn 003

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 16 plusmn 01

[21]b sdot sdot sdot 027+002

minus003027

+003

minus002081

+012

minus008

[15]c sdot sdot sdot 040 (061) 044 (068) 117 (182)[17]d sdot sdot sdot 055 plusmn 002 077

+009

minus007214

+015

minus011

[17]e sdot sdot sdot 054 080 221

120578119888(1119878) 120595(1119878) rarr 119863

119904

This work 090 plusmn 001 090 plusmn 001 081 plusmn 001 155 plusmn 004

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 18

[21]b sdot sdot sdot 037 plusmn 002 038+002

minus001107

+005

minus002

[15]c sdot sdot sdot 047 (066) 055 (078) 125 (180)[17]d sdot sdot sdot 071

+004

minus002094 plusmn 007 230

+009

minus006

[17]e sdot sdot sdot 069 096 236120578119888(2119878) 120595(2119878) rarr 119863

119906119889This work 062 plusmn 001 061 plusmn 001 054 plusmn 001 100 plusmn 004

120578119888(2119878) 120595(2119878) rarr 119863

119904This work 065 plusmn 001 064 plusmn 001 059 plusmn 002 083 plusmn 004

aThe form factors are computed with the covariant light-front quark model where uncertainties come from the decay constant of charmed mesonbThe form factors are computed with QCD sum rules where uncertainties are from the Borel parameterscThe form factors are computed with parameter 120596 = 04 (05) GeV using the WSB modeldThe form factors are computed with flavor dependent parameter 120596 using the WSB modeleThe form factors are computed with parameter 120596 = 119898120572

119904using the WSB model

where the parameters 119860 and 119861 are the normalization coeffi-cients satisfying the normalization condition

intint

1

0

10038161003816100381610038161003816120601 (

119896perp 119909)

10038161003816100381610038161003816

2

119889119909 119889119896perp= 1 (19)

The numerical values of transition form factors at 1199022 = 0

are listed in Table 3 It is found that (1) themodel dependenceof form factors is large (2) isospin-breaking effects arenegligible and flavor breaking effects are small and (3) asstated in [18] 119865

0≃ 119860

0holds within collinear symmetry

3 Numerical Results and Discussion

In the charmonium center-of-mass frame the branchingratio for the charmonium weak decay can be written as

B119903 (120578119888997888rarr 119863119872) =

119901cm4120587119898

2

120578119888

Γ120578119888

1003816100381610038161003816A (120578

119888997888rarr 119863119872)

1003816100381610038161003816

2

B119903 (120595 997888rarr 119863119872) =

119901cm12120587119898

2

120595Γ120595

1003816100381610038161003816A (120595 997888rarr 119863119872)

1003816100381610038161003816

2

(20)

where the common momentum of final states is

119901cm

=

radic[1198982

120578119888120595minus (119898

119863+ 119898

119872)2

] [1198982

120578119888120595minus (119898

119863minus 119898

119872)2

]

2119898120578119888120595

(21)

The decay amplitudes for A(120595 rarr 119863119872) and A(120578119888rarr 119863119872)

are collected in Appendices A and B respectivelyIn our calculation we assume that the light vectormesons

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

the mixing of pseudoscalar 120578 and 1205781015840 meson we will adopt thequark-flavor basis description proposed in [22] and neglectthe contributions from possible gluonium compositions thatis

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (22)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393 plusmn 10)∘ [22] The mass relations are

1198982

120578119902

= 1198982

120578cos2120601 + 1198982

1205781015840sin2120601

minus

radic2119891120578119904

119891120578119902

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

1198982

120578119904

= 1198982

120578sin2120601 + 1198982

1205781015840cos2120601

minus

119891120578119902

radic2119891120578119904

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

(23)

The input parameters including the CKM Wolfensteinparameters decay constants and Gegenbauer moments arecollected in Table 4 If not specified explicitly we will taketheir central values as the default inputs Our numericalresults on branching ratios for the nonleptonic two-body120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 weak decays are displayed in

Tables 5 and 6 where the uncertainties of this work comefrom theCKMparameters the renormalization scale120583 = (1plusmn

02)119898119888 and hadronic parameters including decay constants

and Gegenbauer moments respectively For comparisonprevious results on 119869120595(1119878) weak decays [14 16 17] withparameters 119886

1= 126 and 119886

2= minus051 are also listed in Table 5

The following are some comments

6 Advances in High Energy Physics

Table 4 Numerical values of input parameters

120582 = 022537 plusmn 000061 [12] 119860 = 0814+0023

minus0024[12]

120588 = 0117 plusmn 0021 [12] 120578 = 0353 plusmn 0013 [12]119898

119888= 1275 plusmn 0025GeV [12] 119898

119863119906

= 186484 plusmn 007MeV [12]119898

119863119889

= 186961 plusmn 010MeV [12] 119898119863119904

= 196830 plusmn 011MeV [12]119891120587= 13041 plusmn 020MeV [12] 119891

119870= 1562 plusmn 07MeV [12]

119891120578119902

= (107 plusmn 002) 119891120587[22] 119891

120578119904

= (134 plusmn 006) 119891120587[22]

119891120588= 216 plusmn 3MeV [23] 119891

120596= 187 plusmn 5MeV [23]

119891120601= 215 plusmn 5MeV [23] 119891

119870lowast = 220 plusmn 5MeV [23]

119886120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [24] 119886

120588

2= 119886

120596

2= 015 plusmn 007 [23]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [24] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [24]

119886119870

lowast

1= minus119886

119870lowast

1= 003 plusmn 002 [23] 119886

119870lowast

2= 119886

119870

lowast

2= 011 plusmn 009 [23]

119886120587

1= 119886

120588

1= 119886

120596

1= 119886

120601

1= 0 119886

120601

2= 018 plusmn 008 [23]

Table 5 Branching ratios for the nonleptonic two-body 119869120595(1119878) weak decays where the uncertainties of this work come from the CKMparameters the renormalization scale 120583 = (1 plusmn 02)119898

119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively The results of [14 16 17] are calculated with 1198861= 126 and 119886

2= minus051 The results of [14] are based on QCD sum rules The

numbers in columns of ldquoArdquo ldquoBrdquo ldquoCrdquo and ldquoDrdquo are based on the WSB model with flavor dependent 120596 QCD inspired 120596 = 120572119904119898 and universal

120596 = 04GeV and 05GeV respectively

Final states Case Reference [14] Reference [17] Reference [16] This workA B C D

119863minus

119904120587+ 1-a 20 times 10

minus10

741 times 10minus10

713 times 10minus10

332 times 10minus10

874 times 10minus10

(109+001+010+001

minus001minus006minus001) times 10

minus9

119863minus

119904119870

+ 1-b 16 times 10minus11

53 times 10minus11

52 times 10minus11

24 times 10minus11

55 times 10minus11

(618+003+059+008

minus003minus033minus008) times 10

minus11

119863minus

119889120587+ 1-b 08 times 10

minus11

29 times 10minus11

28 times 10minus11

15 times 10minus11

55 times 10minus11

(637+003+060+003

minus003minus034minus003) times 10

minus11

119863minus

119889119870

+ 1-c sdot sdot sdot 23 times 10minus12

22 times 10minus12

12 times 10minus12

sdot sdot sdot (379+004+036+005

minus004minus020minus005) times 10

minus12

119863

0

1199061205870 2-b sdot sdot sdot 24 times 10

minus12

23 times 10minus12

12 times 10minus12

55 times 10minus12

(350+002+198+006

minus002minus097minus006) times 10

minus12

119863

0

119906119870

0 2-c sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

sdot sdot sdot (416+004+235+011

minus004minus115minus010) times 10

minus13

119863

0

119906119870

0 2-a 36 times 10minus11

139 times 10minus10

134 times 10minus10

72 times 10minus11

28 times 10minus10

(144+001+081+003

minus001minus040minus003) times 10

minus10

119863

0

119906120578 sdot sdot sdot 70 times 10

minus12

67 times 10minus12

36 times 10minus12

16 times 10minus12

(103+001+058+010

minus001minus028minus010) times 10

minus11

119863

0

1199061205781015840

sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

30 times 10minus13

(583+003+329+172

minus003minus161minus150) times 10

minus13

119863minus

119904120588+ 1-a 126 times 10

minus9

511 times 10minus9

532 times 10minus9

177 times 10minus9

363 times 10minus9

(382+001+036+011

minus001minus020minus011) times 10

minus9

119863minus

119904119870

lowast+ 1-b 082 times 10minus10

282 times 10minus10

296 times 10minus10

097 times 10minus10

212 times 10minus10

(200+001+019+010

minus001minus011minus009) times 10

minus10

119863minus

119889120588+ 1-b 042 times 10

minus10

216 times 10minus10

228 times 10minus10

072 times 10minus10

220 times 10minus10

(212+001+020+006

minus001minus011minus006) times 10

minus10

119863minus

119889119870

lowast+ 1-c sdot sdot sdot 13 times 10minus11

13 times 10minus11

42 times 10minus12

sdot sdot sdot (114+001+011+006

minus001minus006minus005) times 10

minus11

119863

0

1199061205880 2-b sdot sdot sdot 18 times 10

minus11

19 times 10minus11

60 times 10minus12

22 times 10minus11

(108+001+061+004

minus001minus030minus004) times 10

minus11

119863

0

119906120596 2-b sdot sdot sdot 16 times 10

minus11

17 times 10minus11

50 times 10minus12

18 times 10minus11

(810+004+456+050

minus004minus225minus048) times 10

minus12

119863

0

119906120601 2-b sdot sdot sdot 42 times 10

minus11

44 times 10minus11

14 times 10minus11

65 times 10minus11

(192+001+108+010

minus001minus053minus010) times 10

minus11

119863

0

119906119870

lowast0 2-c sdot sdot sdot 21 times 10minus12

22 times 10minus12

70 times 10minus13

sdot sdot sdot (119+001+067+007

minus001minus033minus007) times 10

minus12

119863

0

119906119870

lowast0 2-a 154 times 10minus10

761 times 10minus10

812 times 10minus10

251 times 10minus10

103 times 10minus9

(409+001+230+024

minus001minus114minus023) times 10

minus10

(1) There are some differences among the estimates ofbranching ratios for 119869120595(1119878) rarr 119863119872weak decays (seethe numbers in Table 5)These inconsistencies amongprevious works although the same values of param-eters 119886

12are used come principally from different

values of form factors Our results are generally in linewith the numbers in columns ldquoArdquo and ldquoBrdquo which arefavored by [17]

(2) Branching ratios for 119869120595(1119878) weak decay are abouttwo or more times as large as those for 120595(2119878) decayinto the same final states because the decay width of120595(2119878) is about three times as large as that of 119869120595(1119878)

(3) Due to the relatively small decay width and relativelylarge space phases for 120578

119888(2119878) decay branching ratios

for 120578119888(2119878) weak decay are some five (ten) or more

times as large as those for 120578119888(1119878) weak decay into the

same119863119875 (119863119881) final states

(4) Among 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons 120578

119888(1119878) has a

maximal decay width and a minimal mass resultingin a small phase space while 119869120595(1119878) has a minimaldecay width These facts lead to the smallest [or thelargest] branching ratio for 120578

119888(1119878) [or 119869120595(1119878)] weak

decay among 120595(1119878 2119878) 120578119888(1119878 2119878) weak decays into

the same final states

Advances in High Energy Physics 7

Table 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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 2 Numerical values of theWilson coefficients11986212

and parameters 11988612

for 120578119888rarr 119863120587 decay with119898

119888= 1275GeV [12] where 119886

12in [20]

is used in the119863meson weak decay

120583

LO NLO QCDF Previous works119862

1119862

2119862

1119862

21198861

1198862

Ref 1198861

1198862

08119898119888

1335 minus0589 1275 minus0504 1275119890+1198944∘

0503119890minus119894154∘

[14 16 17] 126 minus051

119898119888

1276 minus0505 1222 minus0425 1219119890+1198943∘

0402119890minus119894154∘

[15] 13 plusmn 01 minus055 plusmn 010

12119898119888

1240 minus0450 1190 minus0374 1186119890+1198943∘

0342119890minus119894154∘

[20] 1274 minus0529

where 119865120578119888rarr119863

119894is the weak transition form factor and 119891

119872and

Φ119872(119909) are the decay constant and LCDA of the meson 119872

respectively The leading twist LCDA for the pseudoscalarand longitudinally polarized vector mesons can be expressedin terms of Gegenbauer polynomials [23 24]

Φ119872(119909) = 6119909119909

infin

sum

119899=0

119886119872

11989911986232

119899(119909 minus 119909) (5)

where 119909 = 1 minus 119909 11986232

119899(119911) is the Gegenbauer polynomial

11986232

0(119911) = 1

11986232

1(119911) = 3119911

11986232

2(119911) =

3

2

(51199112

minus 1)

(6)

119886119872

119899is the Gegenbauer moment corresponding to the Gegen-

bauer polynomials 11986232

119899(119911) 119886119872

0equiv 1 for the asymptotic form

and 119886119899= 0 for 119899 = 1 3 5 because of the 119866-parity invari-

ance of the 120587 120578(1015840) 120588 120596 120601 meson distribution amplitudesIn this paper to give a rough estimation the contributionsfrom higher-order 119899 ge 3 Gegenbauer polynomials are notconsidered for the moment

Hard scattering function 119867119894(119909) in (4) is in principle

calculable order by order with the perturbative QCD theoryAt the order of 1205720

119904 119867

119894(119909) = 1 This is the simplest scenario

and one goes back to the naive factorization where there is noinformation about the strong phases and the renormalizationscale hidden in the HME At the order of 120572

119904and higher

orders the renormalization scale dependence of hadronicmatrix elements could be recuperated to partly cancel the 120583-dependence of the Wilson coefficients In addition part ofthe strong phases could be reproduced from nonfactorizablecontributions

Within the QCDF framework amplitudes for 120578119888rarr 119863119872

decays can be expressed as

A (120578119888997888rarr 119863119872) = ⟨119863119872

1003816100381610038161003816Heff

1003816100381610038161003816120578119888⟩

=

119866119865

radic2

119881lowast

1198881199021

1198811199061199022

119886119894⟨119872

100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816120578119888⟩

(7)

In addition the HME for the 120595(1119878 2119878) rarr 119863119881 decays areconventionally expressed as the helicity amplitudes with thedecomposition [27 28]

H120582= ⟨119881

100381610038161003816100381611986912058310038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119869120583

10038161003816100381610038161003816120595⟩ = 120598

lowast120583

119881120598]120595119886119892

120583]

+

119887

119898120595119898

119881

(119901120595+ 119901

119863)

120583

119901]119881

+

119894119888

119898120595119898

119881

120598120583]120572120573119901

120572

119881(119901

120595+ 119901

119863)

120573

(8)

The relations among helicity amplitudes and invariant ampli-tudes 119886 119887 119888 are

H0= minus119886119909 minus 2119887 (119909

2

minus 1)

Hplusmn= 119886 plusmn 2119888radic119909

2minus 1

119909 =

119901120595sdot 119901

119881

119898120595119898

119881

=

1198982

120595minus 119898

2

119863+ 119898

2

119881

2119898120595119898

119881

(9)

where three scalar amplitudes 119886 119887 119888 describe 119904 119889 119901 wavecontributions respectively

The effective coefficient 119886119894at the order of 120572

119904can be

expressed as [19]

1198861= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO2V

1198862= 119862

NLO2

+

1

119873119888

119862NLO1

+

120572119904

4120587

119862119865

119873119888

119862LO1V

(10)

where the color factor 119862119865= 43 the color number 119873

119888= 3

For the transversely polarized light vector meson the factorV = 0 in the helicity H

plusmnamplitudes beyond the leading

twist contributions With the leading twist LCDA for thepseudoscalar and longitudinally polarized vectormesons thefactorV is written as [19]

V = 6 log(119898

2

119888

1205832) minus 18 minus (

1

2

+ 1198943120587)

+ (

11

2

minus 1198943120587) 119886119872

1minus

21

20

119886119872

2+ sdot sdot sdot

(11)

From the numbers in Table 2 it is found that (1) the valuesof coefficients 119886

12agree generally with those used in previous

works [14ndash17 20] (2) the strong phases appear by taking

4 Advances in High Energy Physics

nonfactorizable corrections into account which is necessaryfor119862119875 violation and (3) the strong phase of 119886

1is small due to

the suppression of 120572119904and 1119873

119888The strong phase of 119886

2is large

due to the enhancement from the large Wilson coefficients1198621

23 Form Factors The weak transition form factors betweencharmonium and a charmed meson are defined as follows[18]

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120578119888(119901

1)⟩

= (1199011+ 119901

2)120583minus

1198982

120578119888

minus 1198982

119863

1199022

119902120583119865

1(119902

2

)

+

1198982

120578119888

minus 1198982

119863

1199022

1199021205831198650(119902

2

)

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120595 (119901

1 120598)⟩

= minus120598120583]120572120573120598

]120595119902120572

(1199011+ 119901

2)120573

119881(1199022

)

119898120595+ 119898

119863

minus 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

0(119902

2

)

minus 119894120598120595120583

(119898120595+ 119898

119863)119860

1(119902

2

)

minus 119894

120598120595sdot 119902

119898120595+ 119898

119863

(1199011+ 119901

2)120583119860

2(119902

2

)

+ 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

3(119902

2

)

(12)

where 119902 = 1199011minus119901

2 120598

120595denotes the 120595rsquos polarization vectorThe

form factors 1198650(0) = 119865

1(0) and 119860

0(0) = 119860

3(0) are required

compulsorily to cancel singularities at the pole of 1199022 = 0There is a relation among these form factors

2119898120595119860

3(119902

2

) = (119898120595+ 119898

119863)119860

1(119902

2

)

+ (119898120595minus 119898

119863)119860

2(119902

2

)

(13)

There are four independent transition form factors1198650(0)

11986001(0) and119881(0) at the pole of 1199022 = 0 They could be written

as the overlap integrals of wave functions [18]

1198650(0) = intint

1

0

Φ120578119888

(119896perp 119909 0 0)

sdot Φ119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198600(0) = intint

1

0

Φ120595(119896perp 119909 1 0)

sdot 120590119911Φ

119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198601(0) =

119898119888+ 119898

119902

119898120595+ 119898

119863

119868

119881 (0) =

119898119888minus 119898

119902

119898120595minus 119898

119863

119868

119868 = radic2intint

1

0

Φ120595(119896perp 119909 1 minus1) 119894120590

119910Φ

119863(119896perp 119909 0 0)

sdot

1

119909

119889119909 119889119896perp

(14)

where 120590119910119911

is the Pauli matrix acting on the spin indices ofthe decaying charm quark 119909 and

119896perpdenote the fraction of

the longitudinal momentum and the transverse momentumof the nonspectator quark respectively

With the separation of the spin and spatial variables wavefunctions can be written as

Φ(119896perp 119909 119895 119895

119911) = 120601 (

119896perp 119909)

1003816100381610038161003816119904 119904

119911 119904

1 119904

2⟩ (15)

where the total angular momentum 119895 = + 1199041+ 119904

2= 119904

1+

1199042= 119904 because the orbital angular momentum between the

valence quarks in 120595(1119878 2119878) 120578119888(1119878 2119878)119863mesons in question

have = 0 11990412

denote the spins of valence quarks in meson119904 = 1 and 0 for the 120595 and 120578

119888mesons respectively

The charm quark in the charmonium state is nearlynonrelativistic with an average velocity V ≪ 1 basedon arguments of nonrelativistic quantum chromodynamics(NRQCD) [29ndash31] For the 119863 meson the valence quarks arealso nonrelativistic due to 119898

119863asymp 119898

119888+ 119898

119902 where the light

quark mass 119898119906

asymp 119898119889

asymp 310MeV and 119898119904asymp 510MeV

[32] Here we will take the solution of the Schrodingerequation with a scalar harmonic oscillator potential as thewave functions of the charmonium and119863mesons

1206011119878(119896) sim 119890

minus119896

2

21205722

1206012119878(119896) sim 119890

minus119896

2

21205722

(2119896

2

minus 31205722

)

(16)

where the parameter 120572 determines the average transversequark momentum ⟨120601

1119878|119896

2

perp|120601

1119878⟩ = 120572

2 With the NRQCDpower counting rules [29] | 119896

perp| sim 119898V sim 119898120572

119904for heavy

quarkonium Hence parameter 120572 is approximately taken as119898120572

119904in our calculationUsing the substitution ansatz [33]

119896

2

997888rarr

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

4119909119909

(17)

one can obtain

1206011119878(119896perp 119909) = 119860 exp

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

minus81205722119909119909

1206012119878(119896perp 119909) = 119861120601

1119878(119896perp 119909)

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

61205722119909119909

minus 1

(18)

Advances in High Energy Physics 5

Table 3 The numerical values of transition form factors at 1199022 = 0 where uncertainties of this work come from the charm quark mass

Transition Reference 1198650(0) 119860

0(0) 119860

1(0) 119881(0)

120578119888(1119878) 120595(1119878) rarr 119863

119906119889

This work 085 plusmn 001 085 plusmn 001 072 plusmn 001 176 plusmn 003

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 16 plusmn 01

[21]b sdot sdot sdot 027+002

minus003027

+003

minus002081

+012

minus008

[15]c sdot sdot sdot 040 (061) 044 (068) 117 (182)[17]d sdot sdot sdot 055 plusmn 002 077

+009

minus007214

+015

minus011

[17]e sdot sdot sdot 054 080 221

120578119888(1119878) 120595(1119878) rarr 119863

119904

This work 090 plusmn 001 090 plusmn 001 081 plusmn 001 155 plusmn 004

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 18

[21]b sdot sdot sdot 037 plusmn 002 038+002

minus001107

+005

minus002

[15]c sdot sdot sdot 047 (066) 055 (078) 125 (180)[17]d sdot sdot sdot 071

+004

minus002094 plusmn 007 230

+009

minus006

[17]e sdot sdot sdot 069 096 236120578119888(2119878) 120595(2119878) rarr 119863

119906119889This work 062 plusmn 001 061 plusmn 001 054 plusmn 001 100 plusmn 004

120578119888(2119878) 120595(2119878) rarr 119863

119904This work 065 plusmn 001 064 plusmn 001 059 plusmn 002 083 plusmn 004

aThe form factors are computed with the covariant light-front quark model where uncertainties come from the decay constant of charmed mesonbThe form factors are computed with QCD sum rules where uncertainties are from the Borel parameterscThe form factors are computed with parameter 120596 = 04 (05) GeV using the WSB modeldThe form factors are computed with flavor dependent parameter 120596 using the WSB modeleThe form factors are computed with parameter 120596 = 119898120572

119904using the WSB model

where the parameters 119860 and 119861 are the normalization coeffi-cients satisfying the normalization condition

intint

1

0

10038161003816100381610038161003816120601 (

119896perp 119909)

10038161003816100381610038161003816

2

119889119909 119889119896perp= 1 (19)

The numerical values of transition form factors at 1199022 = 0

are listed in Table 3 It is found that (1) themodel dependenceof form factors is large (2) isospin-breaking effects arenegligible and flavor breaking effects are small and (3) asstated in [18] 119865

0≃ 119860

0holds within collinear symmetry

3 Numerical Results and Discussion

In the charmonium center-of-mass frame the branchingratio for the charmonium weak decay can be written as

B119903 (120578119888997888rarr 119863119872) =

119901cm4120587119898

2

120578119888

Γ120578119888

1003816100381610038161003816A (120578

119888997888rarr 119863119872)

1003816100381610038161003816

2

B119903 (120595 997888rarr 119863119872) =

119901cm12120587119898

2

120595Γ120595

1003816100381610038161003816A (120595 997888rarr 119863119872)

1003816100381610038161003816

2

(20)

where the common momentum of final states is

119901cm

=

radic[1198982

120578119888120595minus (119898

119863+ 119898

119872)2

] [1198982

120578119888120595minus (119898

119863minus 119898

119872)2

]

2119898120578119888120595

(21)

The decay amplitudes for A(120595 rarr 119863119872) and A(120578119888rarr 119863119872)

are collected in Appendices A and B respectivelyIn our calculation we assume that the light vectormesons

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

the mixing of pseudoscalar 120578 and 1205781015840 meson we will adopt thequark-flavor basis description proposed in [22] and neglectthe contributions from possible gluonium compositions thatis

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (22)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393 plusmn 10)∘ [22] The mass relations are

1198982

120578119902

= 1198982

120578cos2120601 + 1198982

1205781015840sin2120601

minus

radic2119891120578119904

119891120578119902

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

1198982

120578119904

= 1198982

120578sin2120601 + 1198982

1205781015840cos2120601

minus

119891120578119902

radic2119891120578119904

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

(23)

The input parameters including the CKM Wolfensteinparameters decay constants and Gegenbauer moments arecollected in Table 4 If not specified explicitly we will taketheir central values as the default inputs Our numericalresults on branching ratios for the nonleptonic two-body120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 weak decays are displayed in

Tables 5 and 6 where the uncertainties of this work comefrom theCKMparameters the renormalization scale120583 = (1plusmn

02)119898119888 and hadronic parameters including decay constants

and Gegenbauer moments respectively For comparisonprevious results on 119869120595(1119878) weak decays [14 16 17] withparameters 119886

1= 126 and 119886

2= minus051 are also listed in Table 5

The following are some comments

6 Advances in High Energy Physics

Table 4 Numerical values of input parameters

120582 = 022537 plusmn 000061 [12] 119860 = 0814+0023

minus0024[12]

120588 = 0117 plusmn 0021 [12] 120578 = 0353 plusmn 0013 [12]119898

119888= 1275 plusmn 0025GeV [12] 119898

119863119906

= 186484 plusmn 007MeV [12]119898

119863119889

= 186961 plusmn 010MeV [12] 119898119863119904

= 196830 plusmn 011MeV [12]119891120587= 13041 plusmn 020MeV [12] 119891

119870= 1562 plusmn 07MeV [12]

119891120578119902

= (107 plusmn 002) 119891120587[22] 119891

120578119904

= (134 plusmn 006) 119891120587[22]

119891120588= 216 plusmn 3MeV [23] 119891

120596= 187 plusmn 5MeV [23]

119891120601= 215 plusmn 5MeV [23] 119891

119870lowast = 220 plusmn 5MeV [23]

119886120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [24] 119886

120588

2= 119886

120596

2= 015 plusmn 007 [23]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [24] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [24]

119886119870

lowast

1= minus119886

119870lowast

1= 003 plusmn 002 [23] 119886

119870lowast

2= 119886

119870

lowast

2= 011 plusmn 009 [23]

119886120587

1= 119886

120588

1= 119886

120596

1= 119886

120601

1= 0 119886

120601

2= 018 plusmn 008 [23]

Table 5 Branching ratios for the nonleptonic two-body 119869120595(1119878) weak decays where the uncertainties of this work come from the CKMparameters the renormalization scale 120583 = (1 plusmn 02)119898

119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively The results of [14 16 17] are calculated with 1198861= 126 and 119886

2= minus051 The results of [14] are based on QCD sum rules The

numbers in columns of ldquoArdquo ldquoBrdquo ldquoCrdquo and ldquoDrdquo are based on the WSB model with flavor dependent 120596 QCD inspired 120596 = 120572119904119898 and universal

120596 = 04GeV and 05GeV respectively

Final states Case Reference [14] Reference [17] Reference [16] This workA B C D

119863minus

119904120587+ 1-a 20 times 10

minus10

741 times 10minus10

713 times 10minus10

332 times 10minus10

874 times 10minus10

(109+001+010+001

minus001minus006minus001) times 10

minus9

119863minus

119904119870

+ 1-b 16 times 10minus11

53 times 10minus11

52 times 10minus11

24 times 10minus11

55 times 10minus11

(618+003+059+008

minus003minus033minus008) times 10

minus11

119863minus

119889120587+ 1-b 08 times 10

minus11

29 times 10minus11

28 times 10minus11

15 times 10minus11

55 times 10minus11

(637+003+060+003

minus003minus034minus003) times 10

minus11

119863minus

119889119870

+ 1-c sdot sdot sdot 23 times 10minus12

22 times 10minus12

12 times 10minus12

sdot sdot sdot (379+004+036+005

minus004minus020minus005) times 10

minus12

119863

0

1199061205870 2-b sdot sdot sdot 24 times 10

minus12

23 times 10minus12

12 times 10minus12

55 times 10minus12

(350+002+198+006

minus002minus097minus006) times 10

minus12

119863

0

119906119870

0 2-c sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

sdot sdot sdot (416+004+235+011

minus004minus115minus010) times 10

minus13

119863

0

119906119870

0 2-a 36 times 10minus11

139 times 10minus10

134 times 10minus10

72 times 10minus11

28 times 10minus10

(144+001+081+003

minus001minus040minus003) times 10

minus10

119863

0

119906120578 sdot sdot sdot 70 times 10

minus12

67 times 10minus12

36 times 10minus12

16 times 10minus12

(103+001+058+010

minus001minus028minus010) times 10

minus11

119863

0

1199061205781015840

sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

30 times 10minus13

(583+003+329+172

minus003minus161minus150) times 10

minus13

119863minus

119904120588+ 1-a 126 times 10

minus9

511 times 10minus9

532 times 10minus9

177 times 10minus9

363 times 10minus9

(382+001+036+011

minus001minus020minus011) times 10

minus9

119863minus

119904119870

lowast+ 1-b 082 times 10minus10

282 times 10minus10

296 times 10minus10

097 times 10minus10

212 times 10minus10

(200+001+019+010

minus001minus011minus009) times 10

minus10

119863minus

119889120588+ 1-b 042 times 10

minus10

216 times 10minus10

228 times 10minus10

072 times 10minus10

220 times 10minus10

(212+001+020+006

minus001minus011minus006) times 10

minus10

119863minus

119889119870

lowast+ 1-c sdot sdot sdot 13 times 10minus11

13 times 10minus11

42 times 10minus12

sdot sdot sdot (114+001+011+006

minus001minus006minus005) times 10

minus11

119863

0

1199061205880 2-b sdot sdot sdot 18 times 10

minus11

19 times 10minus11

60 times 10minus12

22 times 10minus11

(108+001+061+004

minus001minus030minus004) times 10

minus11

119863

0

119906120596 2-b sdot sdot sdot 16 times 10

minus11

17 times 10minus11

50 times 10minus12

18 times 10minus11

(810+004+456+050

minus004minus225minus048) times 10

minus12

119863

0

119906120601 2-b sdot sdot sdot 42 times 10

minus11

44 times 10minus11

14 times 10minus11

65 times 10minus11

(192+001+108+010

minus001minus053minus010) times 10

minus11

119863

0

119906119870

lowast0 2-c sdot sdot sdot 21 times 10minus12

22 times 10minus12

70 times 10minus13

sdot sdot sdot (119+001+067+007

minus001minus033minus007) times 10

minus12

119863

0

119906119870

lowast0 2-a 154 times 10minus10

761 times 10minus10

812 times 10minus10

251 times 10minus10

103 times 10minus9

(409+001+230+024

minus001minus114minus023) times 10

minus10

(1) There are some differences among the estimates ofbranching ratios for 119869120595(1119878) rarr 119863119872weak decays (seethe numbers in Table 5)These inconsistencies amongprevious works although the same values of param-eters 119886

12are used come principally from different

values of form factors Our results are generally in linewith the numbers in columns ldquoArdquo and ldquoBrdquo which arefavored by [17]

(2) Branching ratios for 119869120595(1119878) weak decay are abouttwo or more times as large as those for 120595(2119878) decayinto the same final states because the decay width of120595(2119878) is about three times as large as that of 119869120595(1119878)

(3) Due to the relatively small decay width and relativelylarge space phases for 120578

119888(2119878) decay branching ratios

for 120578119888(2119878) weak decay are some five (ten) or more

times as large as those for 120578119888(1119878) weak decay into the

same119863119875 (119863119881) final states

(4) Among 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons 120578

119888(1119878) has a

maximal decay width and a minimal mass resultingin a small phase space while 119869120595(1119878) has a minimaldecay width These facts lead to the smallest [or thelargest] branching ratio for 120578

119888(1119878) [or 119869120595(1119878)] weak

decay among 120595(1119878 2119878) 120578119888(1119878 2119878) weak decays into

the same final states

Advances in High Energy Physics 7

Table 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

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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

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

nonfactorizable corrections into account which is necessaryfor119862119875 violation and (3) the strong phase of 119886

1is small due to

the suppression of 120572119904and 1119873

119888The strong phase of 119886

2is large

due to the enhancement from the large Wilson coefficients1198621

23 Form Factors The weak transition form factors betweencharmonium and a charmed meson are defined as follows[18]

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120578119888(119901

1)⟩

= (1199011+ 119901

2)120583minus

1198982

120578119888

minus 1198982

119863

1199022

119902120583119865

1(119902

2

)

+

1198982

120578119888

minus 1198982

119863

1199022

1199021205831198650(119902

2

)

⟨119863 (1199012)

10038161003816100381610038161003816119881120583minus 119860

120583

10038161003816100381610038161003816120595 (119901

1 120598)⟩

= minus120598120583]120572120573120598

]120595119902120572

(1199011+ 119901

2)120573

119881(1199022

)

119898120595+ 119898

119863

minus 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

0(119902

2

)

minus 119894120598120595120583

(119898120595+ 119898

119863)119860

1(119902

2

)

minus 119894

120598120595sdot 119902

119898120595+ 119898

119863

(1199011+ 119901

2)120583119860

2(119902

2

)

+ 119894

2119898120595120598120595sdot 119902

1199022

119902120583119860

3(119902

2

)

(12)

where 119902 = 1199011minus119901

2 120598

120595denotes the 120595rsquos polarization vectorThe

form factors 1198650(0) = 119865

1(0) and 119860

0(0) = 119860

3(0) are required

compulsorily to cancel singularities at the pole of 1199022 = 0There is a relation among these form factors

2119898120595119860

3(119902

2

) = (119898120595+ 119898

119863)119860

1(119902

2

)

+ (119898120595minus 119898

119863)119860

2(119902

2

)

(13)

There are four independent transition form factors1198650(0)

11986001(0) and119881(0) at the pole of 1199022 = 0 They could be written

as the overlap integrals of wave functions [18]

1198650(0) = intint

1

0

Φ120578119888

(119896perp 119909 0 0)

sdot Φ119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198600(0) = intint

1

0

Φ120595(119896perp 119909 1 0)

sdot 120590119911Φ

119863(119896perp 119909 0 0) 119889119909 119889

119896perp

1198601(0) =

119898119888+ 119898

119902

119898120595+ 119898

119863

119868

119881 (0) =

119898119888minus 119898

119902

119898120595minus 119898

119863

119868

119868 = radic2intint

1

0

Φ120595(119896perp 119909 1 minus1) 119894120590

119910Φ

119863(119896perp 119909 0 0)

sdot

1

119909

119889119909 119889119896perp

(14)

where 120590119910119911

is the Pauli matrix acting on the spin indices ofthe decaying charm quark 119909 and

119896perpdenote the fraction of

the longitudinal momentum and the transverse momentumof the nonspectator quark respectively

With the separation of the spin and spatial variables wavefunctions can be written as

Φ(119896perp 119909 119895 119895

119911) = 120601 (

119896perp 119909)

1003816100381610038161003816119904 119904

119911 119904

1 119904

2⟩ (15)

where the total angular momentum 119895 = + 1199041+ 119904

2= 119904

1+

1199042= 119904 because the orbital angular momentum between the

valence quarks in 120595(1119878 2119878) 120578119888(1119878 2119878)119863mesons in question

have = 0 11990412

denote the spins of valence quarks in meson119904 = 1 and 0 for the 120595 and 120578

119888mesons respectively

The charm quark in the charmonium state is nearlynonrelativistic with an average velocity V ≪ 1 basedon arguments of nonrelativistic quantum chromodynamics(NRQCD) [29ndash31] For the 119863 meson the valence quarks arealso nonrelativistic due to 119898

119863asymp 119898

119888+ 119898

119902 where the light

quark mass 119898119906

asymp 119898119889

asymp 310MeV and 119898119904asymp 510MeV

[32] Here we will take the solution of the Schrodingerequation with a scalar harmonic oscillator potential as thewave functions of the charmonium and119863mesons

1206011119878(119896) sim 119890

minus119896

2

21205722

1206012119878(119896) sim 119890

minus119896

2

21205722

(2119896

2

minus 31205722

)

(16)

where the parameter 120572 determines the average transversequark momentum ⟨120601

1119878|119896

2

perp|120601

1119878⟩ = 120572

2 With the NRQCDpower counting rules [29] | 119896

perp| sim 119898V sim 119898120572

119904for heavy

quarkonium Hence parameter 120572 is approximately taken as119898120572

119904in our calculationUsing the substitution ansatz [33]

119896

2

997888rarr

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

4119909119909

(17)

one can obtain

1206011119878(119896perp 119909) = 119860 exp

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

minus81205722119909119909

1206012119878(119896perp 119909) = 119861120601

1119878(119896perp 119909)

119896

2

perp+ 119909119898

2

119902+ 119909119898

2

119888

61205722119909119909

minus 1

(18)

Advances in High Energy Physics 5

Table 3 The numerical values of transition form factors at 1199022 = 0 where uncertainties of this work come from the charm quark mass

Transition Reference 1198650(0) 119860

0(0) 119860

1(0) 119881(0)

120578119888(1119878) 120595(1119878) rarr 119863

119906119889

This work 085 plusmn 001 085 plusmn 001 072 plusmn 001 176 plusmn 003

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 16 plusmn 01

[21]b sdot sdot sdot 027+002

minus003027

+003

minus002081

+012

minus008

[15]c sdot sdot sdot 040 (061) 044 (068) 117 (182)[17]d sdot sdot sdot 055 plusmn 002 077

+009

minus007214

+015

minus011

[17]e sdot sdot sdot 054 080 221

120578119888(1119878) 120595(1119878) rarr 119863

119904

This work 090 plusmn 001 090 plusmn 001 081 plusmn 001 155 plusmn 004

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 18

[21]b sdot sdot sdot 037 plusmn 002 038+002

minus001107

+005

minus002

[15]c sdot sdot sdot 047 (066) 055 (078) 125 (180)[17]d sdot sdot sdot 071

+004

minus002094 plusmn 007 230

+009

minus006

[17]e sdot sdot sdot 069 096 236120578119888(2119878) 120595(2119878) rarr 119863

119906119889This work 062 plusmn 001 061 plusmn 001 054 plusmn 001 100 plusmn 004

120578119888(2119878) 120595(2119878) rarr 119863

119904This work 065 plusmn 001 064 plusmn 001 059 plusmn 002 083 plusmn 004

aThe form factors are computed with the covariant light-front quark model where uncertainties come from the decay constant of charmed mesonbThe form factors are computed with QCD sum rules where uncertainties are from the Borel parameterscThe form factors are computed with parameter 120596 = 04 (05) GeV using the WSB modeldThe form factors are computed with flavor dependent parameter 120596 using the WSB modeleThe form factors are computed with parameter 120596 = 119898120572

119904using the WSB model

where the parameters 119860 and 119861 are the normalization coeffi-cients satisfying the normalization condition

intint

1

0

10038161003816100381610038161003816120601 (

119896perp 119909)

10038161003816100381610038161003816

2

119889119909 119889119896perp= 1 (19)

The numerical values of transition form factors at 1199022 = 0

are listed in Table 3 It is found that (1) themodel dependenceof form factors is large (2) isospin-breaking effects arenegligible and flavor breaking effects are small and (3) asstated in [18] 119865

0≃ 119860

0holds within collinear symmetry

3 Numerical Results and Discussion

In the charmonium center-of-mass frame the branchingratio for the charmonium weak decay can be written as

B119903 (120578119888997888rarr 119863119872) =

119901cm4120587119898

2

120578119888

Γ120578119888

1003816100381610038161003816A (120578

119888997888rarr 119863119872)

1003816100381610038161003816

2

B119903 (120595 997888rarr 119863119872) =

119901cm12120587119898

2

120595Γ120595

1003816100381610038161003816A (120595 997888rarr 119863119872)

1003816100381610038161003816

2

(20)

where the common momentum of final states is

119901cm

=

radic[1198982

120578119888120595minus (119898

119863+ 119898

119872)2

] [1198982

120578119888120595minus (119898

119863minus 119898

119872)2

]

2119898120578119888120595

(21)

The decay amplitudes for A(120595 rarr 119863119872) and A(120578119888rarr 119863119872)

are collected in Appendices A and B respectivelyIn our calculation we assume that the light vectormesons

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

the mixing of pseudoscalar 120578 and 1205781015840 meson we will adopt thequark-flavor basis description proposed in [22] and neglectthe contributions from possible gluonium compositions thatis

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (22)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393 plusmn 10)∘ [22] The mass relations are

1198982

120578119902

= 1198982

120578cos2120601 + 1198982

1205781015840sin2120601

minus

radic2119891120578119904

119891120578119902

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

1198982

120578119904

= 1198982

120578sin2120601 + 1198982

1205781015840cos2120601

minus

119891120578119902

radic2119891120578119904

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

(23)

The input parameters including the CKM Wolfensteinparameters decay constants and Gegenbauer moments arecollected in Table 4 If not specified explicitly we will taketheir central values as the default inputs Our numericalresults on branching ratios for the nonleptonic two-body120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 weak decays are displayed in

Tables 5 and 6 where the uncertainties of this work comefrom theCKMparameters the renormalization scale120583 = (1plusmn

02)119898119888 and hadronic parameters including decay constants

and Gegenbauer moments respectively For comparisonprevious results on 119869120595(1119878) weak decays [14 16 17] withparameters 119886

1= 126 and 119886

2= minus051 are also listed in Table 5

The following are some comments

6 Advances in High Energy Physics

Table 4 Numerical values of input parameters

120582 = 022537 plusmn 000061 [12] 119860 = 0814+0023

minus0024[12]

120588 = 0117 plusmn 0021 [12] 120578 = 0353 plusmn 0013 [12]119898

119888= 1275 plusmn 0025GeV [12] 119898

119863119906

= 186484 plusmn 007MeV [12]119898

119863119889

= 186961 plusmn 010MeV [12] 119898119863119904

= 196830 plusmn 011MeV [12]119891120587= 13041 plusmn 020MeV [12] 119891

119870= 1562 plusmn 07MeV [12]

119891120578119902

= (107 plusmn 002) 119891120587[22] 119891

120578119904

= (134 plusmn 006) 119891120587[22]

119891120588= 216 plusmn 3MeV [23] 119891

120596= 187 plusmn 5MeV [23]

119891120601= 215 plusmn 5MeV [23] 119891

119870lowast = 220 plusmn 5MeV [23]

119886120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [24] 119886

120588

2= 119886

120596

2= 015 plusmn 007 [23]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [24] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [24]

119886119870

lowast

1= minus119886

119870lowast

1= 003 plusmn 002 [23] 119886

119870lowast

2= 119886

119870

lowast

2= 011 plusmn 009 [23]

119886120587

1= 119886

120588

1= 119886

120596

1= 119886

120601

1= 0 119886

120601

2= 018 plusmn 008 [23]

Table 5 Branching ratios for the nonleptonic two-body 119869120595(1119878) weak decays where the uncertainties of this work come from the CKMparameters the renormalization scale 120583 = (1 plusmn 02)119898

119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively The results of [14 16 17] are calculated with 1198861= 126 and 119886

2= minus051 The results of [14] are based on QCD sum rules The

numbers in columns of ldquoArdquo ldquoBrdquo ldquoCrdquo and ldquoDrdquo are based on the WSB model with flavor dependent 120596 QCD inspired 120596 = 120572119904119898 and universal

120596 = 04GeV and 05GeV respectively

Final states Case Reference [14] Reference [17] Reference [16] This workA B C D

119863minus

119904120587+ 1-a 20 times 10

minus10

741 times 10minus10

713 times 10minus10

332 times 10minus10

874 times 10minus10

(109+001+010+001

minus001minus006minus001) times 10

minus9

119863minus

119904119870

+ 1-b 16 times 10minus11

53 times 10minus11

52 times 10minus11

24 times 10minus11

55 times 10minus11

(618+003+059+008

minus003minus033minus008) times 10

minus11

119863minus

119889120587+ 1-b 08 times 10

minus11

29 times 10minus11

28 times 10minus11

15 times 10minus11

55 times 10minus11

(637+003+060+003

minus003minus034minus003) times 10

minus11

119863minus

119889119870

+ 1-c sdot sdot sdot 23 times 10minus12

22 times 10minus12

12 times 10minus12

sdot sdot sdot (379+004+036+005

minus004minus020minus005) times 10

minus12

119863

0

1199061205870 2-b sdot sdot sdot 24 times 10

minus12

23 times 10minus12

12 times 10minus12

55 times 10minus12

(350+002+198+006

minus002minus097minus006) times 10

minus12

119863

0

119906119870

0 2-c sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

sdot sdot sdot (416+004+235+011

minus004minus115minus010) times 10

minus13

119863

0

119906119870

0 2-a 36 times 10minus11

139 times 10minus10

134 times 10minus10

72 times 10minus11

28 times 10minus10

(144+001+081+003

minus001minus040minus003) times 10

minus10

119863

0

119906120578 sdot sdot sdot 70 times 10

minus12

67 times 10minus12

36 times 10minus12

16 times 10minus12

(103+001+058+010

minus001minus028minus010) times 10

minus11

119863

0

1199061205781015840

sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

30 times 10minus13

(583+003+329+172

minus003minus161minus150) times 10

minus13

119863minus

119904120588+ 1-a 126 times 10

minus9

511 times 10minus9

532 times 10minus9

177 times 10minus9

363 times 10minus9

(382+001+036+011

minus001minus020minus011) times 10

minus9

119863minus

119904119870

lowast+ 1-b 082 times 10minus10

282 times 10minus10

296 times 10minus10

097 times 10minus10

212 times 10minus10

(200+001+019+010

minus001minus011minus009) times 10

minus10

119863minus

119889120588+ 1-b 042 times 10

minus10

216 times 10minus10

228 times 10minus10

072 times 10minus10

220 times 10minus10

(212+001+020+006

minus001minus011minus006) times 10

minus10

119863minus

119889119870

lowast+ 1-c sdot sdot sdot 13 times 10minus11

13 times 10minus11

42 times 10minus12

sdot sdot sdot (114+001+011+006

minus001minus006minus005) times 10

minus11

119863

0

1199061205880 2-b sdot sdot sdot 18 times 10

minus11

19 times 10minus11

60 times 10minus12

22 times 10minus11

(108+001+061+004

minus001minus030minus004) times 10

minus11

119863

0

119906120596 2-b sdot sdot sdot 16 times 10

minus11

17 times 10minus11

50 times 10minus12

18 times 10minus11

(810+004+456+050

minus004minus225minus048) times 10

minus12

119863

0

119906120601 2-b sdot sdot sdot 42 times 10

minus11

44 times 10minus11

14 times 10minus11

65 times 10minus11

(192+001+108+010

minus001minus053minus010) times 10

minus11

119863

0

119906119870

lowast0 2-c sdot sdot sdot 21 times 10minus12

22 times 10minus12

70 times 10minus13

sdot sdot sdot (119+001+067+007

minus001minus033minus007) times 10

minus12

119863

0

119906119870

lowast0 2-a 154 times 10minus10

761 times 10minus10

812 times 10minus10

251 times 10minus10

103 times 10minus9

(409+001+230+024

minus001minus114minus023) times 10

minus10

(1) There are some differences among the estimates ofbranching ratios for 119869120595(1119878) rarr 119863119872weak decays (seethe numbers in Table 5)These inconsistencies amongprevious works although the same values of param-eters 119886

12are used come principally from different

values of form factors Our results are generally in linewith the numbers in columns ldquoArdquo and ldquoBrdquo which arefavored by [17]

(2) Branching ratios for 119869120595(1119878) weak decay are abouttwo or more times as large as those for 120595(2119878) decayinto the same final states because the decay width of120595(2119878) is about three times as large as that of 119869120595(1119878)

(3) Due to the relatively small decay width and relativelylarge space phases for 120578

119888(2119878) decay branching ratios

for 120578119888(2119878) weak decay are some five (ten) or more

times as large as those for 120578119888(1119878) weak decay into the

same119863119875 (119863119881) final states

(4) Among 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons 120578

119888(1119878) has a

maximal decay width and a minimal mass resultingin a small phase space while 119869120595(1119878) has a minimaldecay width These facts lead to the smallest [or thelargest] branching ratio for 120578

119888(1119878) [or 119869120595(1119878)] weak

decay among 120595(1119878 2119878) 120578119888(1119878 2119878) weak decays into

the same final states

Advances in High Energy Physics 7

Table 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

Table 3 The numerical values of transition form factors at 1199022 = 0 where uncertainties of this work come from the charm quark mass

Transition Reference 1198650(0) 119860

0(0) 119860

1(0) 119881(0)

120578119888(1119878) 120595(1119878) rarr 119863

119906119889

This work 085 plusmn 001 085 plusmn 001 072 plusmn 001 176 plusmn 003

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 16 plusmn 01

[21]b sdot sdot sdot 027+002

minus003027

+003

minus002081

+012

minus008

[15]c sdot sdot sdot 040 (061) 044 (068) 117 (182)[17]d sdot sdot sdot 055 plusmn 002 077

+009

minus007214

+015

minus011

[17]e sdot sdot sdot 054 080 221

120578119888(1119878) 120595(1119878) rarr 119863

119904

This work 090 plusmn 001 090 plusmn 001 081 plusmn 001 155 plusmn 004

[13]a sdot sdot sdot 068 plusmn 001 068 plusmn 001 18

[21]b sdot sdot sdot 037 plusmn 002 038+002

minus001107

+005

minus002

[15]c sdot sdot sdot 047 (066) 055 (078) 125 (180)[17]d sdot sdot sdot 071

+004

minus002094 plusmn 007 230

+009

minus006

[17]e sdot sdot sdot 069 096 236120578119888(2119878) 120595(2119878) rarr 119863

119906119889This work 062 plusmn 001 061 plusmn 001 054 plusmn 001 100 plusmn 004

120578119888(2119878) 120595(2119878) rarr 119863

119904This work 065 plusmn 001 064 plusmn 001 059 plusmn 002 083 plusmn 004

aThe form factors are computed with the covariant light-front quark model where uncertainties come from the decay constant of charmed mesonbThe form factors are computed with QCD sum rules where uncertainties are from the Borel parameterscThe form factors are computed with parameter 120596 = 04 (05) GeV using the WSB modeldThe form factors are computed with flavor dependent parameter 120596 using the WSB modeleThe form factors are computed with parameter 120596 = 119898120572

119904using the WSB model

where the parameters 119860 and 119861 are the normalization coeffi-cients satisfying the normalization condition

intint

1

0

10038161003816100381610038161003816120601 (

119896perp 119909)

10038161003816100381610038161003816

2

119889119909 119889119896perp= 1 (19)

The numerical values of transition form factors at 1199022 = 0

are listed in Table 3 It is found that (1) themodel dependenceof form factors is large (2) isospin-breaking effects arenegligible and flavor breaking effects are small and (3) asstated in [18] 119865

0≃ 119860

0holds within collinear symmetry

3 Numerical Results and Discussion

In the charmonium center-of-mass frame the branchingratio for the charmonium weak decay can be written as

B119903 (120578119888997888rarr 119863119872) =

119901cm4120587119898

2

120578119888

Γ120578119888

1003816100381610038161003816A (120578

119888997888rarr 119863119872)

1003816100381610038161003816

2

B119903 (120595 997888rarr 119863119872) =

119901cm12120587119898

2

120595Γ120595

1003816100381610038161003816A (120595 997888rarr 119863119872)

1003816100381610038161003816

2

(20)

where the common momentum of final states is

119901cm

=

radic[1198982

120578119888120595minus (119898

119863+ 119898

119872)2

] [1198982

120578119888120595minus (119898

119863minus 119898

119872)2

]

2119898120578119888120595

(21)

The decay amplitudes for A(120595 rarr 119863119872) and A(120578119888rarr 119863119872)

are collected in Appendices A and B respectivelyIn our calculation we assume that the light vectormesons

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

the mixing of pseudoscalar 120578 and 1205781015840 meson we will adopt thequark-flavor basis description proposed in [22] and neglectthe contributions from possible gluonium compositions thatis

(

120578

1205781015840) = (

cos120601 minus sin120601sin120601 cos120601

)(

120578119902

120578119904

) (22)

where 120578119902= (119906119906 + 119889119889)radic2 and 120578

119904= 119904119904 the mixing angle 120601 =

(393 plusmn 10)∘ [22] The mass relations are

1198982

120578119902

= 1198982

120578cos2120601 + 1198982

1205781015840sin2120601

minus

radic2119891120578119904

119891120578119902

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

1198982

120578119904

= 1198982

120578sin2120601 + 1198982

1205781015840cos2120601

minus

119891120578119902

radic2119891120578119904

(1198982

1205781015840 minus 119898

2

120578) cos120601 sin120601

(23)

The input parameters including the CKM Wolfensteinparameters decay constants and Gegenbauer moments arecollected in Table 4 If not specified explicitly we will taketheir central values as the default inputs Our numericalresults on branching ratios for the nonleptonic two-body120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 weak decays are displayed in

Tables 5 and 6 where the uncertainties of this work comefrom theCKMparameters the renormalization scale120583 = (1plusmn

02)119898119888 and hadronic parameters including decay constants

and Gegenbauer moments respectively For comparisonprevious results on 119869120595(1119878) weak decays [14 16 17] withparameters 119886

1= 126 and 119886

2= minus051 are also listed in Table 5

The following are some comments

6 Advances in High Energy Physics

Table 4 Numerical values of input parameters

120582 = 022537 plusmn 000061 [12] 119860 = 0814+0023

minus0024[12]

120588 = 0117 plusmn 0021 [12] 120578 = 0353 plusmn 0013 [12]119898

119888= 1275 plusmn 0025GeV [12] 119898

119863119906

= 186484 plusmn 007MeV [12]119898

119863119889

= 186961 plusmn 010MeV [12] 119898119863119904

= 196830 plusmn 011MeV [12]119891120587= 13041 plusmn 020MeV [12] 119891

119870= 1562 plusmn 07MeV [12]

119891120578119902

= (107 plusmn 002) 119891120587[22] 119891

120578119904

= (134 plusmn 006) 119891120587[22]

119891120588= 216 plusmn 3MeV [23] 119891

120596= 187 plusmn 5MeV [23]

119891120601= 215 plusmn 5MeV [23] 119891

119870lowast = 220 plusmn 5MeV [23]

119886120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [24] 119886

120588

2= 119886

120596

2= 015 plusmn 007 [23]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [24] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [24]

119886119870

lowast

1= minus119886

119870lowast

1= 003 plusmn 002 [23] 119886

119870lowast

2= 119886

119870

lowast

2= 011 plusmn 009 [23]

119886120587

1= 119886

120588

1= 119886

120596

1= 119886

120601

1= 0 119886

120601

2= 018 plusmn 008 [23]

Table 5 Branching ratios for the nonleptonic two-body 119869120595(1119878) weak decays where the uncertainties of this work come from the CKMparameters the renormalization scale 120583 = (1 plusmn 02)119898

119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively The results of [14 16 17] are calculated with 1198861= 126 and 119886

2= minus051 The results of [14] are based on QCD sum rules The

numbers in columns of ldquoArdquo ldquoBrdquo ldquoCrdquo and ldquoDrdquo are based on the WSB model with flavor dependent 120596 QCD inspired 120596 = 120572119904119898 and universal

120596 = 04GeV and 05GeV respectively

Final states Case Reference [14] Reference [17] Reference [16] This workA B C D

119863minus

119904120587+ 1-a 20 times 10

minus10

741 times 10minus10

713 times 10minus10

332 times 10minus10

874 times 10minus10

(109+001+010+001

minus001minus006minus001) times 10

minus9

119863minus

119904119870

+ 1-b 16 times 10minus11

53 times 10minus11

52 times 10minus11

24 times 10minus11

55 times 10minus11

(618+003+059+008

minus003minus033minus008) times 10

minus11

119863minus

119889120587+ 1-b 08 times 10

minus11

29 times 10minus11

28 times 10minus11

15 times 10minus11

55 times 10minus11

(637+003+060+003

minus003minus034minus003) times 10

minus11

119863minus

119889119870

+ 1-c sdot sdot sdot 23 times 10minus12

22 times 10minus12

12 times 10minus12

sdot sdot sdot (379+004+036+005

minus004minus020minus005) times 10

minus12

119863

0

1199061205870 2-b sdot sdot sdot 24 times 10

minus12

23 times 10minus12

12 times 10minus12

55 times 10minus12

(350+002+198+006

minus002minus097minus006) times 10

minus12

119863

0

119906119870

0 2-c sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

sdot sdot sdot (416+004+235+011

minus004minus115minus010) times 10

minus13

119863

0

119906119870

0 2-a 36 times 10minus11

139 times 10minus10

134 times 10minus10

72 times 10minus11

28 times 10minus10

(144+001+081+003

minus001minus040minus003) times 10

minus10

119863

0

119906120578 sdot sdot sdot 70 times 10

minus12

67 times 10minus12

36 times 10minus12

16 times 10minus12

(103+001+058+010

minus001minus028minus010) times 10

minus11

119863

0

1199061205781015840

sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

30 times 10minus13

(583+003+329+172

minus003minus161minus150) times 10

minus13

119863minus

119904120588+ 1-a 126 times 10

minus9

511 times 10minus9

532 times 10minus9

177 times 10minus9

363 times 10minus9

(382+001+036+011

minus001minus020minus011) times 10

minus9

119863minus

119904119870

lowast+ 1-b 082 times 10minus10

282 times 10minus10

296 times 10minus10

097 times 10minus10

212 times 10minus10

(200+001+019+010

minus001minus011minus009) times 10

minus10

119863minus

119889120588+ 1-b 042 times 10

minus10

216 times 10minus10

228 times 10minus10

072 times 10minus10

220 times 10minus10

(212+001+020+006

minus001minus011minus006) times 10

minus10

119863minus

119889119870

lowast+ 1-c sdot sdot sdot 13 times 10minus11

13 times 10minus11

42 times 10minus12

sdot sdot sdot (114+001+011+006

minus001minus006minus005) times 10

minus11

119863

0

1199061205880 2-b sdot sdot sdot 18 times 10

minus11

19 times 10minus11

60 times 10minus12

22 times 10minus11

(108+001+061+004

minus001minus030minus004) times 10

minus11

119863

0

119906120596 2-b sdot sdot sdot 16 times 10

minus11

17 times 10minus11

50 times 10minus12

18 times 10minus11

(810+004+456+050

minus004minus225minus048) times 10

minus12

119863

0

119906120601 2-b sdot sdot sdot 42 times 10

minus11

44 times 10minus11

14 times 10minus11

65 times 10minus11

(192+001+108+010

minus001minus053minus010) times 10

minus11

119863

0

119906119870

lowast0 2-c sdot sdot sdot 21 times 10minus12

22 times 10minus12

70 times 10minus13

sdot sdot sdot (119+001+067+007

minus001minus033minus007) times 10

minus12

119863

0

119906119870

lowast0 2-a 154 times 10minus10

761 times 10minus10

812 times 10minus10

251 times 10minus10

103 times 10minus9

(409+001+230+024

minus001minus114minus023) times 10

minus10

(1) There are some differences among the estimates ofbranching ratios for 119869120595(1119878) rarr 119863119872weak decays (seethe numbers in Table 5)These inconsistencies amongprevious works although the same values of param-eters 119886

12are used come principally from different

values of form factors Our results are generally in linewith the numbers in columns ldquoArdquo and ldquoBrdquo which arefavored by [17]

(2) Branching ratios for 119869120595(1119878) weak decay are abouttwo or more times as large as those for 120595(2119878) decayinto the same final states because the decay width of120595(2119878) is about three times as large as that of 119869120595(1119878)

(3) Due to the relatively small decay width and relativelylarge space phases for 120578

119888(2119878) decay branching ratios

for 120578119888(2119878) weak decay are some five (ten) or more

times as large as those for 120578119888(1119878) weak decay into the

same119863119875 (119863119881) final states

(4) Among 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons 120578

119888(1119878) has a

maximal decay width and a minimal mass resultingin a small phase space while 119869120595(1119878) has a minimaldecay width These facts lead to the smallest [or thelargest] branching ratio for 120578

119888(1119878) [or 119869120595(1119878)] weak

decay among 120595(1119878 2119878) 120578119888(1119878 2119878) weak decays into

the same final states

Advances in High Energy Physics 7

Table 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

Table 4 Numerical values of input parameters

120582 = 022537 plusmn 000061 [12] 119860 = 0814+0023

minus0024[12]

120588 = 0117 plusmn 0021 [12] 120578 = 0353 plusmn 0013 [12]119898

119888= 1275 plusmn 0025GeV [12] 119898

119863119906

= 186484 plusmn 007MeV [12]119898

119863119889

= 186961 plusmn 010MeV [12] 119898119863119904

= 196830 plusmn 011MeV [12]119891120587= 13041 plusmn 020MeV [12] 119891

119870= 1562 plusmn 07MeV [12]

119891120578119902

= (107 plusmn 002) 119891120587[22] 119891

120578119904

= (134 plusmn 006) 119891120587[22]

119891120588= 216 plusmn 3MeV [23] 119891

120596= 187 plusmn 5MeV [23]

119891120601= 215 plusmn 5MeV [23] 119891

119870lowast = 220 plusmn 5MeV [23]

119886120587

2= 119886

120578119902

2= 119886

120578119904

2= 025 plusmn 015 [24] 119886

120588

2= 119886

120596

2= 015 plusmn 007 [23]

119886119870

1= minus119886

119870

1= 006 plusmn 003 [24] 119886

119870

2= 119886

119870

2= 025 plusmn 015 [24]

119886119870

lowast

1= minus119886

119870lowast

1= 003 plusmn 002 [23] 119886

119870lowast

2= 119886

119870

lowast

2= 011 plusmn 009 [23]

119886120587

1= 119886

120588

1= 119886

120596

1= 119886

120601

1= 0 119886

120601

2= 018 plusmn 008 [23]

Table 5 Branching ratios for the nonleptonic two-body 119869120595(1119878) weak decays where the uncertainties of this work come from the CKMparameters the renormalization scale 120583 = (1 plusmn 02)119898

119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively The results of [14 16 17] are calculated with 1198861= 126 and 119886

2= minus051 The results of [14] are based on QCD sum rules The

numbers in columns of ldquoArdquo ldquoBrdquo ldquoCrdquo and ldquoDrdquo are based on the WSB model with flavor dependent 120596 QCD inspired 120596 = 120572119904119898 and universal

120596 = 04GeV and 05GeV respectively

Final states Case Reference [14] Reference [17] Reference [16] This workA B C D

119863minus

119904120587+ 1-a 20 times 10

minus10

741 times 10minus10

713 times 10minus10

332 times 10minus10

874 times 10minus10

(109+001+010+001

minus001minus006minus001) times 10

minus9

119863minus

119904119870

+ 1-b 16 times 10minus11

53 times 10minus11

52 times 10minus11

24 times 10minus11

55 times 10minus11

(618+003+059+008

minus003minus033minus008) times 10

minus11

119863minus

119889120587+ 1-b 08 times 10

minus11

29 times 10minus11

28 times 10minus11

15 times 10minus11

55 times 10minus11

(637+003+060+003

minus003minus034minus003) times 10

minus11

119863minus

119889119870

+ 1-c sdot sdot sdot 23 times 10minus12

22 times 10minus12

12 times 10minus12

sdot sdot sdot (379+004+036+005

minus004minus020minus005) times 10

minus12

119863

0

1199061205870 2-b sdot sdot sdot 24 times 10

minus12

23 times 10minus12

12 times 10minus12

55 times 10minus12

(350+002+198+006

minus002minus097minus006) times 10

minus12

119863

0

119906119870

0 2-c sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

sdot sdot sdot (416+004+235+011

minus004minus115minus010) times 10

minus13

119863

0

119906119870

0 2-a 36 times 10minus11

139 times 10minus10

134 times 10minus10

72 times 10minus11

28 times 10minus10

(144+001+081+003

minus001minus040minus003) times 10

minus10

119863

0

119906120578 sdot sdot sdot 70 times 10

minus12

67 times 10minus12

36 times 10minus12

16 times 10minus12

(103+001+058+010

minus001minus028minus010) times 10

minus11

119863

0

1199061205781015840

sdot sdot sdot 40 times 10minus13

40 times 10minus13

20 times 10minus13

30 times 10minus13

(583+003+329+172

minus003minus161minus150) times 10

minus13

119863minus

119904120588+ 1-a 126 times 10

minus9

511 times 10minus9

532 times 10minus9

177 times 10minus9

363 times 10minus9

(382+001+036+011

minus001minus020minus011) times 10

minus9

119863minus

119904119870

lowast+ 1-b 082 times 10minus10

282 times 10minus10

296 times 10minus10

097 times 10minus10

212 times 10minus10

(200+001+019+010

minus001minus011minus009) times 10

minus10

119863minus

119889120588+ 1-b 042 times 10

minus10

216 times 10minus10

228 times 10minus10

072 times 10minus10

220 times 10minus10

(212+001+020+006

minus001minus011minus006) times 10

minus10

119863minus

119889119870

lowast+ 1-c sdot sdot sdot 13 times 10minus11

13 times 10minus11

42 times 10minus12

sdot sdot sdot (114+001+011+006

minus001minus006minus005) times 10

minus11

119863

0

1199061205880 2-b sdot sdot sdot 18 times 10

minus11

19 times 10minus11

60 times 10minus12

22 times 10minus11

(108+001+061+004

minus001minus030minus004) times 10

minus11

119863

0

119906120596 2-b sdot sdot sdot 16 times 10

minus11

17 times 10minus11

50 times 10minus12

18 times 10minus11

(810+004+456+050

minus004minus225minus048) times 10

minus12

119863

0

119906120601 2-b sdot sdot sdot 42 times 10

minus11

44 times 10minus11

14 times 10minus11

65 times 10minus11

(192+001+108+010

minus001minus053minus010) times 10

minus11

119863

0

119906119870

lowast0 2-c sdot sdot sdot 21 times 10minus12

22 times 10minus12

70 times 10minus13

sdot sdot sdot (119+001+067+007

minus001minus033minus007) times 10

minus12

119863

0

119906119870

lowast0 2-a 154 times 10minus10

761 times 10minus10

812 times 10minus10

251 times 10minus10

103 times 10minus9

(409+001+230+024

minus001minus114minus023) times 10

minus10

(1) There are some differences among the estimates ofbranching ratios for 119869120595(1119878) rarr 119863119872weak decays (seethe numbers in Table 5)These inconsistencies amongprevious works although the same values of param-eters 119886

12are used come principally from different

values of form factors Our results are generally in linewith the numbers in columns ldquoArdquo and ldquoBrdquo which arefavored by [17]

(2) Branching ratios for 119869120595(1119878) weak decay are abouttwo or more times as large as those for 120595(2119878) decayinto the same final states because the decay width of120595(2119878) is about three times as large as that of 119869120595(1119878)

(3) Due to the relatively small decay width and relativelylarge space phases for 120578

119888(2119878) decay branching ratios

for 120578119888(2119878) weak decay are some five (ten) or more

times as large as those for 120578119888(1119878) weak decay into the

same119863119875 (119863119881) final states

(4) Among 120595(1119878 2119878) and 120578119888(1119878 2119878)mesons 120578

119888(1119878) has a

maximal decay width and a minimal mass resultingin a small phase space while 119869120595(1119878) has a minimaldecay width These facts lead to the smallest [or thelargest] branching ratio for 120578

119888(1119878) [or 119869120595(1119878)] weak

decay among 120595(1119878 2119878) 120578119888(1119878 2119878) weak decays into

the same final states

Advances in High Energy Physics 7

Table 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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 6 Branching ratios for the nonleptonic two-body 120595(2119878) 120578119888(1119878) and 120578

119888(2119878) weak decays where the uncertainties come from the CKM

parameters the renormalization scale 120583 = (1 plusmn 02)119898119888 and hadronic parameters including decay constants and Gegenbauer moments

respectively

Case Final states 120595(2119878) decay 120578119888(1119878) decay 120578

119888(2119878) decay

1-a 119863minus

119904120587+

(507+001+048+003

minus001minus027minus002) times 10

minus10

(735+001+069+004

minus001minus039minus004) times 10

minus12

(390+001+037+002

minus001minus021minus002) times 10

minus11

1-b 119863minus

119904119870

+

(343+002+033+004

minus002minus018minus004) times 10

minus11

(497+003+048+006

minus003minus027minus006) times 10

minus13

(287+001+027+004

minus001minus015minus004) times 10

minus12

1-b 119863minus

119889120587+

(276+001+026+001

minus001minus015minus001) times 10

minus11

(439+002+041+002

minus002minus023minus002) times 10

minus13

(213+001+020+001

minus001minus011minus001) times 10

minus12

1-c 119863minus

119889119870

+

(190+002+018+002

minus002minus010minus002) times 10

minus12

(304+003+029+004

minus003minus016minus004) times 10

minus14

(158+002+015+002

minus002minus008minus002) times 10

minus13

2-b 119863

0

1199061205870

(151+001+085+002

minus001minus042minus002) times 10

minus12

(241+001+136+004

minus001minus067minus004) times 10

minus14

(116+001+066+002

minus001minus032minus002) times 10

minus13

2-c 119863

0

119906119870

0

(207+002+117+005

minus002minus057minus005) times 10

minus13

(335+004+189+009

minus004minus093minus008) times 10

minus15

(173+002+097+004

minus002minus048minus004) times 10

minus14

2-a 119863

0

119906119870

0

(715+001+404+017

minus001minus198minus016) times 10

minus11

(116+001+065+003

minus001minus032minus003) times 10

minus12

(596+001+337+014

minus001minus165minus014) times 10

minus12

119863

0

119906120578 (535

+003+302+054

minus003minus148minus050) times 10

minus12

(866+004+489+088

minus004minus240minus082) times 10

minus14

(455+002+257+046

minus002minus126minus043) times 10

minus13

119863

0

1199061205781015840

(563+003+318+168

minus003minus156minus146) times 10

minus13

(766+004+432+228

minus004minus212minus198) times 10

minus15

(602+003+340+179

minus003minus167minus156) times 10

minus14

1-a 119863minus

119904120588+

(167+001+015+005

minus001minus009minus005) times 10

minus9

(528+001+050+015

minus001minus028minus015) times 10

minus12

(724+001+068+021

minus001minus038minus021) times 10

minus11

1-b 119863minus

119904119870

lowast+

(959+005+089+046

minus005minus050minus045) times 10

minus11

(118+001+011+006

minus001minus006minus006) times 10

minus13

(347+002+033+017

minus002minus018minus016) times 10

minus12

1-b 119863minus

119889120588+

(899+005+083+026

minus005minus047minus026) times 10

minus11

(432+002+041+012

minus002minus023minus012) times 10

minus13

(413+002+039+012

minus002minus022minus012) times 10

minus12

1-c 119863minus

119889119870

lowast+

(515+006+048+025

minus005minus027minus024) times 10

minus12

(138+001+013+007

minus001minus007minus007) times 10

minus14

(202+002+019+010

minus002minus011minus010) times 10

minus13

2-b 119863

0

1199061205880

(436+002+244+015

minus002minus121minus015) times 10

minus12

(238+001+135+008

minus001minus066minus008) times 10

minus14

(224+001+127+008

minus001minus062minus008) times 10

minus13

2-b 119863

0

119906120596 (328

+002+184+020

minus002minus091minus019) times 10

minus12

(174+001+098+011

minus001minus048minus010) times 10

minus14

(167+001+094+010

minus001minus046minus010) times 10

minus13

2-b 119863

0

119906120601 (940

+005+528+052

minus005minus261minus050) times 10

minus12

(857+004+484+047

minus004minus238minus045) times 10

minus15

(328+002+185+018

minus002minus091minus017) times 10

minus13

2-c 119863

0

119906119870

lowast0

(509+005+286+031

minus005minus142minus030) times 10

minus13

(150+002+085+008

minus002minus042minus008) times 10

minus15

(218+002+123+012

minus002minus060minus012) times 10

minus14

2-a 119863

0

119906119870

lowast0

(174+001+098+011

minus001minus049minus010) times 10

minus10

(520+001+294+029

minus001minus144minus028) times 10

minus13

(757+001+427+042

minus001minus210minus040) times 10

minus12

Table 7 Classification of the nonleptonic charmonium weakdecays

Case Parameter CKM factor1-a 119886

1|119881

119906119889119881

lowast

119888119904| sim 1

1-b 1198861

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

1-c 1198861

|119881119906119904119881

lowast

119888119889| sim 120582

2

2-a 1198862

|119881119906119889119881

lowast

119888119904| sim 1

2-b 1198862

|119881119906119889119881

lowast

119888119889| |119881

119906119904119881

lowast

119888119904| sim 120582

2-c 1198862

|119881119906119904119881

lowast

119888119889| sim 120582

2

(5) Compared with 120595(1119878 2119878) rarr 119863119881 decays the cor-responding 120595(1119878 2119878) rarr 119863119875 decays where 119875 and119881 have the same flavor structures are suppressed bythe orbital angular momentum and so have relativelysmall branching ratiosThere are some approximativerelations B119903(119869120595(1119878) rarr 119863119881) asymp 3B119903(119869120595(1119878) rarr

119863119875) andB119903(120595(2119878) rarr 119863119881) asymp 3B119903(120595(2119878) rarr 119863119875)

(6) According to the CKM factors and parameters 11988612

nonleptonic charmonium weak decays could be sub-divided into six cases (see Table 7) Case ldquoi-ardquo is theCabibbo-favored one so it generally has large branch-ing ratios relative to cases ldquoi-brdquo and ldquoi-crdquo The 119886

2-

dominated charmonium weak decays are suppressedby a color factor relative to 119886

1-dominated onesHence

the charmonium weak decays into119863119904120588 and119863

119904120587 final

states belonging to case ldquo1-ardquo usually have relativelylarge branching ratios the charmonium weak decaysinto the 119863

0

119906119870

lowast0 final states belonging to case ldquo2-crdquo usually have relatively small branching ratios In

addition the branching ratio of case ldquo2-ardquo (or ldquo2-brdquo)is usually larger than that of case ldquo1-brdquo (or ldquo1-crdquo) dueto |119886

2119886

1| ge 120582

(7) Branching ratios for the Cabibbo-favored 120595(1119878

2119878) rarr 119863minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 decays can reach up to10

minus10 whichmight be measurable in the forthcomingdays For example 119869120595(1119878) production cross sectioncan reach up to a few 120583119887 with the LHCb and ALICEdetectors at LHC [7 8] Therefore over 1012 119869120595(1119878)samples are in principle available per 100 fbminus1 datacollected by LHCb and ALICE corresponding to afew tens of 119869120595(1119878) rarr 119863

minus

119904120588+ 119863minus

119904120587+ 1198630

119906119870

lowast0 eventsfor about 10 reconstruction efficiency

(8) There is a large cancellation between the CKM factors119881119906119889119881

lowast

119888119889and 119881

119906119904119881

lowast

119888119904 which results in a very small

branching ratio for charmonium weak decays into119863

1199061205781015840 state

(9) There are many uncertainties in our results Thefirst uncertainty from the CKM factors is small dueto high precision on the Wolfenstein parameter 120582with only 03 relative errors now [12] The seconduncertainty from the renormalization scale 120583 couldin principle be reduced by the inclusion of higherorder 120572

119904corrections For example it has been shown

[34] that tree amplitudes incorporating with theNNLO corrections are relatively less sensitive to therenormalization scale than the NLO amplitudes Thethird uncertainty comes from hadronic parameterswhich is expected to be cancelled or reduced with therelative ratio of branching ratios

8 Advances in High Energy Physics

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

(10) The numbers in Tables 5 and 6 just provide an orderof magnitude estimate Many other factors such asthe final state interactions and 1199022 dependence of formfactors which are not considered here deserve manydedicated studies

4 Summary

With the anticipation of abundant data samples on char-monium at high-luminosity heavy-flavor experiments westudied the nonleptonic two-body 120595(1119878 2119878) and 120578

119888(1119878 2119878)

weak decays into one ground-state charmed meson plus oneground-state light meson based on the low energy effectiveHamiltonian By considering QCD radiative corrections tohadronic matrix elements of tree operators we got theeffective coefficients 119886

12containing partial information of

strong phasesThemagnitude of 11988612

agrees comfortably withthose used in previous works [14ndash17] The transition formfactors between the charmonium and charmed meson arecalculated by using the nonrelativistic wave functions withisotropic harmonic oscillator potential Branching ratios for120595(1119878 2119878) 120578

119888(1119878 2119878) rarr 119863119872 decays are estimated roughly It

is found that the Cabibbo-favored 120595(1119878 2119878) rarr 119863minus

119904120588+119863minus

119904120587+

119863

0

119906119870

lowast0 decays have large branching ratios ≳ 10minus10 which are

promisingly detected in the forthcoming years

Appendices

A The Amplitudes for 120595rarr 119863119872 Decays

ConsiderA (120595 997888rarr 119863

minus

119904120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061198891198861

A (120595 997888rarr 119863minus

119904119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119904

0119881

lowast

1198881199041198811199061199041198861

A (120595 997888rarr 119863minus

119889120587+

) = radic2119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061198891198861

A (120595 997888rarr 119863minus

119889119870

+

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119889

0119881

lowast

1198881198891198811199061199041198861

A (120595 997888rarr 119863

0

1199061205870

) = minus119866119865119898

120595(120598

120595sdot 119901

120587)

sdot 119891120587119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881198891198811199061199041198862

A (120595 997888rarr 119863

0

119906119870

0

) = radic2119866119865119898

120595(120598

120595sdot 119901

119870)

sdot 119891119870119860

120595rarr119863119906

0119881

lowast

1198881199041198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119902) = 119866

119865119898

120595(120598

120595sdot 119901

120578119902

)

sdot 119891120578119902

119860120595rarr119863

119906

0119881

lowast

1198881198891198811199061198891198862

A (120595 997888rarr 119863

0

119906120578119904) = radic2119866

119865119898

120595(120598

120595sdot 119901

120578119904

)

sdot 119891120578119904

119860120595rarr119863

119906

0119881

lowast

1198881199041198811199061199041198862

A (120595 997888rarr 119863

0

119906120578) = cos120601A (120595 997888rarr 119863

0

119906120578119902) minus sin120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863

0

1199061205781015840

) = sin120601A (120595 997888rarr 119863

0

119906120578119902) + cos120601

sdotA (120595 997888rarr 119863

0

119906120578119904)

A (120595 997888rarr 119863minus

119904120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881199041198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119904

)119860120595rarr119863

119904

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119904119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119904

)119860120595rarr119863

119904

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119904

2

119898120595+ 119898

119863119904

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119904

119898120595+ 119898

119863119904

A (120595 997888rarr 119863minus

119889120588+

) = minus119894

119866119865

radic2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198861(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119889

)119860120595rarr119863

119889

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

A (120595 997888rarr 119863minus

119889119870

lowast+

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198861(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119889

)119860120595rarr119863

119889

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119889

2

119898120595+ 119898

119863119889

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119889

119898120595+ 119898

119863119889

Advances in High Energy Physics 9

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

A (120595 997888rarr 119863

0

1199061205880

) = +119894

119866119865

2

119891120588119898

120588119881

lowast

1198881198891198811199061198891198862(120598

lowast

120588sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120588sdot 119901

120595) (120598

120595sdot 119901

120588)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120588120598]120595119901120572

120588119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120596) = minus119894

119866119865

2

119891120596119898

120596119881

lowast

1198881198891198811199061198891198862(120598

lowast

120596sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120596sdot 119901

120595) (120598

120595sdot 119901

120596)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120596120598]120595119901120572

120596119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906120601) = minus119894

119866119865

radic2

119891120601119898

120601119881

lowast

1198881199041198811199061199041198862(120598

lowast

120601sdot 120598

120595)

sdot (119898120595+ 119898

119863119906

)119860120595rarr119863

119906

1+ (120598

lowast

120601sdot 119901

120595) (120598

120595sdot 119901

120601)

sdot

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

120601120598]120595119901120572

120601119901120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881198891198811199061199041198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

A (120595 997888rarr 119863

0

119906119870

lowast0

) = minus119894

119866119865

radic2

sdot 119891119870lowast119898

119870lowast119881

lowast

1198881199041198811199061198891198862(120598

lowast

119870lowast sdot 120598

120595) (119898

120595+ 119898

119863119906

)119860120595rarr119863

119906

1

+ (120598lowast

119870lowast sdot 119901

120595) (120598

120595sdot 119901

119870lowast)

2119860120595rarr119863

119906

2

119898120595+ 119898

119863119906

minus 119894120598120583]120572120573120598

lowast120583

119870lowast120598

]120595119901120572

119870lowast119901

120573

120595

2119881120595rarr119863

119906

119898120595+ 119898

119863119906

(A1)

B The Amplitudes for the 120578119888rarr 119863119872 Decays

ConsiderA (120578

119888997888rarr 119863

minus

119904120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891120587119865120578119888rarr119863119904

0119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119904

) 119891119870119865120578119888rarr119863119904

0119881119906119904119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119889120587+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891120587119865120578119888rarr119863119889

0119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

+

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119889

) 119891119870119865120578119888rarr119863119889

0119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205870

)

= minus119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120587119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

0

)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891119870119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578119902)

= 119894

119866119865

2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119902

119865120578119888rarr119863119906

0119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120578119904)

= 119894

119866119865

radic2

(1198982

120578119888

minus 1198982

119863119906

) 119891120578119904

119865120578119888rarr119863119906

0119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906120578)

= cos120601A (120578119888997888rarr 119863

0

119906120578119902)

minus sin120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

0

1199061205781015840

)

= sin120601A (120578119888997888rarr 119863

0

119906120578119902)

+ cos120601A (120578119888997888rarr 119863

0

119906120578119904)

A (120578119888997888rarr 119863

minus

119904120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119904

1119881119906119889119881

lowast

1198881199041198861

A (120578119888997888rarr 119863

minus

119904119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119904

1119881119906119904119881

lowast

1198881199041198861

10 Advances in High Energy Physics

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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

A (120578119888997888rarr 119863

minus

119889120588+

)

= radic2119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119889

1119881119906119889119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

minus

119889119870

lowast+

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119889

1119881119906119904119881

lowast

1198881198891198861

A (120578119888997888rarr 119863

0

1199061205880

)

= minus119866119865119898

120588(120598

lowast

120588sdot 119901

120578119888

) 119891120588119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120596)

= 119866119865119898

120596(120598

lowast

120596sdot 119901

120578119888

) 119891120596119865120578119888rarr119863119906

1119881119906119889119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906120601)

= radic2119866119865119898

120601(120598

lowast

120601sdot 119901

120578119888

) 119891120601119865120578119888rarr119863119906

1119881119906119904119881

lowast

1198881199041198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119904119881

lowast

1198881198891198862

A (120578119888997888rarr 119863

0

119906119870

lowast0

)

= radic2119866119865119898

119870lowast (120598

lowast

119870lowast sdot 119901

120578119888

) 119891119870lowast119865

120578119888rarr119863119906

1119881119906119889119881

lowast

1198881199041198862

(B1)

Conflict of Interests

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

Acknowledgment

Thework is supported by the National Natural Science Foun-dation of China (Grants nos 11547014 11275057 11475055U1232101 and U1332103)

References

[1] V Novikov L Okun M Shifman et al ldquoCharmonium andgluonsrdquo Physics Reports vol 41 no 1 pp 1ndash133 1978

[2] S Okubo ldquoSome consequences of unitary symmetry modelrdquoPhysics Letters B vol 4 pp 14ndash16 1963

[3] G Zweig ldquoCERN-TH-401 402 412rdquo 1964[4] J Iizuka ldquoA systematics and phenomenology of meson familyrdquo

Progress of Theoretical Physics Supplement vol 37-38 pp 21ndash341966

[5] httpbes3ihepaccndatasetsdatasetshtm[6] H Li and S Zhu ldquoMini-review of rare charmonium decays at

BESIIIrdquo Chinese Physics C vol 36 no 10 pp 932ndash940 2012[7] R Aaij B Adeva M Adinolfi et al ldquoMeasurement of forward

119869120595 production cross-sections in pp collisions at radic119878 = 13TeVrdquoJournal of High Energy Physics vol 2015 no 10 article 172 2015

[8] G Aad B Abbott J Abdallah et al ldquoMeasurement of thedifferential cross-sections of inclusive prompt and non-prompt119869120595 production in protonndashproton collisions at radic119904 = 7 TeVrdquoNuclear Physics B vol 850 no 3 pp 387ndash444 2011

[9] V Khachatryan A Apresyan A Bornheim et al ldquoMeasure-ment of 119869120595 and 120595(2119878) Prompt Double-Differential CrossSections in pp Collisions at radic119904 = 7 TeVrdquo Physical ReviewLetters vol 114 no 19 Article ID 191802 2015

[10] M Ablikim M N Achasov X C Ai et al ldquoSearch for the raredecays 119869120595 rarr 119863

0

119870

lowast0rdquo Physical Review D vol 89 no 7 ArticleID 071101 2014

[11] M A Sanchis-Lozano ldquoOn the search for weak decays of heavyquarkonium in dedicated heavy-quark factoriesrdquo Zeitschrift furPhysik C Particles and Fields vol 62 no 2 pp 271ndash279 1994

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

[13] Y Shen and Y Wang ldquo119869120595 weak decays in the covariant light-front quark modelrdquo Physical Review D vol 78 no 7 Article ID074012 2008

[14] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoWeakdecays of 119869120595 the non-leptonic caserdquo The European PhysicalJournal C vol 55 no 4 pp 607ndash613 2008

[15] R C Verma A N Kamal and A Czarnecki ldquoHadronic weakdecays of120595rdquo Physics Letters B vol 252 no 4 pp 690ndash694 1990

[16] K K Sharma and R C Verma ldquoRare decays of 120595 and ΥrdquoInternational Journal ofModern Physics A vol 14 no 6 pp 937ndash945 1999

[17] R Dhir R C Verma and A Sharma ldquoEffects of flavordependence on weak decays of 119869120595 and Υrdquo Advances in HighEnergy Physics vol 2013 Article ID 706543 12 pages 2013

[18] 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

[19] M Beneke G Buchallab M Neubertc and C T SachrajdadldquoQCD factorization for exclusive non-leptonic B-meson decaysgeneral arguments and the case of heavy-light final statesrdquoNuclear Physics B vol 591 no 1-2 pp 313ndash418 2000

[20] H Cheng and C Chiang ldquoTwo-body hadronic charmedmesondecaysrdquo Physical Review D vol 81 Article ID 074021 2010

[21] Y Wang H Zou Z-T Wei X-Q Li and C-D Lu ldquoThetransition form factors for semi-leptonic weak decays of 119869120595 inQCD sum rulesrdquo The European Physical Journal C vol 54 pp107ndash121 2008

[22] T Feldmann P Kroll andB Stech ldquoMixing and decay constantsof pseudoscalar mesonsrdquo Physical Review D vol 58 no 11Article ID 114006 1998

[23] P Ball and G W Jones ldquoTwist-3 distribution amplitudes of 119870lowast

and120601mesonsrdquo Journal of High Energy Physics vol 2007 no 3 p

69 2007[24] P Ball V M Braun and A Lenz ldquoHigher-twist distribution

amplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 2006 no 5 article 004 2006

[25] G Buchalla A J Buras and M E Lautenbacher ldquoWeak decaysbeyond leading logarithmsrdquo Reviews of Modern Physics vol 68no 4 pp 1125ndash1244 1996

[26] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical Review D vol 22article 2157 1980

Advances in High Energy Physics 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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 11

[27] G Valencia ldquoAngular correlations in the decay 119861 rarr 119881119881 andCP violationrdquo Physical Review D vol 39 no 11 pp 3339ndash33451989

[28] G Kramer and W F Palmer ldquoBranching ratios and CP asym-metries in the decay 119861 rarr VVrdquo Physical Review D vol 45 no 1pp 193ndash216 1992

[29] G Legage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 p 4052 1992

[30] G Bodwin E Braaten and G Legage ldquoRigorous QCD analysisof inclusive annihilation and production of heavy quarkoniumrdquoPhysical Review D vol 51 no 3 p 1125 1995

[31] N Brambilla A Pineda J Soto and A Vairo ldquoEffective-fieldtheories for heavy quarkoniumrdquoReviews ofModern Physics vol77 no 4 pp 1423ndash1496 2005

[32] A Kamal Particle Physics Springer Berlin Germany 2014[33] B Xiao X Qin and B Ma ldquoThe kaon form factor in the light-

cone quark modelrdquoThe European Physical Journal A vol 15 pp523ndash527 2002

[34] 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