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
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
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
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Journal of
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Soft MatterJournal of
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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
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