Research Article Study of Decays with QCD Factorization ...downloads.hindawi.com/journals/ahep/2016/3863725.pdf · contribute to heavy-light nal states at leading power in the heavy-quark

Research ArticleStudy of 119861

lowast0

119889119904rarr 119863

+

119889119904119872minus Decays with QCD Factorization Approach

Qin Chang1 Xiaohui Hu1 Zhe Chang2 Junfeng Sun1 and Yueling Yang1

1 Institute of Particle and Nuclear Physics Henan Normal University Xinxiang 453007 China2University of Electronic Science and Technology of China Chengdu Sichuan 610054 China

Correspondence should be addressed to Qin Chang changqinhtueducn

Received 7 December 2015 Revised 18 February 2016 Accepted 3 March 2016

Academic Editor Chao-Qiang Geng

Copyright copy 2016 Qin Chang 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

Motivated by the 119887-physics experiments at running LHC and upcoming SuperKEKBBelle-II the nonleptonic 119861lowast0119902rarr 119863

+

119902119872

minus (119902 =119889 119904 and119872 = 120587119870 120588 119870

lowast) weak decays are studied within QCD factorization framework The observables of these decay modesare first predicted It is found that the tree-dominated and CKM-favored 119861lowast0

119902rarr 119863

+

119902120588minus decays have the largest branching fractions

simO(10minus8) and thus are hopefully to be measured The 119861lowast0119902rarr 119863

+

119902119881

minus decays are dominated by the longitudinal polarization statesIn addition associating with the relevant 119861 meson decays some interesting phenomena and relations are discussed in detail forexample 119877B equivB(119861

119902rarr 119863

lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃ 3120591

119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881)

1 Introduction

Thanks to the fruitful running of BABAR Belle CDF andD0 experiments in the past decade most of the 119861 mesondecays with branching fractions ≳O(10minus7) are measuredprecisely which provide a very fertile testing ground forthe Standard Model (SM) pictures of flavor physics charge-parity (CP) asymmetry andQCDmechanism As the particlephysics enter a new era of precision more experimentalinformation about 119887-physics will be explored at ongoing LHCand forthcoming SuperKEKBBelle-II In the (119887119902) systembesides the bound state 119861

119902meson some excited states decays

are also hopefully to be well measured in the futureThe 119861lowast

119902meson with quantum number of 1198992119904+1119871

119869=

13

1198781and 119869119875 = 1

minus is the first excited state in the spectraof heavy-light (119887119902) system [1ndash4] In the past years the 119861lowast

119902

masses (or the differences 119898119861lowast

119902

minus 119898119861119902

) are well measuredby many collaborations [5ndash10] However the experimentalinformation about the 119861lowast

119902meson decays is very limited due to

the following facts (i) At 119890+119890minus collider119861lowast119906119889

and119861lowast119904mesons are

produced mainly through Υ(5119878) resonance decays while thepast 119887-physics experiments (BABAR and Belle) run mainly

around Υ(4119878) resonance for producing 119861119906119889

mesons and thecollected data sample of 119890+119890minus collisions in the vicinity ofΥ(5119878)resonance is not sufficient enough to probe 119861lowast rare decays(ii) the decays of unstable 119861lowast mesons are dominated by theelectromagnetic transition 119861lowast rarr 119861120574 and thus the otherdecay modes are too rare to be detected soon Fortunatelysuch situation will possibly be improved by the running LHCand upgrading Belle-II experiments in the near future

For the Belle-II experiment at SuperKEKB one of themain goals of theΥ(5119878) physics program is to study the decaysof 119861

119904meson Meanwhile it also should be noted that a plenty

of 119861lowast samples would be produced simultaneously With thetarget luminosity 8 times 1035 cmminus2sminus1 the annual integratedluminosity is expected to be 13 abminus1year after 2018 [11]Usingthe cross section of Υ(5119878) production 120590(119890+119890minus rarr Υ(5119878)) =

0301 nb [12] and the branching fractions of Υ(5119878) decaysrelated to 119861lowast final states [13] we find that about 119873(119861lowast

119906119889+

119861

lowast

119906119889)year sim 4 times 10

9 and 119873(119861lowast119904+ 119861

lowast

119904)year sim 2 times 10

9

samples could be collected per year As a result the 119861lowastdecays with branching fractions ≳O(10minus9) are hopefully to beobserved by Belle-II Moreover because of the much largerbeauty production cross section of 119901119901 collisions [14ndash16]

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 3863725 9 pageshttpdxdoiorg10115520163863725

2 Advances in High Energy Physics

the LHC experiments may also provide a lot of experimentalinformation for 119861lowast decays For instance as analyzed in[17] even though 119861lowast

119904rarr 119897

+

119897minus decay having branching

fraction simO(10minus11) is very rare and obviously out of thescope of Belle-II it is still possible to be measured by LHCafter high-luminosity upgrade (Run-III) So with the rapiddevelopment of experiment the theoretical studies of 119861lowastweak decays are worthwhile then

Recently a few interesting theoretical studies of119861lowast decayshave been performed For instance in [18ndash20] the theoreticalestimates of the semileptonic 119861lowast

119888decays are made within

QCD sum rules framework In [17] the pure leptonic119861lowast119904rarr 119897119897

and 119861lowast119906119888rarr 119897] decays are studied and the detectability of

LHC on these decays is analyzed in detail Besides the (semi-)leptonic119861lowast decays the study of nonleptonic119861lowast weak decayswhich involve much more decay channels is also essentialIn this paper we will focus on the 119861lowast0

119889119904rarr 119863

+

119889119904119872

minus (119872 =

120587119870 120588 and 119870lowast) decays which are tree-dominated and thusprincipally have relative large branching fractions In the 119861meson system in order to evaluate the QCD correctionssome calculation approaches such as the QCD factorization(QCDF) [21 22] the perturbative QCD [23 24] and thesoft-collinear effective theory [25ndash28] are presented Thenthe nonleptonic 119861lowast decays provide another testing groundfor these approaches In our following calculation the QCDfactorization approach is employed

Our paper is organized as follows In Section 2 the basictheoretical framework and the amplitudes of 119861lowast0

119889119904rarr 119863

+

119889119904119872

minus

decays calculated through theQCDF approach are presentedSection 3 is devoted to the numerical results and discussionFinally a short summary is given

2 Theoretical Framework

The low energy effective Hamiltonian responsible for the119861

lowast0

119889119904rarr 119863

+

119889119904119872

minus decays could be written as [31 32]

Heff =119866119865

radic2

sum

119902=119889119904

119881119888119887119881

lowast

119906119902119862

1(120583)119876

1(120583) + 119862

2(120583)119876

2(120583)

+ hc

(1)

where 119866119865is the Fermi coupling and 119881

119888119887119881

lowast

119906119902is product of

Cabibbo-Kobayashi-Maskawa (CKM) matrix elements TheWilson coefficients119862

12(120583) summarize the physical contribu-

tions above scale of 120583 and are calculable with the perturbationtheoryTheir values at scales of120583 sim O(119898

1198872119898

119887 2119898

119887) in naive

dimensional regularization scheme are listed in Table 2 1198761

and 1198762are local tree four-quark operators and defined as

1198761= [119888

119894120574120583(1 minus 120574

5) 119887

119894] [119902

119895120574120583

(1 minus 1205745) 119906

119895]

1198762= [119888

119894120574120583(1 minus 120574

5) 119887

119895] [119902

119895120574120583

(1 minus 1205745) 119906

119894]

(2)

where 119894 and 119895 are color indices and the sum over repeatedindices is understood One can refer to [31 32] for detailsof this part To obtain the decay amplitudes the remaining

works are to accurately calculate the hadronic matrix ele-ments of local operators

The simplest way to deal with the hadronic matrixelements is the naive factorization (NF) scheme [33 34] basedon the color transparency mechanism [35ndash37] Within theNF scheme the hadronic matrix element is approximatedby the product of two current matrix elements which arefurther parameterized by decay constants and transition formfactors Explicitly for the 119861lowast0

119902rarr 119863

+

119902119872

minus (119902 = 119889 119904) decays thehadronic matrix element could be written as

119867119863119872= ⟨119872119863 |119876| 119861

lowast

⟩

= ⟨119872

1003816100381610038161003816119902120574

120583

(1 minus 1205745) 119906

10038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119888120574

120583(1 minus 120574

5) 119887

10038161003816100381610038161003816119861

lowast

⟩

(3)

For the case that119872 is a light pseudoscalar (119875) evaluating (3)we get

119867119863119875= 119894

119866119865

radic2

(minus2119898119861lowast) (120578 sdot 119901

2) 119860

119861lowast

rarr119863

0(0) 119891

119875 (4)

where 120578 is the polarization four-vector of 119861lowast meson 1199012is the

momentum of 119875 and their production could be simplified byreplacing

120578 sdot 1199012997888rarr 119901

119888

=

radic[1198982

119861lowast minus (119898

119863+ 119898

119875)

2

] [1198982


119863minus 119898

119875)

2

]

2119898119861lowast

(5)

For the case that119872 is a light vector (119881) corresponding to thedifferent helicity amplitudes (3) could be written as

1198670

119863119881= minus119894119891

119881

1198982

119861lowast minus 119898

2

119863

2119898119861lowast

[(119898119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

+ (119898119861lowast minus 119898

119863) 119860

119861lowast

rarr119863

2(0)] = minus119894119891

119881(119898

2


2

119863)

sdot 119860119861lowast

rarr119863

0(0)

119867plusmn

119863119881= minus119894119891

119881119898

119881[(119898

119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

∓ (119898119861lowast minus 119898

119863) 119881

119861lowast

rarr119863

(0)]

(6)

In the evaluations the definition of decay constants

⟨119875 (1199012)

10038161003816100381610038161199021120574120583

12057451199022

10038161003816100381610038160⟩ = minus119894119891

119875119901

120583

2

⟨119881 (1199012 120576

lowast

)10038161003816100381610038161199021120574120583

1199022

10038161003816100381610038160⟩ = minus119894119891

119881119898

119881120576lowast120583

(7)

the form factors (the expression of parameterization of thehadronic matrix ⟨119875|1199021015840Γ119902|119881⟩ such as (8) could be obtained

Advances in High Energy Physics 3

through taking the Hermitian conjugate of ⟨119881|119902Γ1199021015840|119875⟩ forthe latter of which we take the same conventions as [38])

⟨119863 (1199011)

100381610038161003816100381610038161199023120574120583119887

10038161003816100381610038161003816119861lowast

(119901 120578)⟩

= minus

2119894119881 (1199022

)

119898119861

lowast + 119898119863

120576120583]120588120590120578

]119901120588

119901120590

1

⟨119863 (1199011)

1003816100381610038161003816100381611990231205741205831205745119887

10038161003816100381610038161003816119861lowast

(119901)⟩ = 2119898119861

lowast1198600(119902

2

)

120578 sdot 119902

1199022

119902120583

+ (119898119875+ 119898

119861

lowast)1198601(119902

2

) (120578120583

minus

120578 sdot 119902

1199022

119902120583

)

+ 1198602(119902

2

)

120578 sdot 119902

119898119863+ 119898

119861

lowast

[(119901 + 1199011)

120583

minus

1198982


2

119863

1199022

119902120583

]

(8)

and the relation

119860119861lowast

rarr119863

0(0) =

1

2119898119861lowast

[(119898119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

+ (119898119861lowast minus 119898

119863) 119860

119861lowast

rarr119863

2(0)]

(9)

are used Then one can easily get the amplitudes of 119861lowast0119902rarr

119863+

119902119872

minus decays within NF which are proportional to 119867119863119875

or119867

0plusmn

119863119881However in the NF framework the amplitudes are

renormalization scale dependence and the nonfactorizablecontributions dominated by the hard gluon exchange arelost As a result the amplitude is unphysical and the strongphase which is essential for evaluating CP asymmetrycannot be calculated In order to remedy these deficienciesand take into account the nonfactorizable contribution theQCDF approach is proposed by BBNS [21 22] and has beenwidely used to deal with the hadronic matrix elements (eg[39ndash49]) Within the framework of QCDF in the heavy-quark limit (119898

119887≫ ΛQCD) the hadronic matrix elements

⟨119863119872|119876119894|119861

lowast

⟩ are expressed by the factorization formula [2122]

⟨1198721198631003816100381610038161003816119876

119894

1003816100381610038161003816119861lowast

⟩ = sum

119895

119865119861lowast

rarr119863

119895int119889119909T

119894119895(119909)Φ

119872(119909) (10)

Here119865119861lowast

rarr119863

119895is119861lowast rarr 119863 form factorT

119894119895(119909) is hard-scattering

function which is perturbatively calculable Φ119872(119909) is the

light-cone distribution amplitude for the quark-antiquarkFock state of meson 119872 The leading-twist distributionamplitudes of pseudoscalars (120587 and 119870) and longitudinalpolarized vectors (120588 and119870lowast) are conventionally expanded inGegenbauer polynomials [29 30 50]

Φ119872(119909) = 6119909 (1 minus 119909) [1 +

infin

sum

119899=1

119886119872

11989911986232

119899(2119909 minus 1)] (11)

where 119886119872119899

is the Gegenbauer moment In factorization for-mula (10) the spectator scattering contribution does notappear due to the fact that it is not only 120572

119904-suppressed but

also power-suppressed by the factor of ΛQCD119898119887relative to

the LO contribution for case of heavy-light final states whilethe vertex correction included in (10) is only 120572

119904-suppressed

In fact after a detailed analysis for 119861 decays the authors of[22] have concluded that the spectator interaction does notcontribute to heavy-light final states at leading power in theheavy-quark expansion [22]

Applying the QCDF formula the matrix elements of theeffective weak Hamiltonian for 119861lowast0

119902rarr 119863

+

119902119872

minus (119902 = 119889 119904)decays could be written as

A (119861lowast0

119902997888rarr 119863

+

119902119872

minus

) = ⟨119872minus

119863+

119902

1003816100381610038161003816Heff

1003816100381610038161003816119861

lowast0

119902⟩ =

119866119865

radic2

sdot 119881119888119887119881

lowast

1199061199021205721⟨119872

1003816100381610038161003816119902120574

120583

(1 minus 1205745) 119906

10038161003816100381610038160⟩

sdot ⟨119863

10038161003816100381610038161003816119888120574

120583(1 minus 120574

5) 119887

10038161003816100381610038161003816119861

lowast

⟩

(12)

where the products of matrix elements of two current⟨119872|119902120574

120583

(1 minus 1205745)119906|0⟩⟨119863|119888120574

120583(1 minus 120574

5)119887|119861

lowast

⟩ have been givenexplicitly by (4) and (6) The effective coefficient 120572

1in

the amplitude including nonfactorizable contributions fromQCD radiative vertex corrections is defined as [22 51]

1205721= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO21198811 (13)

Obviously theNF result is recovered if theQCD-loop correc-tion (the third term in (13)) is neglected With the modifiedminimal subtraction (MS) scheme for the pseudoscalar 119872and the longitudinally polarized vector meson the function1198811is written as

1198811= 3 log(

1198982

119887

1205832

) + 3 log(119898

2

119888

1205832

) minus 18

+ int

1

0

119889119909119879 (119909)Φ119872(119909)

(14)

where the loop function 119879(119909) is

119879 (119909) =

119888119886

1 minus 119888119886

log (119888119886) minus

119888119887

1 minus 119888119887

log (119888119887)

+

119888119889

1 minus 119888119889

log (119888119889) minus

119888119888

1 minus 119888119888

log (119888119888)

minus 119903119888[

119888119886

(1 minus 119888119886)

2

log (119888119886) +

1

1 minus 119888119886

]

minus 119903minus1

119888[

119888119889

(1 minus 119888119889)

2log (119888

119889) +

1

1 minus 119888119889

] + 119891 (119888119886)

minus 119891 (119888119887) minus 119891 (119888

119888) + 119891 (119888

119889)

+ 2 log (1199032119888) [log (119888

119886) minus log (119888

119887)]

(15)

with the definitions 119903119888= 119898

119888119898

119887 119888

119886= 119909(1minus119903

2

119888) 119888

119887= 119909(1minus119903

2

119888)

119888119888= minus119888

119886119903

2

119888 and 119888

119889= minus119888

119887119903

2

119888 For the transversely polarized


vectormeson the leading-twist contribution to1198811is zeroThe

result of 1198811(see (14)) is exactly the same as the result in 119861 rarr

119863(lowast)

119871 decays where 119871 is a light meson given in [22]With the amplitudes given in (12) we can evaluate the

observables of 119861lowast0119902rarr 119863

119902119872 decays In the rest frame of

119861

lowast0

119902meson the spin-averaged branching fractions could be

written as

B (119861lowast0

119902997888rarr 119863

119902119875)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902

Γtot (119861lowast0

119902)

100381610038161003816100381610038161003816

A (119861lowast0

119902997888rarr 119863

119902119875)

100381610038161003816100381610038161003816

2

(16)

B (119861lowast0

119902997888rarr 119863

119902119881)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

sum

120582

100381610038161003816100381610038161003816

A120582(119861

lowast0

119902997888rarr 119863

119902119881)

100381610038161003816100381610038161003816

2

(17)

where Γtot(119861lowast0

119902) is the total decay width of 119861lowast0

119902 Besides of the

branching fraction the polarization fractions of 119861lowast0119902rarr 119863

119902119881

decays are also important observables They are defined as

119891119871perp =

1003816100381610038161003816A

0perp1003816100381610038161003816

2

1003816100381610038161003816A

0

1003816100381610038161003816

2

+

1003816100381610038161003816A

1003816100381610038161003816

2

+

1003816100381610038161003816A

perp

1003816100381610038161003816

2 (18)

whereAandA

perpare parallel and perpendicular amplitudes

and could be easily obtained throughAperp = (Aminus

plusmnA+)radic2

3 Numerical Results and Discussions

With the theoretical framework given in Section 2 we thenpresent our numerical results and discussions Firstly wewould like to clarify the input parameters in our numericalevaluation The input values of Wolfenstein parametersmasses of quarks decay constants andGegenbauermomentsare summarized in Table 1 Our numerical results of theWilson coefficients 119862

1and 119862

2at different scales are listed

in Table 2 Besides that to evaluate the branching fractionsof 119861lowast0

119902rarr 119863

119902119872 decays the total decay widths Γtot(119861

lowast

119902) are

essential Unfortunately there are no available experimentalor theoretical results until now In our numerical evaluationthe approximation Γtot(119861

lowast

119902) ≃ Γ(119861

lowast

119902rarr 119861

119902120574) is taken because

of the known fact that the radiative process 119861lowast119902rarr 119861

119902120574

dominates119861lowast119902meson decaysTheoretically the predictions on

Γ(119861lowast

rarr 119861120574) have been widely evaluated in various modelssuch as relativistic quarkmodel [52 53] QCD sum rules [54]light-cone QCD sum rules [55] light front quark model [56]heavy-quark effective theory with vector meson dominancehypothesis [57] or covariant model [58] In this paper weemploy the most recent results [56 58]

Γ (119861lowast0

997888rarr 1198610

120574) = (148 plusmn 20) eV

Γ (119861lowast0

119904997888rarr 119861

0

119904120574) = (68 plusmn 17) eV

(19)

which are consistent with the results in the other models

Table 1 The values of input parameters

TheWolfenstein parameters120582 = 022548

+000068

minus000034[13] 119860 = 0810

+0018

minus0024[13]

Masses of quarks119898

119888= 167 plusmn 007GeV [13] 119898

119887= 478 plusmn 006GeV [13]

Decay constants119891120587= 13041 plusmn 020MeV [13] 119891

119870= 1562 plusmn 07MeV [13]

119891120588= 216 plusmn 3MeV [29] 119891

119870lowast = 220 plusmn 5MeV [29]

The Gegenbauer moments at the scale 120583 = 1GeV119886

120588

1= 0 [29] 119886

120588

2= 015 plusmn 007 [29]

119886119870lowast

1= 003 plusmn 002 [29] 119886

119870lowast

2= 011 plusmn 009 [29]

119886120587

1= 0 [30] 119886

120587

2= 025 plusmn 015 [30]

119886119870

1= 006 plusmn 003 [30] 119886

119870

2= 025 plusmn 015 [30]

In addition the values of 119861lowast119902rarr 119863

119902transition form

factors are also unknown In this paper the Bauer-Stech-Wirbel (BSW) model [59] is employed to evaluate the valuesof 119860

0(0) 119860

1(0) and 119881(0) which could be written as the

overlap integrals of wave functions of mesons [59] With themeson wave function 120593

119872(

perp 119909) as solution of a relativistic

scalar harmonic oscillator potential and 120596 = 04GeV whichdetermines the average transverse quark momentum we get

119860

119861lowast

119889rarr119863119889

0(0) = 071

119860

119861lowast

119889rarr119863119889

1(0) = 075

119881119861lowast

119889rarr119863119889

(0) = 076

119860

119861lowast

119904rarr119863119904

0(0) = 066

119860

119861lowast

119904rarr119863119904

1(0) = 069

119881119861lowast

119904rarr119863119904

(0) = 072

(20)

In our numerical evaluation these numbers and 15of themare treated as default inputs and uncertainties respectively

Using the given values of input parameters and thetheoretical formula we then present QCDF predictions ofthe CP-averaged branching ratios of 119861lowast0

119902rarr 119863

+

119902119872

minus (119872 =

120587119870 120588 119870lowast) decays in Table 3 in which the three theoretical

uncertainties are induced by the CKM parameters hadronicparameters (decay constants and form factors) and totaldecay widths respectively In comparison the NF results arealso listed in Table 3 The followings are some analyses anddiscussions

(1) In Table 2 the values of effective coefficient 1205721within

NF and QCDF are summarized It could be foundthat information of strong phases is obtained byconsidering gluon radiative corrections to vertexwhich plays an important role in exploring the directCP violation However due to lack of interferencethe direct CP asymmetries of 119861lowast0

119902rarr 119863

+

119902119872

minus (119872 =

120587119870 120588 119870lowast) decays are zero


Table 2 The values of Wilsonrsquos coefficients 11986212(120583) and effective coefficient 120572

1

120583 119862LO1

119862LO2

119862NLO1

119862NLO2

1205721(NF) 120572

1(QCDF)

1198981198872 1166 minus0335 1126 minus0266 1037 1075 minus 0027119894

119898119887

111 minus0236 1076 minus0173 1018 1054 minus 0016119894

2119898119887


Table 3 The results of branching fractions

Decay modes CKM NF QCDF120583 = 119898

119887120583 = 119898

1198872 120583 = 119898

119887120583 = 2119898

119887

119861

lowast0

rarr 119863+

119870minus

[10minus10

] 1205823

33+02+11+05

minus02minus10minus0437

+02+12+06


+02+12+06


+02+11+05


119861

lowast0

119904rarr 119863

+

119904119870

minus

[10minus10

] 1205823

63+03+21+21


+03+23+06


+03+22+23


+03+21+22


119861

lowast0

rarr 119863+

120587minus

[10minus9

] 1205822

44+02+15+07


+02+16+08


+02+16+07


+02+15+07


119861

lowast0

119904rarr 119863

+

119904120587minus

[10minus9

] 1205822

85+04+28+28


+04+31+32


+04+30+30


+04+29+29


119861

lowast0

rarr 119863+

119870lowastminus

[10minus10

] 1205823

76+04+19+12


+04+21+13


+04+20+13


+04+19+12


119861

lowast0

119904rarr 119863

+

119904119870

lowastminus

[10minus9

] 1205823

15+01+04+05


+01+04+05


+01+04+05


+01+04+05


119861

lowast0

rarr 119863+

120588minus

[10minus8

] 1205822

13+01+03+02


+01+04+02


+01+03+02


+01+03+02


119861

lowast0

119904rarr 119863

+

119904120588minus

[10minus8

] 1205822

26+01+06+09


+01+07+09


+01+07+09


+01+06+09


In Figure 1 the dependence of tree coefficient 1205721on

the renormalization scale 120583 is shown As Figure 1(b)shows the imaginary part Im(120572

1) which is zero at LO

(NF result) arises after taking into account the NLOcorrections For the real part Re(120572

1) as Figure 1(a)

shows the scale dependence has been reduced partlyat low scales when the NLO corrections are takeninto account To further clarify such partial reductionwe define the quantity 119896(120583) = |119889Re[120572

1(120583)]119889120583|

which is equal to zero if Re[1205721(120583)] is totally scale-

independent It is found that the value of 119896 at NLOis a little bit smaller than the one at LO for instance119896 (2GeV)times103 = 21 (LO) 14 (NLO) and 119896 (3GeV)times10

3

= 99 (LO) 97 (NLO) as found from Figure 1(a)However one also should note that the reduction ofscale dependence is not very obvious as one expectedwhich could be attributed to the fact that NLOcorrection is color-suppressed [60] while the scaledependence reduction effect becomes very significantwhen the NNLO correction which is no longer color-suppressed is taken into account as found in [60]

(2) From Table 3 one may find a clear hierarchy ofbranching fractions B(119861lowast0

119902rarr 119863

+

119902120588minus

) gt B(119861lowast0

119902rarr

119863+

119902120587minus


119902rarr 119863

+

119902119870

lowastminus


119902rarr 119863

+

119902119870

minus

)It is mainly induced by the following two reasons(i) The CKM element 119881

119888119887119881

lowast

119906119904responsible for 119861lowast0

119902rarr

119863+

119902119870

(lowast)minus decays is suppressed by factor of 120582 comparedwith the one 119881

119888119887119881

lowast

119906119889for 119861lowast0

119902rarr 119863

+

119902120587minus

(120588minus

) decays and(ii) the 119861lowast

119902rarr 119863

119902119875 decays are suppressed relatively

by the orbital angular momentum compared with thecorresponding 119861lowast

119902rarr 119863

119902119881 decays

In addition one also may find that the 119861lowast0119904

decay isalways about two times larger than the corresponding

119861

lowast0

119889decay for instance B(119861

lowast0

119904rarr 119863

+

119904119870

minus

) asymp

2B(119861lowast0

rarr 119863+

119870minus

) It is mainly induced by thetheoretical prediction Γ(119861lowast0 rarr 119861

0

120574)Γ(119861lowast0

119904rarr

1198610

119904120574) asymp 2 (see (19)) and the assumption Γtot(119861

lowast

119902) ≃

Γ(119861lowast

119902rarr 119861

119902120574) Explicitly such relation could be

expressed as

119877119904119889equiv

B (119861lowast0

119904997888rarr 119863

119904119872)

B (119861lowast0

119889997888rarr 119863

119889119872)

≃

Γ (119861lowast0

119889997888rarr 119861

0

119889120574)

Γ (119861lowast0

119904997888rarr 119861

0

119904120574)

theo≃ 2 (21)

which is a useful observable for measuring 120591119861

lowast0

119889

120591119861

lowast0

119904

experimentally and further testing the theoreticalpredictions of Γ(119861lowast

119902rarr 119861

119902120574)

From Table 3 it could be found that 119861lowast0119902rarr 119863

+

119902120588minus

decays have the largest branching fractions aboutO(10minus8) and thus are hopefully to be well measuredby Belle-II experiment in the near future In additionthe processes that 119861lowast0

119902decays into two light mesons

such as 120587120587 and 120587119870 final states generally have muchmore interesting phenomena However they are gen-erally CKM- andor loop-suppressed and thereforehard to be observed soon

(3) Besides the branching ratio the polarization fractionsare also important observables In the 119861 rarr 119881119881

decays the hierarchy pattern of helicity amplitudes

A0 A

minus A

+= 1

ΛQCD

119898119887

(

ΛQCD

119898119887

)

2

(22)

is expected [61ndash63] especially for the tree-dominateddecays For the 119861lowast0

119902rarr 119863

119902119881 decays such hierarchical

relation is also naively expected due to the following


Re(120572

1)

100

102

104

106

108

3 4 5 6 7 82120583 (GeV)

(a)

minus001

000

001

002

003

004

005

Im(120572

1)

3 4 5 6 7 82120583 (GeV)

(b)

Figure 1 Dependence of the tree coefficient 1205721(119863120587) on the renormalization scale 120583 with asymptotic light-cone distribution amplitudes

120601120587(119909) = 6119909119909 The solid red and dotted blue lines denote 120572

1(119863120587) within QCDF and NF frameworks respectively

taking 119861lowast0 rarr 119863+

120588minus decay (119887 rarr 119888119906119889 transition)

as an example for convenience of discussion in thelongitudinal transition the quark and antiquark ineach meson have opposite helicities in which thecase (ℎ

119902 ℎ

119902) = (minus12 12) is favored by (119881 minus 119860)

interaction Relative to A0 for A

minusto occur the 119906

quark has to flip its helicity which results in the so-called ldquohelicity-fliprdquo suppression ForA

+ in addition

to the ldquohelicity-fliprdquo suppression a further chiralitysuppression appears since the 119888 quark in the (119881 minus 119860)interaction has a ldquowrongrdquo helicity at this momentExactly from (6) it could be found that the trans-verse amplitudes 119867plusmn

119863119881are suppressed by a factor

2119898119861lowast119898

119881(119898

2


2

119863) sim ΛQCD119898119887

relative to 1198670

119863119881

In addition the axial-vector and vector contributionto 119867+

119863119881cancel in the heavy-quark limit As a result

the hierarchy pattern of helicity amplitudes (see (22))is still fulfilled by the 119861lowast0

119902rarr 119863

119902119881 decays within NF

framework Further considering that the QCD NLOcorrection in120572

1(see (13)) ismuch smaller than the LO

one the very large longitudinal polarization fractionsof 119861lowast0

119902rarr 119863

119902119881 decays are generally expected in both

NF and QCDF frameworks Numerically within theQCDF using the default values of input parametersand taking 120583 = 119898

119887 we get

119891119871(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

)

= (85 86 89 89)

119891(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

) = (12 12 9 9)

(23)

(4) In order to explore the relation between 119861119902rarr 119863

lowast

119902119875

and 119861lowast119902rarr 119863

119902119875 decays we define the ratio

119877A equiv

10038161003816100381610038161003816A (119861

119902997888rarr 119863

lowast

119902119875)

10038161003816100381610038161003816

10038161003816100381610038161003816A (119861

lowast

119902997888rarr 119863

119902119875)

10038161003816100381610038161003816

=

(1198982

119861minus 119898

2

119863lowast)119860

119861rarr119863lowast

0

(1198982


2

119863) 119860

119861lowastrarr119863

0

(24)

which is independent of the decay constants 119891119875and

the coefficient 1205721and close to 1 Further evaluating

the branching fractions we get

119877B equiv

B (119861119902997888rarr 119863

lowast

119902119875)

B (119861lowast

119902997888rarr 119863

119902119875)

= 3

120591119861

120591119861lowast

(1198982

119861minus 119898

2

119863lowast)

2

1198983

119861lowast

(1198982


2

119863)

2

1198983

119861

119860119861rarr119863

lowast

0

119860119861lowastrarr119863

0

(25)

in which the prefactor of 3 corresponds to the factor13 in (16) caused by averaging over the initial119861lowast spinWith the values ofmasses given by PDG [13] the ratioofmasses in (25) is equal to 094 for 119902 = 119889 and 119904More-over the ratio of form factors in (25) is generally closeto 1 for instance 063071(058066) ≃ 089(088)for 119902 = 119889(119904) within WSB model So the relation119877B ≃ 3120591

119861120591

119861lowast is expected Numerically with the

assumption Γtot(119861lowast

119902) ≃ Γ(119861

lowast

119902rarr 119861

119902120574) and the values

of Γ(119861lowast119902rarr 119861

119902120574) given by (19) we get 119877B ≃ 102 times 10

6

for 119902 = 119889 and 047 times 106 for 119902 = 119904 which could betested experimentally

For the 119861119902rarr 119863

lowast

119902119881 and 119861lowast

119902rarr 119863

119902119881 decays the

relation between their polarization fractions is muchinteresting It could be found that the relation

119891119871 (119861119902 997888rarr 119863

lowast

119902119881) ≃ 119891

119871 (119861lowast

119902997888rarr 119863

119902119881) (26)

is generally expected because (i) the expressions oftheir helicity amplitudes are very similar to eachother except for the replacements 119861lowast harr 119861 and119863 harr 119863

lowast everywhere in (6) and (ii) different frombranching ratio (see (26)) the polarization fractionis sensitive to the relative strengths of form factorsrather than the absolute ones In order to test therelation we take 119861lowast0 rarr 119863

+

120588minus and 1198610 rarr 119863

lowast+

120588minus

decays for example One may find that our prediction


119891119871(119861

lowast0

rarr 119863+

120588minus

) = (89 plusmn 1) numerically agreeswell with 119891

119871(119861

0

rarr 119863lowast+

120588minus

) = 87 predicted in [64]which is consistent with experimental results sim885[65]

4 Summary

In this paper detailed analyses of the 119861lowast0119889119904

rarr 119863+

119889119904119872

minus

(119872 = 120587119870 120588 119870lowast) weak decays are performed within QCD

factorization framework The theoretical predictions for thebranching fractions and polarization fractions are presentedinTable 3 and (23) respectively Some interesting phenomenaand relations are discussed It is found that (i) there is aclear hierarchy of branching fractions in 119861lowast0

119889119904rarr 119863

+

119889119904119872

minus

decays in which the 119861lowast0119902rarr 119863

+

119902120588minus decays have the largest

branching fractions simO(10minus8) and thus are very hopefullyto be observed by Belle-II experiment in the near future (ii)The 119861lowast0

119889119904rarr 119863

+

119889119904119881

minus decays are dominated by the longitudinalpolarization states numerically 119891

119871sim [80 90] (iii) Some

interesting and useful correlations between 119861lowast119902rarr 119863

119902119872

and its corresponding 119861119902rarr 119863

lowast

119902119872 decays are presented

For instance 119877B equiv B(119861119902rarr 119863

lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are

generally expected All of above findings are waiting for theexperimental test at LHC and SuperKEKBBelle-II

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11475055 11275057 U1232101 andU1332103) Q Chang is also supported by the Foundationfor the Author of National Excellent Doctoral Dissertation ofChina (Grant no 201317) the Program for Science and Tech-nology Innovation Talents in Universities of Henan Province(Grant no 14HASTIT036) and Foundation for UniversityKey Teacher of Henan Province (Grant no 2013GGJS-58)

References

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[2] S Godfrey and R Kokoski ldquoProperties of P-wave mesons withone heavy quarkrdquo Physical Review D vol 43 no 5 pp 1679ndash1687 1991

[3] E J Eichten C T Hill and C Quigg ldquoProperties of orbitallyexcited heavy-light (Qqminus) mesonsrdquo Physical Review Letters vol71 pp 4116ndash4119 1993

[4] D Ebert V O Galkin and R N Faustov ldquoMass spectrumof orbitally and radially excited heavy-light mesons in therelativistic quark modelrdquo Physical Review D vol 57 no 9 pp

5663ndash5669 1998 Erratum in Physical Review D vol 59 ArticleID 019902 1998

[5] R Aaij C Abellan Beteta A Adametz et al ldquoFirst observationof the decay 119861

1199042(5840) rarr 119861

+ 119870

minus and studies of excited 1198610119904

mesonsrdquo Physical Review Letters vol 110 no 15 Article ID151803 2013

[6] R Louvot J Wicht O Schneider et al ldquoMeasurement of thedecay 1198610

119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870

plusmn in 119890+119890minusannihilation at radic119904 asymp 1087 GeVrdquo Physical Review Letters vol102 no 2 Article ID 021801 2009

[7] O Aquines Z Li A Lopez et al ldquoMeasurements of theexclusive decays of the Υ(5S) to B meson final states andimproved 119861

119904mass measurementrdquo Physical Review Letters vol

96 Article ID 152001 2006[8] K Ackerstaff G Alexander J Allison et al ldquoBlowast production in

Z0 decaysrdquo Zeitschrift fur Physik C Particles and Fields vol 74no 3 pp 413ndash423 1997

[9] D Buskulic D Casper I De Bonis et al ldquoProduction of excitedbeauty states in Z decaysrdquo Zeitschrift fur Physik C Particles andFields vol 96 no 3 pp 393ndash404 1996

[10] P Abreu W Adam T Adye et al ldquo119861lowast production in Z decaysrdquoZeitschrift fur Physik C Particles and Fields vol 68 no 3 pp353ndash362 1995

[11] T Abe I Adachi K Adamczyk et al ldquoBelle II technicaldesignreportrdquo httparxivorgabs10110352

[12] G S Huang D H Miller V Pavlunin et al ldquoMeasurementof 119861(Υ(5119878) rarr 119861

(lowast)

119904119861119904

(lowast)

) using 120601 mesonsrdquo httparxivorgabshep-ex0607080v1

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

[14] A Bharucha I I Bigi C Bobeth et al ldquoImplications of LHCbmeasurements and future prospectsrdquo The European PhysicalJournal C vol 73 article 2373 2013

[15] R Aaij C Abellan Beteta B Adeva et al ldquoMeasurement of120590(119901119901 rarr 119887119887119883) at radics = 7TeV in the forward regionrdquo PhysicsLetters B vol 694 pp 209ndash216 2010

[16] R Aaij B Adeva M Adinolfi et al ldquoLHCb detector perfor-mancerdquo International Journal of Modern Physics A vol 30 no7 Article ID 1530022 2015

[17] B Grinstein and J M Camalich ldquoWeak decays ofunstable b-mesonsrdquo httparxivorgabs150905049

[18] Z-G Wang ldquoSemileptonic decays 119861lowast119888rarr 120578

119888119897V119897 with QCD sum

rulesrdquo Communications in Theoretical Physics vol 61 no 1article 81 2014

[19] K Zeynali V Bashiry and F Zolfagharpour ldquoForm factors anddecay rate of Blowast

C rarr Dsl+lminus decays in the QCD sum rulesrdquo TheEuropean Physical Journal A vol 50 article 127 2014

[20] V Bashiry ldquoInvestigation of the rare exclusive 119861119888rarr 119863

119904]]

decays in the framework of the QCD sum rulesrdquo Advances inHigh Energy Physics vol 2014 Article ID 503049 10 pages 2014

[21] M Beneke G Buchalla M Neubert and C Sachrajda ldquoQCDfactorization for 119861 rarr 120587120587 decays strong phases and CPviolation in the heavy quark limitrdquo Physical Review Letters vol83 no 10 pp 1914ndash1917 1999

[22] M Beneke G Buchalla M Neubert and C T SachrajdaldquoQCD 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


[23] Y Y Keum H N Li and A I Sanda ldquoFat penguins andimaginary penguins in perturbativeQCDrdquoPhysics Letters B vol504 no 1-2 pp 6ndash14 2001

[24] Y-Y Keum H-N Li and A I Sanda ldquoPenguin enhancementand

119861119870120587 decays in perturbative QCDrdquo Physical Review D vol63 no 5 Article ID 054008 2001

[25] C W Bauer S Fleming and M Luke ldquoSumming Sudakovlogarithms in

119861119883119904120574 in effective field theoryrdquo Physical Review D

vol 63 no 1 Article ID 014006 2000[26] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAn

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

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

[28] C W Bauer D Pirjol and I W Stewart ldquoSoft-collinearfactorization in effective field theoryrdquo Physical Review D vol65 no 5 Article ID 054022 2002

[29] P Ball G W Jones and R Zwicky ldquo119861 rarr 119881120574beyond QCD

factorizationrdquo Physical Review D vol 75 no 5 Article ID054004 2007

[30] P Ball V M Braun and A Lenz ldquoHigher-twist distributionamplitudes of the K meson in QCDrdquo Journal of High EnergyPhysics vol 5 article 4 2006

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

[32] A J Buras ldquoWeak hamiltonian CPviolation and rare decaysrdquohttparxivorgabshep-ph9806471

[33] D Fakirov and B Stech ldquoF- and D-decaysrdquo Nuclear Physics Bvol 133 no 2 pp 315ndash326 1978

[34] N Cabibbo and L Maiani ldquoTwo-body decays of charmedmesonsrdquo Physics Letters B vol 73 no 4-5 pp 418ndash422 1978Erratum in Physics Letters B vol 76 p 663 1978

[35] J D Bjorken ldquoTopics in B-physicsrdquo Nuclear Physics BmdashProceedings Supplements vol 11 pp 325ndash341 1989

[36] M J Dugan and B Grinstein ldquoQCD basis for factorization indecays of heavy mesonsrdquo Physics Letters B vol 255 no 4 pp583ndash588 1991

[37] P Jain B Pire and J P Ralston ldquoQuantum color transparencyand nuclear filteringrdquo Physics Report vol 271 no 2-3 pp 67ndash179 1996

[38] M Beneke and T Feldmann ldquoSymmetry-breaking correctionsto heavy-to-light B meson form factors at large recoilrdquo NuclearPhysics B vol 592 no 1-2 pp 3ndash34 2001

[39] M Beneke andM Neubert ldquoFlavor-singlet B-decay amplitudesinQCD factorizationrdquoNuclear Physics B vol 651 no 3 pp 225ndash248 2003

[40] M Beneke and M Neubert ldquoQCD factorization for 119861 rarr 119875119875

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

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

[42] D S Du H J Gong J F Sun D S Yang and G H ZhuldquoPhenomenological analysis of 119861 rarr 119875119875 decays with QCDfactorizationrdquo Physical Review D vol 65 no 7 Article ID074001 2002

[43] J F Sun G H Zhu and D S Du ldquoPhenomenological analysisof charmless decays 119861

119904rarr 119875119875 119875119881 with QCD factorizationrdquo

Physical Review D vol 68 no 5 Article ID 054003 2003

[44] J Sun L Chen N Wang Q Chang J Huang and Y YangldquoStudy on the Υ(1119878) rarr 119861

119888119872 weak decaysrdquo Advances in High

Energy Physics vol 2015 Article ID 691261 8 pages 2015[45] J Sun N Wang Q Chang and Y Yang ldquo119861

119888rarr 119861119875 BV decays

with the QCD factorization approachrdquoAdvances in High EnergyPhysics vol 2015 Article ID 104378 10 pages 2015

[46] H-Y Cheng and C-K Chua ldquoRevisiting charmless hadronic119861119906119889

decays in QCD factorizationrdquo Physical Review D vol 80no 11 Article ID 114008 2009

[47] H-Y Cheng and C-K Chua ldquoQCD factorization for charmlesshadronic 119861

119904decays revisitedrdquo Physical Review D vol 80 no 11

Article ID 114026 2009[48] Q Chang J Sun Y Yang and X Li ldquoA combined fit on the

annihilation corrections in 119861119906119889119904

rarr 119875119875 decays within QCDFrdquoPhysics Letters B vol 740 pp 56ndash60 2015

[49] J Sun Q Chang X Hu and Y Yang ldquoConstraints on hardspectator scattering and annihilation corrections in 119861

119906119889rarr 119875119881

decays withinQCD factorizationrdquo Physics Letters B vol 743 pp444ndash450 2015

[50] P Ball ldquoTheoretical update of pseudoscalar meson distributionamplitudes of higher twist the nonsinglet caserdquo Journal of HighEnergy Physics vol 1999 no 1 article 010 1999

[51] J Sun G Xue Y Yang G Lu andDDu ldquoStudy of119861119888rarr 119869120595120587

minus120578119888120587minus decays withQCD factorizationrdquo Physical ReviewD vol 77

no 7 Article ID 074013 9 pages 2008[52] J L Goity and W Roberts ldquoRadiative transitions in heavy

mesons in a relativistic quark modelrdquo Physical Review D vol64 no 9 Article ID 094007 2001

[53] D Ebert R N Faustov and V O Galkin ldquoRadiative M1-decaysof heavy-light mesons in the relativistic quark modelrdquo PhysicsLetters B vol 537 no 3-4 pp 241ndash248 2002

[54] S L Zhu W Y P Hwang and Z S Yang ldquo119863rarr 119863120574 and 119861 rarr

119861120574 as derived from QCD Sum rulesrdquoModern Physics Letters Avol 12 no 39 pp 3027ndash3035 1997

[55] TMAlievDADemir E Iltan andNK Pak ldquoRadiative119861 rarr119861120574 and119863

rarr 119863120574 decays in light-coneQCD sum rulesrdquo PhysicalReview D vol 54 no 1 pp 857ndash862 1996

[56] H-M Choi ldquoDecay constants and radiative decays of heavymesons in light-front quark modelrdquo Physical Review D vol 75no 7 Article ID 073016 2007

[57] P Colangelo F De Fazio and G Nardulli ldquoRadiative heavymeson transitionsrdquo Physics Letters B vol 316 no 4 pp 555ndash5601993

[58] C-Y Cheung and C-W Hwang ldquoStrong and radiative decaysof heavy mesons in a covariant modelrdquo Journal of High EnergyPhysics vol 2014 article 177 2014

[59] 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

[60] 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

[61] A Ali J G Korner G Kramer and J Willrodt ldquoNonleptonicweak decays of bottom mesonsrdquo Zeitschrift fur Physik CParticles and Fields vol 1 no 3 pp 269ndash277 1979

[62] J G Korner and G R Goldstein ldquoQuark and particle helicitiesin hadronic charmed particle decaysrdquo Physics Letters B vol 89no 1 pp 105ndash110 1979

[63] A L Kagan ldquoPolarization in 119861 rarr 119881119881 decaysrdquo Physics LettersB vol 601 no 3-4 pp 151ndash163 2004


[64] G Kramer T Mannel and W F Palmer ldquoAngular correlationsin the decays 119861 rarr 119881119881 using heavy quark symmetryrdquo Zeitschriftfur Physik C vol 55 no 3 pp 497ndash501 1992

[65] S E Csorna I Danko G Bonvicini et al ldquoMeasurements of thebranching fractions and helicity amplitudes in

119861119863120588 decaysrdquo


Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

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OpticsInternational Journal of



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Journal of



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AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


the LHC experiments may also provide a lot of experimentalinformation for 119861lowast decays For instance as analyzed in[17] even though 119861lowast

119904rarr 119897

+

119897minus decay having branching

fraction simO(10minus11) is very rare and obviously out of thescope of Belle-II it is still possible to be measured by LHCafter high-luminosity upgrade (Run-III) So with the rapiddevelopment of experiment the theoretical studies of 119861lowastweak decays are worthwhile then

Recently a few interesting theoretical studies of119861lowast decayshave been performed For instance in [18ndash20] the theoreticalestimates of the semileptonic 119861lowast

119888decays are made within

QCD sum rules framework In [17] the pure leptonic119861lowast119904rarr 119897119897

and 119861lowast119906119888rarr 119897] decays are studied and the detectability of

LHC on these decays is analyzed in detail Besides the (semi-)leptonic119861lowast decays the study of nonleptonic119861lowast weak decayswhich involve much more decay channels is also essentialIn this paper we will focus on the 119861lowast0

119889119904rarr 119863

+

119889119904119872

minus (119872 =

120587119870 120588 and 119870lowast) decays which are tree-dominated and thusprincipally have relative large branching fractions In the 119861meson system in order to evaluate the QCD correctionssome calculation approaches such as the QCD factorization(QCDF) [21 22] the perturbative QCD [23 24] and thesoft-collinear effective theory [25ndash28] are presented Thenthe nonleptonic 119861lowast decays provide another testing groundfor these approaches In our following calculation the QCDfactorization approach is employed

Our paper is organized as follows In Section 2 the basictheoretical framework and the amplitudes of 119861lowast0

119889119904rarr 119863

+

119889119904119872

minus

decays calculated through theQCDF approach are presentedSection 3 is devoted to the numerical results and discussionFinally a short summary is given

2 Theoretical Framework

The low energy effective Hamiltonian responsible for the119861

lowast0

119889119904rarr 119863

+

119889119904119872

minus decays could be written as [31 32]

Heff =119866119865

radic2

sum

119902=119889119904

119881119888119887119881

lowast

119906119902119862

1(120583)119876

1(120583) + 119862

2(120583)119876

2(120583)

+ hc

(1)

where 119866119865is the Fermi coupling and 119881

119888119887119881

lowast

119906119902is product of

Cabibbo-Kobayashi-Maskawa (CKM) matrix elements TheWilson coefficients119862

12(120583) summarize the physical contribu-

tions above scale of 120583 and are calculable with the perturbationtheoryTheir values at scales of120583 sim O(119898

1198872119898

119887 2119898

119887) in naive

dimensional regularization scheme are listed in Table 2 1198761

and 1198762are local tree four-quark operators and defined as

1198761= [119888

119894120574120583(1 minus 120574

5) 119887

119894] [119902

119895120574120583

(1 minus 1205745) 119906

119895]

1198762= [119888

119894120574120583(1 minus 120574

5) 119887

119895] [119902

119895120574120583

(1 minus 1205745) 119906

119894]

(2)

where 119894 and 119895 are color indices and the sum over repeatedindices is understood One can refer to [31 32] for detailsof this part To obtain the decay amplitudes the remaining

works are to accurately calculate the hadronic matrix ele-ments of local operators

The simplest way to deal with the hadronic matrixelements is the naive factorization (NF) scheme [33 34] basedon the color transparency mechanism [35ndash37] Within theNF scheme the hadronic matrix element is approximatedby the product of two current matrix elements which arefurther parameterized by decay constants and transition formfactors Explicitly for the 119861lowast0

119902rarr 119863

+

119902119872

minus (119902 = 119889 119904) decays thehadronic matrix element could be written as

119867119863119872= ⟨119872119863 |119876| 119861

lowast

⟩

= ⟨119872

1003816100381610038161003816119902120574

120583

(1 minus 1205745) 119906

10038161003816100381610038160⟩ ⟨119863

10038161003816100381610038161003816119888120574

120583(1 minus 120574

5) 119887

10038161003816100381610038161003816119861

lowast

⟩

(3)

For the case that119872 is a light pseudoscalar (119875) evaluating (3)we get

119867119863119875= 119894

119866119865

radic2

(minus2119898119861lowast) (120578 sdot 119901

2) 119860

119861lowast

rarr119863

0(0) 119891

119875 (4)

where 120578 is the polarization four-vector of 119861lowast meson 1199012is the

momentum of 119875 and their production could be simplified byreplacing

120578 sdot 1199012997888rarr 119901

119888

=

radic[1198982


119863+ 119898

119875)

2

] [1198982


119863minus 119898

119875)

2

]

2119898119861lowast

(5)

For the case that119872 is a light vector (119881) corresponding to thedifferent helicity amplitudes (3) could be written as

1198670

119863119881= minus119894119891

119881

1198982


2

119863

2119898119861lowast

[(119898119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

+ (119898119861lowast minus 119898

119863) 119860

119861lowast

rarr119863

2(0)] = minus119894119891

119881(119898

2


2

119863)

sdot 119860119861lowast

rarr119863

0(0)

119867plusmn

119863119881= minus119894119891

119881119898

119881[(119898

119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

∓ (119898119861lowast minus 119898

119863) 119881

119861lowast

rarr119863

(0)]

(6)

In the evaluations the definition of decay constants

⟨119875 (1199012)

10038161003816100381610038161199021120574120583

12057451199022

10038161003816100381610038160⟩ = minus119894119891

119875119901

120583

2

⟨119881 (1199012 120576

lowast

)10038161003816100381610038161199021120574120583

1199022

10038161003816100381610038160⟩ = minus119894119891

119881119898

119881120576lowast120583

(7)

the form factors (the expression of parameterization of thehadronic matrix ⟨119875|1199021015840Γ119902|119881⟩ such as (8) could be obtained



⟨119863 (1199011)

100381610038161003816100381610038161199023120574120583119887

10038161003816100381610038161003816119861lowast

(119901 120578)⟩

= minus

2119894119881 (1199022

)

119898119861

lowast + 119898119863

120576120583]120588120590120578

]119901120588

119901120590

1

⟨119863 (1199011)

1003816100381610038161003816100381611990231205741205831205745119887

10038161003816100381610038161003816119861lowast

(119901)⟩ = 2119898119861

lowast1198600(119902

2

)

120578 sdot 119902

1199022

119902120583

+ (119898119875+ 119898

119861

lowast)1198601(119902

2

) (120578120583

minus

120578 sdot 119902

1199022

119902120583

)

+ 1198602(119902

2

)

120578 sdot 119902

119898119863+ 119898

119861

lowast

[(119901 + 1199011)

120583

minus

1198982


2

119863

1199022

119902120583

]

(8)

and the relation

119860119861lowast

rarr119863

0(0) =

1

2119898119861lowast

[(119898119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

+ (119898119861lowast minus 119898

119863) 119860

119861lowast

rarr119863

2(0)]

(9)


119863+

119902119872


or119867

0plusmn




⟨119863119872|119876119894|119861

lowast


⟨1198721198631003816100381610038161003816119876

119894

1003816100381610038161003816119861lowast

⟩ = sum

119895

119865119861lowast

rarr119863

119895int119889119909T

119894119895(119909)Φ

119872(119909) (10)


rarr119863





Φ119872(119909) = 6119909 (1 minus 119909) [1 +

infin

sum

119899=1

119886119872

11989911986232

119899(2119909 minus 1)] (11)

where 119886119872119899





119904-suppressed



119902rarr 119863

+

119902119872


A (119861lowast0

119902997888rarr 119863

+

119902119872

minus

) = ⟨119872minus

119863+

119902

1003816100381610038161003816Heff

1003816100381610038161003816119861

lowast0

119902⟩ =

119866119865

radic2

sdot 119881119888119887119881

lowast

1199061199021205721⟨119872

1003816100381610038161003816119902120574

120583

(1 minus 1205745) 119906

10038161003816100381610038160⟩

sdot ⟨119863

10038161003816100381610038161003816119888120574

120583(1 minus 120574

5) 119887

10038161003816100381610038161003816119861

lowast

⟩

(12)


120583

(1 minus 1205745)119906|0⟩⟨119863|119888120574

120583(1 minus 120574

5)119887|119861

lowast


1in


1205721= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO21198811 (13)


1198811= 3 log(

1198982

119887

1205832

) + 3 log(119898

2

119888

1205832

) minus 18

+ int

1

0

119889119909119879 (119909)Φ119872(119909)

(14)


119879 (119909) =

119888119886

1 minus 119888119886

log (119888119886) minus

119888119887

1 minus 119888119887

log (119888119887)

+

119888119889

1 minus 119888119889

log (119888119889) minus

119888119888

1 minus 119888119888

log (119888119888)

minus 119903119888[

119888119886

(1 minus 119888119886)

2

log (119888119886) +

1

1 minus 119888119886

]

minus 119903minus1

119888[

119888119889

(1 minus 119888119889)

2log (119888

119889) +

1

1 minus 119888119889

] + 119891 (119888119886)

minus 119891 (119888119887) minus 119891 (119888

119888) + 119891 (119888

119889)

+ 2 log (1199032119888) [log (119888

119886) minus log (119888

119887)]

(15)


119888119898

119887 119888

119886= 119909(1minus119903

2

119888) 119888

119887= 119909(1minus119903

2

119888)

119888119888= minus119888

119886119903

2

119888 and 119888

119889= minus119888

119887119903

2





119863(lowast)




119861

lowast0


written as

B (119861lowast0

119902997888rarr 119863

119902119875)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

100381610038161003816100381610038161003816

A (119861lowast0

119902997888rarr 119863

119902119875)

100381610038161003816100381610038161003816

2

(16)

B (119861lowast0

119902997888rarr 119863

119902119881)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

sum

120582

100381610038161003816100381610038161003816

A120582(119861

lowast0

119902997888rarr 119863

119902119881)

100381610038161003816100381610038161003816

2

(17)





119902119881


119891119871perp =

1003816100381610038161003816A

0perp1003816100381610038161003816

2

1003816100381610038161003816A

0

1003816100381610038161003816

2

+

1003816100381610038161003816A

1003816100381610038161003816

2

+

1003816100381610038161003816A

perp

1003816100381610038161003816

2 (18)

whereAandA



plusmnA+)radic2



1and 119862



119902rarr 119863


lowast

119902) are


lowast

119902) ≃ Γ(119861

lowast

119902rarr 119861



119902120574


Γ(119861lowast


Γ (119861lowast0

997888rarr 1198610

120574) = (148 plusmn 20) eV

Γ (119861lowast0

119904997888rarr 119861

0

119904120574) = (68 plusmn 17) eV

(19)




+000068

minus000034[13] 119860 = 0810

+0018

minus0024[13]


119888= 167 plusmn 007GeV [13] 119898

119887= 478 plusmn 006GeV [13]


119870= 1562 plusmn 07MeV [13]

119891120588= 216 plusmn 3MeV [29] 119891



120588

1= 0 [29] 119886

120588

2= 015 plusmn 007 [29]

119886119870lowast

1= 003 plusmn 002 [29] 119886

119870lowast

2= 011 plusmn 009 [29]

119886120587

1= 0 [30] 119886

120587

2= 025 plusmn 015 [30]

119886119870

1= 006 plusmn 003 [30] 119886

119870

2= 025 plusmn 015 [30]




0(0) 119860



119872(



119860

119861lowast

119889rarr119863119889

0(0) = 071

119860

119861lowast

119889rarr119863119889

1(0) = 075

119881119861lowast

119889rarr119863119889

(0) = 076

119860

119861lowast

119904rarr119863119904

0(0) = 066

119860

119861lowast

119904rarr119863119904

1(0) = 069

119881119861lowast

119904rarr119863119904

(0) = 072

(20)



119902rarr 119863

+

119902119872

minus (119872 =





119902rarr 119863

+

119902119872

minus (119872 =




1

120583 119862LO1

119862LO2

119862NLO1

119862NLO2

1205721(NF) 120572

1(QCDF)


119898119887


2119898119887




119887120583 = 119898

1198872 120583 = 119898

119887120583 = 2119898

119887

119861

lowast0

rarr 119863+

119870minus

[10minus10

] 1205823

33+02+11+05


+02+12+06


+02+12+06


+02+11+05


119861

lowast0

119904rarr 119863

+

119904119870

minus

[10minus10

] 1205823

63+03+21+21


+03+23+06


+03+22+23


+03+21+22


119861

lowast0

rarr 119863+

120587minus

[10minus9

] 1205822

44+02+15+07


+02+16+08


+02+16+07


+02+15+07


119861

lowast0

119904rarr 119863

+

119904120587minus

[10minus9

] 1205822

85+04+28+28


+04+31+32


+04+30+30


+04+29+29


119861

lowast0

rarr 119863+

119870lowastminus

[10minus10

] 1205823

76+04+19+12


+04+21+13


+04+20+13


+04+19+12


119861

lowast0

119904rarr 119863

+

119904119870

lowastminus

[10minus9

] 1205823

15+01+04+05


+01+04+05


+01+04+05


+01+04+05


119861

lowast0

rarr 119863+

120588minus

[10minus8

] 1205822

13+01+03+02


+01+04+02


+01+03+02


+01+03+02


119861

lowast0

119904rarr 119863

+

119904120588minus

[10minus8

] 1205822

26+01+06+09


+01+07+09


+01+07+09


+01+06+09






1) as Figure 1(a)


1(120583)]119889120583|



3



119902rarr 119863

+

119902120588minus


119902rarr

119863+

119902120587minus


119902rarr 119863

+

119902119870

lowastminus


119902rarr 119863

+

119902119870

minus


119888119887119881

lowast


119902rarr

119863+

119902119870


119888119887119881

lowast

119906119889for 119861lowast0

119902rarr 119863

+

119902120587minus

(120588minus


119902rarr 119863



119902rarr 119863

119902119881 decays



119861

lowast0


lowast0

119904rarr 119863

+

119904119870

minus

) asymp

2B(119861lowast0

rarr 119863+

119870minus


0

120574)Γ(119861lowast0

119904rarr

1198610


lowast

119902) ≃

Γ(119861lowast

119902rarr 119861


expressed as

119877119904119889equiv

B (119861lowast0

119904997888rarr 119863

119904119872)

B (119861lowast0

119889997888rarr 119863

119889119872)

≃

Γ (119861lowast0

119889997888rarr 119861

0

119889120574)

Γ (119861lowast0

119904997888rarr 119861

0

119904120574)

theo≃ 2 (21)


lowast0

119889

120591119861

lowast0

119904


119902rarr 119861

119902120574)


+

119902120588minus






A0 A

minus A

+= 1

ΛQCD

119898119887

(

ΛQCD

119898119887

)

2

(22)


119902rarr 119863




Re(120572

1)

100

102

104

106

108

3 4 5 6 7 82120583 (GeV)

(a)

minus001

000

001

002

003

004

005

Im(120572

1)

3 4 5 6 7 82120583 (GeV)

(b)







119902 ℎ





+ in addition



2119898119861lowast119898

119881(119898

2


2

119863) sim ΛQCD119898119887

relative to 1198670

119863119881




119902rarr 119863





119902rarr 119863



119887 we get

119891119871(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

)

= (85 86 89 89)

119891(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

) = (12 12 9 9)

(23)


lowast

119902119875

and 119861lowast119902rarr 119863


119877A equiv

10038161003816100381610038161003816A (119861

119902997888rarr 119863

lowast

119902119875)

10038161003816100381610038161003816

10038161003816100381610038161003816A (119861

lowast

119902997888rarr 119863

119902119875)

10038161003816100381610038161003816

=

(1198982

119861minus 119898

2

119863lowast)119860


0

(1198982


2

119863) 119860


0

(24)




119877B equiv

B (119861119902997888rarr 119863

lowast

119902119875)

B (119861lowast

119902997888rarr 119863

119902119875)

= 3

120591119861

120591119861lowast

(1198982

119861minus 119898

2

119863lowast)

2

1198983

119861lowast

(1198982


2

119863)

2

1198983

119861

119860119861rarr119863

lowast

0

119860119861lowastrarr119863

0

(25)


119861120591



119902) ≃ Γ(119861

lowast

119902rarr 119861


of Γ(119861lowast119902rarr 119861


6


For the 119861119902rarr 119863

lowast

119902119881 and 119861lowast

119902rarr 119863

119902119881 decays the


119891119871 (119861119902 997888rarr 119863

lowast

119902119881) ≃ 119891

119871 (119861lowast

119902997888rarr 119863

119902119881) (26)



+

120588minus and 1198610 rarr 119863

lowast+

120588minus



119891119871(119861

lowast0

rarr 119863+

120588minus


119871(119861

0

rarr 119863lowast+

120588minus


4 Summary


rarr 119863+

119889119904119872

minus



119889119904rarr 119863

+

119889119904119872

minus


+



119889119904rarr 119863

+

119889119904119881


119871sim [80 90] (iii) Some


119902119872


lowast



lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are


Competing Interests


Acknowledgments


References







1199042(5840) rarr 119861

+ 119870




119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870










(lowast)

119904119861119904

(lowast)








119888119897V119897 with QCD sum





119904]]




































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





⟨119863 (1199011)

100381610038161003816100381610038161199023120574120583119887

10038161003816100381610038161003816119861lowast

(119901 120578)⟩

= minus

2119894119881 (1199022

)

119898119861

lowast + 119898119863

120576120583]120588120590120578

]119901120588

119901120590

1

⟨119863 (1199011)

1003816100381610038161003816100381611990231205741205831205745119887

10038161003816100381610038161003816119861lowast

(119901)⟩ = 2119898119861

lowast1198600(119902

2

)

120578 sdot 119902

1199022

119902120583

+ (119898119875+ 119898

119861

lowast)1198601(119902

2

) (120578120583

minus

120578 sdot 119902

1199022

119902120583

)

+ 1198602(119902

2

)

120578 sdot 119902

119898119863+ 119898

119861

lowast

[(119901 + 1199011)

120583

minus

1198982


2

119863

1199022

119902120583

]

(8)

and the relation

119860119861lowast

rarr119863

0(0) =

1

2119898119861lowast

[(119898119861lowast + 119898

119863) 119860

119861lowast

rarr119863

1(0)

+ (119898119861lowast minus 119898

119863) 119860

119861lowast

rarr119863

2(0)]

(9)


119863+

119902119872


or119867

0plusmn




⟨119863119872|119876119894|119861

lowast


⟨1198721198631003816100381610038161003816119876

119894

1003816100381610038161003816119861lowast

⟩ = sum

119895

119865119861lowast

rarr119863

119895int119889119909T

119894119895(119909)Φ

119872(119909) (10)


rarr119863





Φ119872(119909) = 6119909 (1 minus 119909) [1 +

infin

sum

119899=1

119886119872

11989911986232

119899(2119909 minus 1)] (11)

where 119886119872119899





119904-suppressed



119902rarr 119863

+

119902119872


A (119861lowast0

119902997888rarr 119863

+

119902119872

minus

) = ⟨119872minus

119863+

119902

1003816100381610038161003816Heff

1003816100381610038161003816119861

lowast0

119902⟩ =

119866119865

radic2

sdot 119881119888119887119881

lowast

1199061199021205721⟨119872

1003816100381610038161003816119902120574

120583

(1 minus 1205745) 119906

10038161003816100381610038160⟩

sdot ⟨119863

10038161003816100381610038161003816119888120574

120583(1 minus 120574

5) 119887

10038161003816100381610038161003816119861

lowast

⟩

(12)


120583

(1 minus 1205745)119906|0⟩⟨119863|119888120574

120583(1 minus 120574

5)119887|119861

lowast


1in


1205721= 119862

NLO1

+

1

119873119888

119862NLO2

+

120572119904

4120587

119862119865

119873119888

119862LO21198811 (13)


1198811= 3 log(

1198982

119887

1205832

) + 3 log(119898

2

119888

1205832

) minus 18

+ int

1

0

119889119909119879 (119909)Φ119872(119909)

(14)


119879 (119909) =

119888119886

1 minus 119888119886

log (119888119886) minus

119888119887

1 minus 119888119887

log (119888119887)

+

119888119889

1 minus 119888119889

log (119888119889) minus

119888119888

1 minus 119888119888

log (119888119888)

minus 119903119888[

119888119886

(1 minus 119888119886)

2

log (119888119886) +

1

1 minus 119888119886

]

minus 119903minus1

119888[

119888119889

(1 minus 119888119889)

2log (119888

119889) +

1

1 minus 119888119889

] + 119891 (119888119886)

minus 119891 (119888119887) minus 119891 (119888

119888) + 119891 (119888

119889)

+ 2 log (1199032119888) [log (119888

119886) minus log (119888

119887)]

(15)


119888119898

119887 119888

119886= 119909(1minus119903

2

119888) 119888

119887= 119909(1minus119903

2

119888)

119888119888= minus119888

119886119903

2

119888 and 119888

119889= minus119888

119887119903

2





119863(lowast)




119861

lowast0


written as

B (119861lowast0

119902997888rarr 119863

119902119875)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

100381610038161003816100381610038161003816

A (119861lowast0

119902997888rarr 119863

119902119875)

100381610038161003816100381610038161003816

2

(16)

B (119861lowast0

119902997888rarr 119863

119902119881)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

sum

120582

100381610038161003816100381610038161003816

A120582(119861

lowast0

119902997888rarr 119863

119902119881)

100381610038161003816100381610038161003816

2

(17)





119902119881


119891119871perp =

1003816100381610038161003816A

0perp1003816100381610038161003816

2

1003816100381610038161003816A

0

1003816100381610038161003816

2

+

1003816100381610038161003816A

1003816100381610038161003816

2

+

1003816100381610038161003816A

perp

1003816100381610038161003816

2 (18)

whereAandA



plusmnA+)radic2



1and 119862



119902rarr 119863


lowast

119902) are


lowast

119902) ≃ Γ(119861

lowast

119902rarr 119861



119902120574


Γ(119861lowast


Γ (119861lowast0

997888rarr 1198610

120574) = (148 plusmn 20) eV

Γ (119861lowast0

119904997888rarr 119861

0

119904120574) = (68 plusmn 17) eV

(19)




+000068

minus000034[13] 119860 = 0810

+0018

minus0024[13]


119888= 167 plusmn 007GeV [13] 119898

119887= 478 plusmn 006GeV [13]


119870= 1562 plusmn 07MeV [13]

119891120588= 216 plusmn 3MeV [29] 119891



120588

1= 0 [29] 119886

120588

2= 015 plusmn 007 [29]

119886119870lowast

1= 003 plusmn 002 [29] 119886

119870lowast

2= 011 plusmn 009 [29]

119886120587

1= 0 [30] 119886

120587

2= 025 plusmn 015 [30]

119886119870

1= 006 plusmn 003 [30] 119886

119870

2= 025 plusmn 015 [30]




0(0) 119860



119872(



119860

119861lowast

119889rarr119863119889

0(0) = 071

119860

119861lowast

119889rarr119863119889

1(0) = 075

119881119861lowast

119889rarr119863119889

(0) = 076

119860

119861lowast

119904rarr119863119904

0(0) = 066

119860

119861lowast

119904rarr119863119904

1(0) = 069

119881119861lowast

119904rarr119863119904

(0) = 072

(20)



119902rarr 119863

+

119902119872

minus (119872 =





119902rarr 119863

+

119902119872

minus (119872 =




1

120583 119862LO1

119862LO2

119862NLO1

119862NLO2

1205721(NF) 120572

1(QCDF)


119898119887


2119898119887




119887120583 = 119898

1198872 120583 = 119898

119887120583 = 2119898

119887

119861

lowast0

rarr 119863+

119870minus

[10minus10

] 1205823

33+02+11+05


+02+12+06


+02+12+06


+02+11+05


119861

lowast0

119904rarr 119863

+

119904119870

minus

[10minus10

] 1205823

63+03+21+21


+03+23+06


+03+22+23


+03+21+22


119861

lowast0

rarr 119863+

120587minus

[10minus9

] 1205822

44+02+15+07


+02+16+08


+02+16+07


+02+15+07


119861

lowast0

119904rarr 119863

+

119904120587minus

[10minus9

] 1205822

85+04+28+28


+04+31+32


+04+30+30


+04+29+29


119861

lowast0

rarr 119863+

119870lowastminus

[10minus10

] 1205823

76+04+19+12


+04+21+13


+04+20+13


+04+19+12


119861

lowast0

119904rarr 119863

+

119904119870

lowastminus

[10minus9

] 1205823

15+01+04+05


+01+04+05


+01+04+05


+01+04+05


119861

lowast0

rarr 119863+

120588minus

[10minus8

] 1205822

13+01+03+02


+01+04+02


+01+03+02


+01+03+02


119861

lowast0

119904rarr 119863

+

119904120588minus

[10minus8

] 1205822

26+01+06+09


+01+07+09


+01+07+09


+01+06+09






1) as Figure 1(a)


1(120583)]119889120583|



3



119902rarr 119863

+

119902120588minus


119902rarr

119863+

119902120587minus


119902rarr 119863

+

119902119870

lowastminus


119902rarr 119863

+

119902119870

minus


119888119887119881

lowast


119902rarr

119863+

119902119870


119888119887119881

lowast

119906119889for 119861lowast0

119902rarr 119863

+

119902120587minus

(120588minus


119902rarr 119863



119902rarr 119863

119902119881 decays



119861

lowast0


lowast0

119904rarr 119863

+

119904119870

minus

) asymp

2B(119861lowast0

rarr 119863+

119870minus


0

120574)Γ(119861lowast0

119904rarr

1198610


lowast

119902) ≃

Γ(119861lowast

119902rarr 119861


expressed as

119877119904119889equiv

B (119861lowast0

119904997888rarr 119863

119904119872)

B (119861lowast0

119889997888rarr 119863

119889119872)

≃

Γ (119861lowast0

119889997888rarr 119861

0

119889120574)

Γ (119861lowast0

119904997888rarr 119861

0

119904120574)

theo≃ 2 (21)


lowast0

119889

120591119861

lowast0

119904


119902rarr 119861

119902120574)


+

119902120588minus






A0 A

minus A

+= 1

ΛQCD

119898119887

(

ΛQCD

119898119887

)

2

(22)


119902rarr 119863




Re(120572

1)

100

102

104

106

108

3 4 5 6 7 82120583 (GeV)

(a)

minus001

000

001

002

003

004

005

Im(120572

1)

3 4 5 6 7 82120583 (GeV)

(b)







119902 ℎ





+ in addition



2119898119861lowast119898

119881(119898

2


2

119863) sim ΛQCD119898119887

relative to 1198670

119863119881




119902rarr 119863





119902rarr 119863



119887 we get

119891119871(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

)

= (85 86 89 89)

119891(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

) = (12 12 9 9)

(23)


lowast

119902119875

and 119861lowast119902rarr 119863


119877A equiv

10038161003816100381610038161003816A (119861

119902997888rarr 119863

lowast

119902119875)

10038161003816100381610038161003816

10038161003816100381610038161003816A (119861

lowast

119902997888rarr 119863

119902119875)

10038161003816100381610038161003816

=

(1198982

119861minus 119898

2

119863lowast)119860


0

(1198982


2

119863) 119860


0

(24)




119877B equiv

B (119861119902997888rarr 119863

lowast

119902119875)

B (119861lowast

119902997888rarr 119863

119902119875)

= 3

120591119861

120591119861lowast

(1198982

119861minus 119898

2

119863lowast)

2

1198983

119861lowast

(1198982


2

119863)

2

1198983

119861

119860119861rarr119863

lowast

0

119860119861lowastrarr119863

0

(25)


119861120591



119902) ≃ Γ(119861

lowast

119902rarr 119861


of Γ(119861lowast119902rarr 119861


6


For the 119861119902rarr 119863

lowast

119902119881 and 119861lowast

119902rarr 119863

119902119881 decays the


119891119871 (119861119902 997888rarr 119863

lowast

119902119881) ≃ 119891

119871 (119861lowast

119902997888rarr 119863

119902119881) (26)



+

120588minus and 1198610 rarr 119863

lowast+

120588minus



119891119871(119861

lowast0

rarr 119863+

120588minus


119871(119861

0

rarr 119863lowast+

120588minus


4 Summary


rarr 119863+

119889119904119872

minus



119889119904rarr 119863

+

119889119904119872

minus


+



119889119904rarr 119863

+

119889119904119881


119871sim [80 90] (iii) Some


119902119872


lowast



lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are


Competing Interests


Acknowledgments


References







1199042(5840) rarr 119861

+ 119870




119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870










(lowast)

119904119861119904

(lowast)








119888119897V119897 with QCD sum





119904]]




































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119863(lowast)




119861

lowast0


written as

B (119861lowast0

119902997888rarr 119863

119902119875)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

100381610038161003816100381610038161003816

A (119861lowast0

119902997888rarr 119863

119902119875)

100381610038161003816100381610038161003816

2

(16)

B (119861lowast0

119902997888rarr 119863

119902119881)

=

1

3

1

8120587

119901119888

1198982

119861

lowast0

119902


119902)

sum

120582

100381610038161003816100381610038161003816

A120582(119861

lowast0

119902997888rarr 119863

119902119881)

100381610038161003816100381610038161003816

2

(17)





119902119881


119891119871perp =

1003816100381610038161003816A

0perp1003816100381610038161003816

2

1003816100381610038161003816A

0

1003816100381610038161003816

2

+

1003816100381610038161003816A

1003816100381610038161003816

2

+

1003816100381610038161003816A

perp

1003816100381610038161003816

2 (18)

whereAandA



plusmnA+)radic2



1and 119862



119902rarr 119863


lowast

119902) are


lowast

119902) ≃ Γ(119861

lowast

119902rarr 119861



119902120574


Γ(119861lowast


Γ (119861lowast0

997888rarr 1198610

120574) = (148 plusmn 20) eV

Γ (119861lowast0

119904997888rarr 119861

0

119904120574) = (68 plusmn 17) eV

(19)




+000068

minus000034[13] 119860 = 0810

+0018

minus0024[13]


119888= 167 plusmn 007GeV [13] 119898

119887= 478 plusmn 006GeV [13]


119870= 1562 plusmn 07MeV [13]

119891120588= 216 plusmn 3MeV [29] 119891



120588

1= 0 [29] 119886

120588

2= 015 plusmn 007 [29]

119886119870lowast

1= 003 plusmn 002 [29] 119886

119870lowast

2= 011 plusmn 009 [29]

119886120587

1= 0 [30] 119886

120587

2= 025 plusmn 015 [30]

119886119870

1= 006 plusmn 003 [30] 119886

119870

2= 025 plusmn 015 [30]




0(0) 119860



119872(



119860

119861lowast

119889rarr119863119889

0(0) = 071

119860

119861lowast

119889rarr119863119889

1(0) = 075

119881119861lowast

119889rarr119863119889

(0) = 076

119860

119861lowast

119904rarr119863119904

0(0) = 066

119860

119861lowast

119904rarr119863119904

1(0) = 069

119881119861lowast

119904rarr119863119904

(0) = 072

(20)



119902rarr 119863

+

119902119872

minus (119872 =





119902rarr 119863

+

119902119872

minus (119872 =




1

120583 119862LO1

119862LO2

119862NLO1

119862NLO2

1205721(NF) 120572

1(QCDF)


119898119887


2119898119887




119887120583 = 119898

1198872 120583 = 119898

119887120583 = 2119898

119887

119861

lowast0

rarr 119863+

119870minus

[10minus10

] 1205823

33+02+11+05


+02+12+06


+02+12+06


+02+11+05


119861

lowast0

119904rarr 119863

+

119904119870

minus

[10minus10

] 1205823

63+03+21+21


+03+23+06


+03+22+23


+03+21+22


119861

lowast0

rarr 119863+

120587minus

[10minus9

] 1205822

44+02+15+07


+02+16+08


+02+16+07


+02+15+07


119861

lowast0

119904rarr 119863

+

119904120587minus

[10minus9

] 1205822

85+04+28+28


+04+31+32


+04+30+30


+04+29+29


119861

lowast0

rarr 119863+

119870lowastminus

[10minus10

] 1205823

76+04+19+12


+04+21+13


+04+20+13


+04+19+12


119861

lowast0

119904rarr 119863

+

119904119870

lowastminus

[10minus9

] 1205823

15+01+04+05


+01+04+05


+01+04+05


+01+04+05


119861

lowast0

rarr 119863+

120588minus

[10minus8

] 1205822

13+01+03+02


+01+04+02


+01+03+02


+01+03+02


119861

lowast0

119904rarr 119863

+

119904120588minus

[10minus8

] 1205822

26+01+06+09


+01+07+09


+01+07+09


+01+06+09






1) as Figure 1(a)


1(120583)]119889120583|



3



119902rarr 119863

+

119902120588minus


119902rarr

119863+

119902120587minus


119902rarr 119863

+

119902119870

lowastminus


119902rarr 119863

+

119902119870

minus


119888119887119881

lowast


119902rarr

119863+

119902119870


119888119887119881

lowast

119906119889for 119861lowast0

119902rarr 119863

+

119902120587minus

(120588minus


119902rarr 119863



119902rarr 119863

119902119881 decays



119861

lowast0


lowast0

119904rarr 119863

+

119904119870

minus

) asymp

2B(119861lowast0

rarr 119863+

119870minus


0

120574)Γ(119861lowast0

119904rarr

1198610


lowast

119902) ≃

Γ(119861lowast

119902rarr 119861


expressed as

119877119904119889equiv

B (119861lowast0

119904997888rarr 119863

119904119872)

B (119861lowast0

119889997888rarr 119863

119889119872)

≃

Γ (119861lowast0

119889997888rarr 119861

0

119889120574)

Γ (119861lowast0

119904997888rarr 119861

0

119904120574)

theo≃ 2 (21)


lowast0

119889

120591119861

lowast0

119904


119902rarr 119861

119902120574)


+

119902120588minus






A0 A

minus A

+= 1

ΛQCD

119898119887

(

ΛQCD

119898119887

)

2

(22)


119902rarr 119863




Re(120572

1)

100

102

104

106

108

3 4 5 6 7 82120583 (GeV)

(a)

minus001

000

001

002

003

004

005

Im(120572

1)

3 4 5 6 7 82120583 (GeV)

(b)







119902 ℎ





+ in addition



2119898119861lowast119898

119881(119898

2


2

119863) sim ΛQCD119898119887

relative to 1198670

119863119881




119902rarr 119863





119902rarr 119863



119887 we get

119891119871(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

)

= (85 86 89 89)

119891(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

) = (12 12 9 9)

(23)


lowast

119902119875

and 119861lowast119902rarr 119863


119877A equiv

10038161003816100381610038161003816A (119861

119902997888rarr 119863

lowast

119902119875)

10038161003816100381610038161003816

10038161003816100381610038161003816A (119861

lowast

119902997888rarr 119863

119902119875)

10038161003816100381610038161003816

=

(1198982

119861minus 119898

2

119863lowast)119860


0

(1198982


2

119863) 119860


0

(24)




119877B equiv

B (119861119902997888rarr 119863

lowast

119902119875)

B (119861lowast

119902997888rarr 119863

119902119875)

= 3

120591119861

120591119861lowast

(1198982

119861minus 119898

2

119863lowast)

2

1198983

119861lowast

(1198982


2

119863)

2

1198983

119861

119860119861rarr119863

lowast

0

119860119861lowastrarr119863

0

(25)


119861120591



119902) ≃ Γ(119861

lowast

119902rarr 119861


of Γ(119861lowast119902rarr 119861


6


For the 119861119902rarr 119863

lowast

119902119881 and 119861lowast

119902rarr 119863

119902119881 decays the


119891119871 (119861119902 997888rarr 119863

lowast

119902119881) ≃ 119891

119871 (119861lowast

119902997888rarr 119863

119902119881) (26)



+

120588minus and 1198610 rarr 119863

lowast+

120588minus



119891119871(119861

lowast0

rarr 119863+

120588minus


119871(119861

0

rarr 119863lowast+

120588minus


4 Summary


rarr 119863+

119889119904119872

minus



119889119904rarr 119863

+

119889119904119872

minus


+



119889119904rarr 119863

+

119889119904119881


119871sim [80 90] (iii) Some


119902119872


lowast



lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are


Competing Interests


Acknowledgments


References







1199042(5840) rarr 119861

+ 119870




119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870










(lowast)

119904119861119904

(lowast)








119888119897V119897 with QCD sum





119904]]




































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1

120583 119862LO1

119862LO2

119862NLO1

119862NLO2

1205721(NF) 120572

1(QCDF)


119898119887


2119898119887




119887120583 = 119898

1198872 120583 = 119898

119887120583 = 2119898

119887

119861

lowast0

rarr 119863+

119870minus

[10minus10

] 1205823

33+02+11+05


+02+12+06


+02+12+06


+02+11+05


119861

lowast0

119904rarr 119863

+

119904119870

minus

[10minus10

] 1205823

63+03+21+21


+03+23+06


+03+22+23


+03+21+22


119861

lowast0

rarr 119863+

120587minus

[10minus9

] 1205822

44+02+15+07


+02+16+08


+02+16+07


+02+15+07


119861

lowast0

119904rarr 119863

+

119904120587minus

[10minus9

] 1205822

85+04+28+28


+04+31+32


+04+30+30


+04+29+29


119861

lowast0

rarr 119863+

119870lowastminus

[10minus10

] 1205823

76+04+19+12


+04+21+13


+04+20+13


+04+19+12


119861

lowast0

119904rarr 119863

+

119904119870

lowastminus

[10minus9

] 1205823

15+01+04+05


+01+04+05


+01+04+05


+01+04+05


119861

lowast0

rarr 119863+

120588minus

[10minus8

] 1205822

13+01+03+02


+01+04+02


+01+03+02


+01+03+02


119861

lowast0

119904rarr 119863

+

119904120588minus

[10minus8

] 1205822

26+01+06+09


+01+07+09


+01+07+09


+01+06+09






1) as Figure 1(a)


1(120583)]119889120583|



3



119902rarr 119863

+

119902120588minus


119902rarr

119863+

119902120587minus


119902rarr 119863

+

119902119870

lowastminus


119902rarr 119863

+

119902119870

minus


119888119887119881

lowast


119902rarr

119863+

119902119870


119888119887119881

lowast

119906119889for 119861lowast0

119902rarr 119863

+

119902120587minus

(120588minus


119902rarr 119863



119902rarr 119863

119902119881 decays



119861

lowast0


lowast0

119904rarr 119863

+

119904119870

minus

) asymp

2B(119861lowast0

rarr 119863+

119870minus


0

120574)Γ(119861lowast0

119904rarr

1198610


lowast

119902) ≃

Γ(119861lowast

119902rarr 119861


expressed as

119877119904119889equiv

B (119861lowast0

119904997888rarr 119863

119904119872)

B (119861lowast0

119889997888rarr 119863

119889119872)

≃

Γ (119861lowast0

119889997888rarr 119861

0

119889120574)

Γ (119861lowast0

119904997888rarr 119861

0

119904120574)

theo≃ 2 (21)


lowast0

119889

120591119861

lowast0

119904


119902rarr 119861

119902120574)


+

119902120588minus






A0 A

minus A

+= 1

ΛQCD

119898119887

(

ΛQCD

119898119887

)

2

(22)


119902rarr 119863




Re(120572

1)

100

102

104

106

108

3 4 5 6 7 82120583 (GeV)

(a)

minus001

000

001

002

003

004

005

Im(120572

1)

3 4 5 6 7 82120583 (GeV)

(b)







119902 ℎ





+ in addition



2119898119861lowast119898

119881(119898

2


2

119863) sim ΛQCD119898119887

relative to 1198670

119863119881




119902rarr 119863





119902rarr 119863



119887 we get

119891119871(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

)

= (85 86 89 89)

119891(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

) = (12 12 9 9)

(23)


lowast

119902119875

and 119861lowast119902rarr 119863


119877A equiv

10038161003816100381610038161003816A (119861

119902997888rarr 119863

lowast

119902119875)

10038161003816100381610038161003816

10038161003816100381610038161003816A (119861

lowast

119902997888rarr 119863

119902119875)

10038161003816100381610038161003816

=

(1198982

119861minus 119898

2

119863lowast)119860


0

(1198982


2

119863) 119860


0

(24)




119877B equiv

B (119861119902997888rarr 119863

lowast

119902119875)

B (119861lowast

119902997888rarr 119863

119902119875)

= 3

120591119861

120591119861lowast

(1198982

119861minus 119898

2

119863lowast)

2

1198983

119861lowast

(1198982


2

119863)

2

1198983

119861

119860119861rarr119863

lowast

0

119860119861lowastrarr119863

0

(25)


119861120591



119902) ≃ Γ(119861

lowast

119902rarr 119861


of Γ(119861lowast119902rarr 119861


6


For the 119861119902rarr 119863

lowast

119902119881 and 119861lowast

119902rarr 119863

119902119881 decays the


119891119871 (119861119902 997888rarr 119863

lowast

119902119881) ≃ 119891

119871 (119861lowast

119902997888rarr 119863

119902119881) (26)



+

120588minus and 1198610 rarr 119863

lowast+

120588minus



119891119871(119861

lowast0

rarr 119863+

120588minus


119871(119861

0

rarr 119863lowast+

120588minus


4 Summary


rarr 119863+

119889119904119872

minus



119889119904rarr 119863

+

119889119904119872

minus


+



119889119904rarr 119863

+

119889119904119881


119871sim [80 90] (iii) Some


119902119872


lowast



lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are


Competing Interests


Acknowledgments


References







1199042(5840) rarr 119861

+ 119870




119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870










(lowast)

119904119861119904

(lowast)








119888119897V119897 with QCD sum





119904]]




































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Re(120572

1)

100

102

104

106

108

3 4 5 6 7 82120583 (GeV)

(a)

minus001

000

001

002

003

004

005

Im(120572

1)

3 4 5 6 7 82120583 (GeV)

(b)







119902 ℎ





+ in addition



2119898119861lowast119898

119881(119898

2


2

119863) sim ΛQCD119898119887

relative to 1198670

119863119881




119902rarr 119863





119902rarr 119863



119887 we get

119891119871(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

)

= (85 86 89 89)

119891(119863

+

119870lowastminus

119863+

119904119870

lowastminus

119863+

120588minus

119863+

119904120588minus

) = (12 12 9 9)

(23)


lowast

119902119875

and 119861lowast119902rarr 119863


119877A equiv

10038161003816100381610038161003816A (119861

119902997888rarr 119863

lowast

119902119875)

10038161003816100381610038161003816

10038161003816100381610038161003816A (119861

lowast

119902997888rarr 119863

119902119875)

10038161003816100381610038161003816

=

(1198982

119861minus 119898

2

119863lowast)119860


0

(1198982


2

119863) 119860


0

(24)




119877B equiv

B (119861119902997888rarr 119863

lowast

119902119875)

B (119861lowast

119902997888rarr 119863

119902119875)

= 3

120591119861

120591119861lowast

(1198982

119861minus 119898

2

119863lowast)

2

1198983

119861lowast

(1198982


2

119863)

2

1198983

119861

119860119861rarr119863

lowast

0

119860119861lowastrarr119863

0

(25)


119861120591



119902) ≃ Γ(119861

lowast

119902rarr 119861


of Γ(119861lowast119902rarr 119861


6


For the 119861119902rarr 119863

lowast

119902119881 and 119861lowast

119902rarr 119863

119902119881 decays the


119891119871 (119861119902 997888rarr 119863

lowast

119902119881) ≃ 119891

119871 (119861lowast

119902997888rarr 119863

119902119881) (26)



+

120588minus and 1198610 rarr 119863

lowast+

120588minus



119891119871(119861

lowast0

rarr 119863+

120588minus


119871(119861

0

rarr 119863lowast+

120588minus


4 Summary


rarr 119863+

119889119904119872

minus



119889119904rarr 119863

+

119889119904119872

minus


+



119889119904rarr 119863

+

119889119904119881


119871sim [80 90] (iii) Some


119902119872


lowast



lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are


Competing Interests


Acknowledgments


References







1199042(5840) rarr 119861

+ 119870




119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870










(lowast)

119904119861119904

(lowast)








119888119897V119897 with QCD sum





119904]]




































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119891119871(119861

lowast0

rarr 119863+

120588minus


119871(119861

0

rarr 119863lowast+

120588minus


4 Summary


rarr 119863+

119889119904119872

minus



119889119904rarr 119863

+

119889119904119872

minus


+



119889119904rarr 119863

+

119889119904119881


119871sim [80 90] (iii) Some


119902119872


lowast



lowast

119902119875)B(119861

lowast

119902rarr 119863

119902119875) ≃

3120591119861120591

119861lowast and 119891

119871(119861119902 rarr 119863lowast

119902119881) ≃ 119891

119871(119861lowast

119902rarr 119863

119902119881) are


Competing Interests


Acknowledgments


References







1199042(5840) rarr 119861

+ 119870




119878rarr 119863

119878120587+ and evidence for 1198610

119878rarr 119863

∓

119878119870










(lowast)

119904119861119904

(lowast)








119888119897V119897 with QCD sum





119904]]




































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



































119888rarr 119861119875 BV decays










119906119889rarr 119875119881























119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119861119863120588 decaysrdquo







FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Study of Decays with QCD Factorization ...downloads.hindawi.com/journals/ahep/2016/3863725.pdf · contribute to heavy-light nal states at leading power in the heavy-quark