Research Article Decays with Perturbative QCD ApproachResearch Article , Decays with Perturbative...

Research ArticleΥ(119899119878) rarr 119861

lowast

119888120587 119861

lowast

119888119870 Decays with Perturbative QCD Approach

Junfeng Sun1 Yueling Yang1 Qin Chang1 Gongru Lu1 and Jinshu Huang2

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

Correspondence should be addressed to Yueling Yang yangyuelinghtueducn

Received 23 March 2016 Accepted 17 May 2016

Academic Editor Chao-Qiang Geng

Copyright copy 2016 Junfeng Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Besides the traditional strong and electromagnetic decay modes Υ(119899119878)meson can also decay through the weak interactions withinthe standard model of elementary particle With anticipation of copious Υ(119899119878) data samples at the running LHC and comingSuperKEKB experiments the two-body nonleptonic bottom-changing Υ(119899119878) rarr 119861

lowast

119888120587 119861

lowast

119888119870 decays (119899 = 1 2 3) are investigated

with perturbative QCD approach firstly The absolute branching ratios for Υ(119899119878) rarr 119861lowast

119888120587 and 119861lowast

119888119870 decays are estimated to reach up

to about 10minus10 and 10minus11 respectively which might possibly be measured by the future experiments

1 Introduction

The upsilon Υ(119899119878) meson is the spin-triplet 119878-wave state ofbottomonium (bound state consisting of bottom quark 119887 andantibottom quark 119887) with well-established quantum numberof 119868119866119869119875119862 = 0minus1minusminus [1] The characteristic narrow decay widthsof Υ(119899119878) mesons for 119899 = 1 2 and 3 provide insight intothe study of strong interactions (see Table 1 and note thatfor simplicity Υ(119899119878) will denote Υ(1119878) Υ(2119878) and Υ(3119878)mesons in the following content if not specified definitely)The mass of Υ(119899119878) meson is below 119861 meson pair thresholdΥ(119899119878) meson decays into bottomed hadrons through strongand electromagnetic interactions are forbidden by the law ofconservation of flavor number The bottom-changing Υ(119899119878)decays can occur only via the weak interactions within thestandard model although with tiny incidence probabilityBoth constituent quarks of upsilons can decay individuallywhich provide an alternative system for investigating theweak decay of heavy-flavored hadrons In this paper wewill study the nonleptonic Υ(119899119878) rarr 119861

lowast

119888119875 (119875 = 120587 and

119870) weak decays with perturbative QCD (pQCD) approach[2ndash4]

Experimentally (1) over 108 Υ(119899119878) data samples havebeen accumulated at Belle and BaBar experiments [5] Moreand more upsilon data samples will be collected at the

running hadron collider LHC and the forthcoming 119890+119890minus col-lider SuperKEKB (the SuperKEKB has started commission-ing test run (httpwwwkekjpenNewsRoomRelease))There seems to exist a realistic possibility to explore Υ(119899119878)weak decay at future experiments (2) Signals of Υ(119899119878) rarr119861lowast

119888120587 119861

lowast

119888119870 decays should be easily distinguished with ldquocharge

tagrdquo technique due to the facts that the back-to-back finalstates with different electric charges have definitemomentumand energy in the rest frame of Υ(119899119878) meson (3) 119861lowast

119888meson

has not been observed experimentally by now 119861lowast119888meson

production via the strong interaction is suppressed due tothe simultaneous presence of two heavy quarks with differentflavors and higher order in QCD coupling constant 120572

119904

Υ(119899119878) rarr 119861lowast

119888120587 119861lowast

119888119870 decays provide a novel pattern to

study 119861lowast119888meson production The identification of a single

explicitly flavored 119861lowast119888meson could be used as an effective

selection criterion to detect upsilon weak decays Moreoverthe radiative decay of 119861lowast

119888meson provides a useful extra

signal and a powerful constraint (the investigation on theradiative decay of 119861lowast

119888meson can be found in eg [6] with

QCD sum rules) Of course any discernible evidences of ananomalous production rate of single bottomed meson fromupsilon decays might be a hint of new physics

Theoretically many attractive QCD-inspired methodshave been developed recently to describe the exclusive

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 4893649 9 pageshttpdxdoiorg10115520164893649

2 Advances in High Energy Physics

Table 1 Summary of mass decay width on(off)-peak luminosity and numbers of Υ(119899119878)

Meson Properties [1] Luminosity (fbminus1) [5] Numbers (106) [5]Mass (MeV) Width (keV) Belle BaBar Belle BaBar

Υ(1119878) 946030 plusmn 026 5402 plusmn 125 57 (18) sdot sdot sdot 102 plusmn 2 sdot sdot sdot

Υ(2119878) 1002326 plusmn 031 3198 plusmn 263 249 (17) 136 (14) 158 plusmn 4 983 plusmn 09


nonleptonic decay of heavy-flavored mesons such as thepQCD approach [2ndash4] the QCD factorization approach[7ndash9] and soft and collinear effective theory [10ndash13] andhave been applied widely to vindicate measurements on 119861meson decaysThe upsilon weak decay permits one to furtherconstrain parameters obtained from 119861 meson decay andcross comparisons provide an opportunity to test variousphenomenologicalmodelsThe upsilonweak decay possessesa unique structure due to the Cabibbo-Kobayashi-Maskawa(CKM) matrix properties which predicts that the channelswith one 119861(lowast)

119888meson are dominant Υ(119899119878) rarr 119861

lowast

119888119875 decay

belongs to the favorable 119887 rarr 119888 transition which should inprinciple have relatively large branching ratio among upsilonweak decays However there is still no theoretical studydevoted to Υ(119899119878) rarr 119861

lowast

119888119875 decay for the moment In this

paper we will present a phenomenological investigation onΥ(119899119878) rarr 119861

lowast

119888119875weak decay with the pQCD approach to supply

a ready reference for the future experimentsThis paper is organized as follows Section 2 focuses on

theoretical framework and decay amplitudes for Υ(119899119878) rarr119861lowast

119888120587 119861

lowast

119888119870 weak decays Section 3 is devoted to numerical

results and discussion The last section is a summary

2 Theoretical Framework

21 The Effective Hamiltonian Theoretically Υ(119899119878) rarr 119861lowast

119888120587

119861lowast

119888119870 weak decays are described by an effective bottom-

changing Hamiltonian based on operator product expansion[19]

Heff =119866119865

radic2

sum

119902=119889119904

119881119888119887119881

lowast

119906119902119862

1(120583)119874

1(120583) + 119862

2(120583)119874

2(120583)

+ hc

(1)

where 119866119865≃ 1166 times 10

minus5 GeVminus2 [1] is the Fermi couplingconstant the CKM factors 119881

119888119887119881

lowast

119906119889and 119881

119888119887119881

lowast

119906119904correspond

to Υ(119899119878) rarr 119861lowast

119888120587 and 119861lowast

119888119870 decays respectively with the

Wolfenstein parameterization theCKM factors are expandedas a power series in a smallWolfenstein parameter 120582 sim 02 [1]

119881119888119887119881

lowast

119906119889= 119860120582

2

minus

1

2

1198601205824

minus

1

8

1198601205826

+ O (1205827

)

119881119888119887119881

lowast

119906119904= 119860120582

3

+ O (1205827

)

(2)

The local tree operators 11987612

are defined as

1198741= [119888

120572120574120583(1 minus 120574

5) 119887

120572] [119902

120573120574120583

(1 minus 1205745) 119906

120573]

1198742= [119888

120572120574120583(1 minus 120574

5) 119887

120573] [119902

120573120574120583

(1 minus 1205745) 119906

120572]

(3)

where 120572 and 120573 are color indices and the sum over repeatedindices is understood

The scale 120583 factorizes physics contributions into short-and long-distance dynamics The Wilson coefficients 119862

119894(120583)

summarize the physics contributions at scale higher than 120583and are calculable with the renormalization group improvedperturbation theory The hadronic matrix elements (HME)where the local operators are inserted between initial andfinal hadron states embrace the physics contributions belowscale of 120583 To obtain decay amplitudes the remaining workis to calculate HME properly by separating from perturbativeand nonperturbative contributions

22 Hadronic Matrix Elements Based on Lepage-Brodskyapproach for exclusive processes [20] HME is commonlyexpressed as a convolution integral of hard scattering subam-plitudes containing perturbative contributions with univer-sal wave functions reflecting nonperturbative contributionsIn order to effectively regulate endpoint singularities andprovide a naturally dynamical cutoff on nonperturbativecontributions transverse momentum of valence quarks isretained and the Sudakov factor is introduced within thepQCD framework [2ndash4] Phenomenologically pQCDrsquos decayamplitude could be divided into three parts theWilson coef-ficients119862

119894incorporating the hard contributions above typical

scale of 119905 process-dependent rescattering subamplitudes 119879accounting for the heavy quark decay and wave functions Φof all participating hadrons which is expressed as

int119889119896119862119894(119905) 119879 (119905 119896)Φ (119896) 119890

minus119878

(4)

where 119896 is the momentum of valence quarks and 119890minus119878 is theSudakov factor

23 Kinematic Variables The light cone kinematic variablesin Υ(119899119878) rest frame are defined as follows

119901Υ= 119901

1=

1198981

radic2

(1 1 0)

119901119861lowast

119888

= 1199012= (119901

+

2 119901

minus

2 0)

1199013= (119901

minus

3 119901

+

3 0)

119901plusmn

119894=

(119864119894plusmn 119901)

radic2

119896119894= 119909

119894119901119894+ (0 0

119896119894perp)

120598

1=

1199011

1198981

minus

1198981

1199011sdot 119899

+

119899+

Advances in High Energy Physics 3

120598

2=

1199012

1198982

minus

1198982

1199012sdot 119899

minus

119899minus

120598perp

12= (0 0 1)

119899+= (1 0 0)

119899minus= (0 1 0)

119904 = 21199012sdot 119901

3

119905 = 21199011sdot 119901

2= 2119898

11198642

119906 = 21199011sdot 119901

3= 2119898

11198643

119901 =

radic[1198982

1minus (119898

2+ 119898

3)

2

] [1198982

1minus (119898

2minus 119898

3)

2

]

21198981

(5)

where119909119894and

119896119894perpare the longitudinalmomentum fraction and

transverse momentum of valence quarks respectively 120598119894and

120598perp

119894are the longitudinal and transverse polarization vectors

respectively and satisfy relations 1205982119894= minus1 and 120598

119894sdot 119901

119894= 0

the subscript 119894 on variables 119901119894 119864

119894 119898

119894 and 120598

119894corresponds

to participating hadrons namely 119894 = 1 for Υ(119899119878) meson119894 = 2 for the recoiled 119861lowast

119888meson and 119894 = 3 for the emitted

pseudoscalar meson 119899+and 119899

minusare positive and negative null

vectors respectively 119904 119905 and 119906 are the Lorentz-invariantvariables 119901 is the common momentum of final states Thenotation of momentum is displayed in Figure 2(a)

24 Wave Functions With the notation in [16 21] wavefunctions are defined as

⟨0

10038161003816100381610038161003816119887119894(119911) 119887

119895(0)

10038161003816100381610038161003816Υ (119901

1 120598

1)⟩ =

119891Υ

4

sdot int 1198891198961119890minus1198941198961sdot119911

120598

1[119898

1Φ

VΥ(119896

1) minus 1199011Φ

119905

Υ(119896

1)]

119895119894

⟨0

10038161003816100381610038161003816119887119894(119911) 119887

119895(0)

10038161003816100381610038161003816Υ (119901

1 120598

perp

1)⟩ =

119891Υ

4


120598perp

1[119898

1Φ

119881

Υ(119896

1) minus 1199011Φ

119879

Υ(119896

1)]

119895119894

⟨119861lowast

119888(119901

2 120598

2)

10038161003816100381610038161003816119888119894(119911) 119887

119895(0)

100381610038161003816100381610038160⟩ =

119891119861lowast

119888

4

sdot int

1

0

11988911989631198901198941198962sdot119911

120598

2[119898

2Φ

V119861lowast

119888

(1198962) + 1199012Φ

119905

119861lowast

119888

(1198962)]

119895119894

⟨119861lowast

119888(119901

2 120598

perp

2)

10038161003816100381610038161003816119888119894(119911) 119887

119895(0)

100381610038161003816100381610038160⟩ =

119891119861lowast

119888

4

sdot int

1

0

11988911989621198901198941198962sdot119911

120598perp

2[119898

2Φ

119881

119861lowast

119888

(1198962) + 1199012Φ

119879

119861lowast

119888

(1198962)]

119895119894

⟨119875 (1199013)

10038161003816100381610038161003816119906119894(0) 119902

119895(119911)

100381610038161003816100381610038160⟩ =

119894119891119875

4

sdot int 11988911989631198901198941198963sdot119911

1205745[1199013Φ

119886

119875(119896

3) + 120583

119875Φ

119901

119875(119896

3)

+ 120583119875(119899minus119899+ minus 1)Φ

119905

119875(119896

3)]

119895119894

(6)

where 119891Υ 119891

119861lowast

119888

and 119891119875are decay constants of Υ(119899119878) 119861lowast

119888 and

119875mesons respectivelyConsidering mass relations of 119898

Υ(119899119878)≃ 2119898

119887and 119898

119861lowast

119888

≃

119898119887+ 119898

119888 it might assume that the motion of heavy valence

quarks in Υ(119899119878) and 119861lowast

119888mesons is nearly nonrelativis-

tic The wave functions of Υ(119899119878) and 119861lowast

119888mesons could

be approximately described with nonrelativistic quantumchromodynamics (NRQCD) [22ndash24] and time-independentSchrodinger equation For an isotropic harmonic oscillatorpotential the eigenfunctions of stationary statewith quantumnumbers 119899119871 are written as [17]

1206011119878(119896) sim 119890

minus119896

2

21205732

1206012119878(119896) sim 119890

minus119896

2

21205732

(2119896

2

minus 31205732

)

1206013119878(119896) sim 119890

minus119896

2

21205732

(4119896

4

minus 20119896

2

1205732

+ 151205734

)

(7)

where parameter 120573 determines the average transversemomentum that is ⟨119899119878|1198962

perp|119899119878⟩ sim 120573

2 Employing the substi-tution ansatz [25]

119896

2

997888rarr

1

4

sum

119894

119896

2

119894perp+ 119898

2

119902119894

119909119894

(8)

where 119909119894and 119898

119902119894

are the longitudinal momentum fractionand mass of valence quark respectively then integrating out119896perpand combining with their asymptotic forms the distribu-

tion amplitudes (DAs) for Υ(119899119878) and 119861lowast119888mesons can be

written as [17]

120601V119879Υ(1119878)

(119909) = 119860119909119909 expminus119898

2

119887

81205732

1119909119909

120601119905

Υ(1119878)(119909) = 119861119905

2 expminus119898

2

119887

81205732

1119909119909

120601119881

Υ(1119878)(119909) = 119862 (1 + 119905

2

) expminus119898

2

119887

81205732

1119909119909

120601V119905119881119879Υ(2119878)

(119909) = 119863120601V119905119881119879Υ(1119878)

(119909) 1 +

1198982

119887

21205732

1119909119909

120601V119905119881119879Υ(3119878)

(119909) = 119864120601V119905119881119879Υ(1119878)

(119909)(1 minus

1198982

119887

21205732

1119909119909

)

2

+ 6

120601V119879119861lowast

119888

(119909) = 119865119909119909 expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909


120601119905

119861lowast

119888

(119909) = 1198661199052 expminus

1199091198982

119888+ 119909119898

2

119887

81205732

2119909119909

120601119881

119861lowast

119888

(119909) = 119867(1 minus 1199052

) expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909

(9)

where 119909 = 1 minus 119909 119905 = 119909 minus 119909 According to NRQCD powercounting rules [22] 120573

119894≃ 120585

119894120572119904(120585

119894) with 120585

119894= 119898

1198942 and QCD

coupling constant 120572119904 The exponential function represents 119896

perp

distribution Parameters of 119860 119861 119862 119863 119864 119865 119866 and 119867 arenormalization coefficients satisfying the conditions

int

1

0

119889119909120601119894

Υ(119899119878)(119909) = int

1

0

119889119909120601119894

119861lowast

119888

(119909) = 1

for 119894 = V 119905 119881 119879(10)

The shape lines of normalized DAs for Υ(119899119878) and 119861lowast119888

mesons are shown in Figure 1 It is clearly seen that (1) DAsforΥ(119899119878) and119861lowast

119888mesons fall quickly down to zero at endpoint

119909 119909 rarr 0 due to suppression from exponential functions (2)DAs for Υ(119899119878) meson are symmetric under the interchangeof momentum fractions 119909 harr 119909 and DAs for 119861lowast

119888meson are

basically consistent with the feature that valence quarks sharemomentum fractions according to their masses

Our study shows that only the leading twist (twist-2) DAsof the emitted light pseudoscalar meson 119875 are involved indecay amplitudes (see Appendix) The twist-2 DAs have theexpansion [16]

120601119886

119875(119909) = 6119909119909sum

119894=0

11988611989411986232

119894(119905) (11)

and are normalized as

int

1

0

120601119886

119875(119909) 119889119909 = 1 (12)

where 11986232

119894(119905) are Gegenbauer polynomials

11986232

0(119905) = 1

11986232

1(119905) = 3119905

11986232

2(119905) =

3

2

(51199052

minus 1)

(13)

and each term corresponds to a nonperturbative Gegenbauermoment 119886

119894 note that 119886

0= 1 due to the normalization

condition equation (12) 119866-parity invariance of the pion DAsrequires Gegenbauer moment 119886

119894= 0 for 119894 = 1 3 5

25 Decay Amplitudes The Feynman diagrams for Υ(119899119878) rarr119861lowast

119888120587 weak decay are shown in Figure 2 There are two types

One is factorizable emission topology where gluon attachesto quarks in the samemeson and the other is nonfactorizable

emission topology where gluon connects to quarks betweendifferent mesons

With the pQCD master formula equation (4) the ampli-tude for Υ(119899119878) rarr 119861

lowast

119888119875 decay can be expressed as [26]

A (Υ (119899119878) 997888rarr 119861lowast

119888119875) = A

119871(120598

1 120598

2) +A

119873(120598

perp

1 120598

perp

2)

+ 119894A119879120598120583V120572120573120598

120583

1120598V2119901120572

1119901

120573

2

(14)

which is conventionally written as the helicity amplitudes[26]

A0= minus119862Asum

119895

A119895

119871(120598

1 120598

2)

A= radic2119862Asum

119895

A119895

119873(120598

perp

1 120598

perp

2)

Aperp= radic2119862A1198981

119901sum

119895

A119895

119879

119862A = 119894119881119888119887119881lowast

119906119902

119866119865

radic2

119862119865

119873119888

120587119891Υ119891119861lowast

119888

119891119875

(15)

where 119862119865= 43 and the color number119873

119888= 3 the subscript 119894

on 119860119895

119894corresponds to three different helicity amplitudes that

is 119894 = 119871119873 119879 the superscript 119895 on 119860119895

119894denotes indices of

Figure 2 The explicit expressions of building blocks A119895

119894are

collected in Appendix

3 Numerical Results and Discussion

In the center-of-mass ofΥ(119899119878)meson branching ratioB119903 forΥ(119899119878) rarr 119861

lowast

119888119875 decay is defined as

B119903 =1

12120587

119901

1198982

ΥΓΥ

1003816100381610038161003816A

0

1003816100381610038161003816

2

+1003816100381610038161003816A

1003816100381610038161003816

2

+1003816100381610038161003816A

perp

1003816100381610038161003816

2

(16)

The input parameters are listed in Tables 1 and 2 If notspecified explicitly we will take their central values as thedefault inputs Our numerical results are collected in Table 3where the first uncertainty comes from scale (1 plusmn 01)119905

119894and

the expression of 119905119894is given in (A4) and (A5) the second

uncertainty is from mass of119898119887and119898

119888 the third uncertainty

is from hadronic parameters including decay constants andGegenbauer moments the fourth uncertainty is from CKMparameters The followings are some comments

(1) Branching ratio for Υ(119899119878) rarr 119861lowast

119888120587 decay is about

O(10minus10) with pQCD approach which is well withinthe measurement potential of LHC and SuperKEKBFor example experimental studies have showed thatproduction cross sections forΥ(119899119878)meson in p-p andp-Pb collisions are a few 120583119887 at the LHCb [27 28] andALICE [29 30] detectors Consequently there willbe more than 1012 Υ(119899119878) data samples per 119886119887minus1 datacollected by the LHCb and ALICE corresponding toa few hundreds of Υ(119899119878) rarr 119861

lowast

119888120587 events Branching

ratio for Υ(119899119878) rarr 119861lowast

119888119870 decay O(10minus11) is generally


060402 08 10000

1

2

3

4

120601Blowast119888(x)

120601Υ(3S)(x)120601Υ(2S)(x)

120601Υ(1S)(x)

(a)

00 02 04 06 08 100

1

2

3

4

5

6

120601tBlowast119888(x)

120601tΥ(3S)(x)120601tΥ(2S)(x)

120601tΥ(1S)(x)

(b)

000

1

2

3

4

04 06 08 1002

120601VBlowast119888(x)

120601VΥ(3S)(x)120601VΥ(2S)(x)

120601VΥ(1S)(x)

(c)

Figure 1 The normalized distribution amplitudes for Υ(119899119878) and 119861lowast

119888mesons

u k3

bb

p3

p2p1

120587

d(k3)

G Blowastc

b(k1) c(k2)

Υ

)(

(a)

bb

u

120587

G Blowastc

b c

d

Υ

(b)

bb

u

G Blowastc

b c

d

120587

Υ

(c)

G

b c

120587

bb

u

Blowastc

d

Υ

(d)

Figure 2 Feynman diagrams for Υ(119899119878) rarr 119861lowast

119888120587 decay with the pQCD approach including factorizable emission diagrams (a b) and

nonfactorizable emission diagrams (c d)

less than that for Υ(119899119878) rarr 119861lowast

119888120587 decay by one order of

magnitude due to the CKM suppression |119881lowast

119906119904119881

lowast

119906119889|2

sim

1205822

(2) As it is well known due to the large mass of119861lowast

119888 the momentum transition in Υ(119899119878) rarr 119861

lowast

119888119875

decay may be not large enough One might naturally

wonder whether the pQCD approach is applicableand whether the perturbative calculation is reliableTherefore it is necessary to check what percentage ofthe contributions comes from the perturbative regionThe contributions to branching ratio for Υ(119899119878) rarr119861lowast

119888120587 decay from different 120572

119904120587 region are shown

in Figure 3 It can be clearly seen that more than


6906

3811 014 017

2477

049 038 00806050402 0701 0803

120572s120587

0

20

40

60

80(

)

(1S)rarrBlowastc 120587Υ

(a)

550

021 013 01606050402 0701 0803

120572s120587

3988

328 088 039 0070

20

40

60

80

()


(b)

011 013

(3S)rarrBlowastc 120587

06050402 0701 0803120572s120587

0

20

40

60

80

()

50674531

247 074 033 019 005

Υ

(c)

Figure 3 The contributions to branching ratios for Υ(1119878) rarr 119861lowast

119888120587 decay (a) Υ(2119878) rarr 119861

lowast

119888120587 decay (b) and Υ(3119878) rarr 119861

lowast

119888120587 decay (c) from

different region of 120572119904120587 (horizontal axes) where the numbers over histogram denote the percentage of the corresponding contributions

Table 2 The numerical values of input parameters

TheWolfenstein parameters119860 = 0814

+0023

minus0024[1] 120582 = 022537 plusmn 000061 [1]

Mass decay constant and Gegenbauer moments119898

119887= 478 plusmn 006 GeV [1] 119891

120587= 13041 plusmn 020MeV [1]

119898119888= 167 plusmn 007 GeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119898119861lowast

119888

= 6332 plusmn 9MeV [14] 119891119861lowast

119888

= 422 plusmn 13MeV [15]c119886119870

1(1 GeV) = minus006 plusmn 003 [16] 119891

Υ(1119878)= 6764 plusmn 107MeV [17]

119886119870

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(2119878)= 4730 plusmn 237MeV [17]

119886120587

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(3119878)= 4095 plusmn 294MeV [17]

cThe decay constant 119891119861lowast

119888

cannot be extracted from the experimental data because of no measurement on 119861lowast

119888weak decay at the present time Theoretically the

value of 119891119861lowast

119888

has been estimated for example in [18] with the QCD sum rules From Table 3 of [18] one can see that the value of 119891119861lowast

119888

is model-dependentIn our calculation we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio forΥ(119899119878) rarr 119861

lowast

119888119875 decays

Table 3 Branching ratio for Υ(119899119878) rarr 119861lowast

119888119875 decays

Modes Υ(1119878) rarr 119861lowast

119888120587 Υ(2119878) rarr 119861

lowast

119888120587 Υ(3119878) rarr 119861

lowast

119888120587

1010

timesB119903 435+029+019+044+017

minus024minus041minus031minus030228

+013+026+040+009


+012+009+048+007



119888119870 Υ(2119878) rarr 119861

lowast

119888119870 Υ(3119878) rarr 119861

lowast

119888119870

1011

timesB119903 345+023+013+038+013


+011+007+036+007


+009+008+040+005



93 (97) contributions come from 120572119904120587 le 02

(03) region implying that Υ(119899119878) rarr 119861lowast

119888120587 decay

is computable with the pQCD approach As thediscussion in [2ndash4] there are many factors for thisfor example the choice of the typical scale retainingthe quark transverse moment and introducing theSudakov factor to suppress the nonperturbative con-tributions which deserve much attention and furtherinvestigation

(3) Because of the relations among masses 119898Υ(3119878)

gt

119898Υ(2119878)

gt 119898Υ(1119878)

resulting in the fact that phase spaceincreases with the radial quantum number 119899 in addi-tion to the relations among decay widths Γ

Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)

in principle there should be relationsamong branching ratios B119903(Υ(3119878) rarr 119861

lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for

the same pseudoscalar meson 119875 But the numericalresults in Table 3 are beyond such expectation WhyThe reason is that the factor of 1199011198982

Υ(119899119878)in (16) has

almost the same value for 119899 le 3 so branching ratio isproportional to factor 1198912

Υ(119899119878)Γ

Υ(119899119878)with the maximal

value 1198912

Υ(1119878)Γ

Υ(1119878)for 119899 le 3 Besides contributions

from 120572119904120587 isin [02 03] regions decrease with 119899 (see

Figure 3) which enhance the decay amplitudes(4) Besides the uncertainties listed in Table 3 other

factors such as the models of wave functions con-tributions of higher order corrections to HME andrelativistic effects deserve the dedicated study Ourresults just provide an order of magnitude estimation

4 Summary

Υ(119899119878) decay via the weak interaction as a complementaryto strong and electromagnetic decay mechanism is allowablewithin the standard model Based on the potential prospectsof Υ(119899119878) physics at high-luminosity collider experimentΥ(119899119878) decay into119861lowast

119888120587 and119861lowast

119888119870 final states is investigated with

the pQCD approach firstly It is found that (1) the dominantcontributions come from perturbative regions 120572

119904120587 le 03

which might imply that the pQCD calculation is practicableandworkable (2) there is a promiseful possibility of searchingfor Υ(119899119878) rarr 119861

lowast

119888120587 (119861lowast

119888119870) decay with branching ratio about

10minus10 (10minus11) at the future experiments

Appendix

Building Blocks for Υrarr 119861lowast

119888119875 Decays

The building blocksA119895

119894 where the superscript 119895 corresponds

to indices of Figure 2 and the subscript 119894 relates to differenthelicity amplitudes are expressed as follows

A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)

sdot 120601V119861lowast

119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909

119894is the longitudinal momentum

fraction of the valence quark 119887119894is the conjugate variable of the

transverse momentum 119896119894perp and 120572

119904(119905) is the QCD coupling at

the scale of 119905 1198861= 119862

1+ 119862

2119873

119888

The function119867119894is defined as follows [17]

119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870

0(radicminus120572119887

119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870

0) are the (modified) Bessel function

of the first and second kind respectively 120572119890(120572

119886) is the

gluon virtuality of the emission (annihilation) topologicaldiagrams the subscript of the quark virtuality 120573

119894corresponds

to the indices of Figure 2 The definition of the particlevirtuality is listed as follows [17]

120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)

The typical scale 119905119894and the Sudakov factor 119864

119894are defined

as follows where the subscript 119894 corresponds to the indices ofFigure 2

119905119886(119887)

= max(radicminus120572radicminus120573119886(119887)

1

1198871

1

1198872

) (A4)

119905119888(119889)

= max(radicminus120572radic1003816100381610038161003816120573119888(119889)

1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572

119904120587 is the quark anomalous dimension

the explicit expression of 119904(119909 119876 1119887) can be found in theappendix of [2]

Competing Interests

The authors declare that there are no competing interestsrelated to this paper

Acknowledgments

Thework is supported by the National Natural Science Foun-dation of China (Grant nos 11547014 11475055 U1332103and 11275057)

References

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

[2] H Li ldquoApplicability of perturbative QCD to 119861 rarr 119863 decaysrdquoPhysical Review D vol 52 no 7 pp 3958ndash3965 1995

[3] C V Chang and H Li ldquoThree-scale factorization theoremand effective field theory analysis of nonleptonic heavy mesondecaysrdquo Physical Review D vol 55 no 9 pp 5577ndash5580 1997

[4] T Yeh and H Li ldquoFactorization theorems effective field theoryand nonleptonic heavy meson decaysrdquo Physical Review D vol56 no 3 pp 1615ndash1631 1997

[5] E A Bevan B Golob T Mannel et al ldquoThe physics of the BfactoriesrdquoThe European Physical Journal C vol 74 article 30262014

[6] Z Wang ldquoThe radiative decays 119861119888

plusmn

rarr 119861plusmn

119888120574 with QCD sum

rulesrdquo The European Physical Journal C vol 73 no 9 article2559 2013

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

[8] M Beneke et al ldquoQCD factorization for exclusive non-leptonicB-meson decays general arguments and the case of heavy-lightfinal statesrdquo Nuclear Physics B vol 591 no 1-2 pp 313ndash4182000

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

[10] C W Bauer S Fleming D Pirjol and I W Stewart ldquoAneffective field theory for collinear and soft gluons heavy to lightdecaysrdquo Physical Review D vol 63 no 11 Article ID 114020 17pages 2001

[11] C Bauer D Pirjol and I Stewart ldquoSoft-collinear factorizationin effective field theoryrdquo Physical Review D vol 65 Article ID054022 2002

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

[13] M Beneke A P ChapovskyMDiehl andTh Feldmann ldquoSoft-collinear effective theory and heavy-to-light currents beyondleading powerrdquoNuclear Physics B vol 643 no 1ndash3 pp 431ndash4762002

[14] R Dowdall C T H Davies T C Hammant and R R HorganldquoPrecise heavy-light meson masses and hyperfine splittingsfrom lattice QCD including charm quarks in the seardquo PhysicalReview D vol 86 no 9 Article ID 094510 19 pages 2012

[15] B Colquhoun C T H Davies J Kettle et al ldquoB-meson decayconstants amore complete picture from full latticeQCDrdquo Phys-ical Review D vol 91 no 11 Article ID 114509 2015

[16] P Ball V Braun and A Lenz ldquoHigher-twist distribution ampli-tudes of the K meson in QCDrdquo Journal of High Energy Physicsvol 2006 no 5 article 004 2006

[17] Y Yang J Sun Y Guo Q Li J Huang and Q Chang ldquoStudy ofΥ(119899119878) rarr 119861

119888119875 decays with perturbative QCD approachrdquo Physics

Letters B vol 751 pp 171ndash176 2015[18] Z Wang ldquoAnalysis of the vector and axialvector Bc mesons

with QCD sum rulesrdquoThe European Physical Journal A vol 49article 131 2013

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

[20] G P Lepage and S J Brodsky ldquoExclusive processes in perturba-tive quantum chromodynamicsrdquo Physical Review D vol 22 no9 pp 2157ndash2198 1980

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

[22] G P Lepage L Magnea C Nakhleh U Magnea and KHornbostel ldquoImproved nonrelativistic QCD for heavy-quarkphysicsrdquo Physical Review D vol 46 no 9 pp 4052ndash4067 1992

[23] G T Bodwin E Braaten and G P Lepage ldquoRigorous QCDanalysis of inclusive annihilation and production of heavyquarkoniumrdquo Physical Review D vol 51 no 3 pp 1125ndash11711995

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

[25] B Xiao X Qian and B Ma ldquoThe kaon form factor in the light-cone quarkmodelrdquoThe European Physical Journal A vol 15 no4 pp 523ndash527 2002

[26] C Chen Y Keum and H Li ldquoPerturbative QCD analysis of119861120593119870

decaysrdquo Physical Review D vol 66 Article ID 0540132002

[27] R Aaij B AdevaM Adinolfi et al ldquoMeasurement ofΥ produc-tion in pp collisions at radic119904 = 276TeVrdquo The European PhysicalJournal C vol 74 article 2835 2014

[28] R Aaij B Adeva M Adinolfi et al ldquoStudy ofΥ production andcold nuclear matter effects in pPb collisions at radic119904

119873119873= 5TeVrdquo

Journal of High Energy Physics vol 2014 no 7 article 094 2014[29] G Aad T Abajyan B Abbott et al ldquoMeasurement of upsilon

production in 7 TeV pp collisions at ATLASrdquo Physical ReviewD vol 87 Article ID 052004 2013

[30] B Abelev J Adam D Adamova et al ldquoProduction of inclusiveΥ(1S) and Υ(2S) in pndashPb collisions atradic119904NN = 502TeVrdquo PhysicsLetters B vol 740 pp 105ndash117 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


Table 1 Summary of mass decay width on(off)-peak luminosity and numbers of Υ(119899119878)

Meson Properties [1] Luminosity (fbminus1) [5] Numbers (106) [5]Mass (MeV) Width (keV) Belle BaBar Belle BaBar

Υ(1119878) 946030 plusmn 026 5402 plusmn 125 57 (18) sdot sdot sdot 102 plusmn 2 sdot sdot sdot



nonleptonic decay of heavy-flavored mesons such as thepQCD approach [2ndash4] the QCD factorization approach[7ndash9] and soft and collinear effective theory [10ndash13] andhave been applied widely to vindicate measurements on 119861meson decaysThe upsilon weak decay permits one to furtherconstrain parameters obtained from 119861 meson decay andcross comparisons provide an opportunity to test variousphenomenologicalmodelsThe upsilonweak decay possessesa unique structure due to the Cabibbo-Kobayashi-Maskawa(CKM) matrix properties which predicts that the channelswith one 119861(lowast)

119888meson are dominant Υ(119899119878) rarr 119861

lowast

119888119875 decay

belongs to the favorable 119887 rarr 119888 transition which should inprinciple have relatively large branching ratio among upsilonweak decays However there is still no theoretical studydevoted to Υ(119899119878) rarr 119861

lowast

119888119875 decay for the moment In this

paper we will present a phenomenological investigation onΥ(119899119878) rarr 119861

lowast

119888119875weak decay with the pQCD approach to supply

a ready reference for the future experimentsThis paper is organized as follows Section 2 focuses on

theoretical framework and decay amplitudes for Υ(119899119878) rarr119861lowast

119888120587 119861

lowast

119888119870 weak decays Section 3 is devoted to numerical

results and discussion The last section is a summary

2 Theoretical Framework

21 The Effective Hamiltonian Theoretically Υ(119899119878) rarr 119861lowast

119888120587

119861lowast

119888119870 weak decays are described by an effective bottom-

changing Hamiltonian based on operator product expansion[19]

Heff =119866119865

radic2

sum

119902=119889119904

119881119888119887119881

lowast

119906119902119862

1(120583)119874

1(120583) + 119862

2(120583)119874

2(120583)

+ hc

(1)

where 119866119865≃ 1166 times 10

minus5 GeVminus2 [1] is the Fermi couplingconstant the CKM factors 119881

119888119887119881

lowast

119906119889and 119881

119888119887119881

lowast

119906119904correspond

to Υ(119899119878) rarr 119861lowast

119888120587 and 119861lowast

119888119870 decays respectively with the

Wolfenstein parameterization theCKM factors are expandedas a power series in a smallWolfenstein parameter 120582 sim 02 [1]

119881119888119887119881

lowast

119906119889= 119860120582

2

minus

1

2

1198601205824

minus

1

8

1198601205826

+ O (1205827

)

119881119888119887119881

lowast

119906119904= 119860120582

3

+ O (1205827

)

(2)

The local tree operators 11987612

are defined as

1198741= [119888

120572120574120583(1 minus 120574

5) 119887

120572] [119902

120573120574120583

(1 minus 1205745) 119906

120573]

1198742= [119888

120572120574120583(1 minus 120574

5) 119887

120573] [119902

120573120574120583

(1 minus 1205745) 119906

120572]

(3)

where 120572 and 120573 are color indices and the sum over repeatedindices is understood

The scale 120583 factorizes physics contributions into short-and long-distance dynamics The Wilson coefficients 119862

119894(120583)

summarize the physics contributions at scale higher than 120583and are calculable with the renormalization group improvedperturbation theory The hadronic matrix elements (HME)where the local operators are inserted between initial andfinal hadron states embrace the physics contributions belowscale of 120583 To obtain decay amplitudes the remaining workis to calculate HME properly by separating from perturbativeand nonperturbative contributions

22 Hadronic Matrix Elements Based on Lepage-Brodskyapproach for exclusive processes [20] HME is commonlyexpressed as a convolution integral of hard scattering subam-plitudes containing perturbative contributions with univer-sal wave functions reflecting nonperturbative contributionsIn order to effectively regulate endpoint singularities andprovide a naturally dynamical cutoff on nonperturbativecontributions transverse momentum of valence quarks isretained and the Sudakov factor is introduced within thepQCD framework [2ndash4] Phenomenologically pQCDrsquos decayamplitude could be divided into three parts theWilson coef-ficients119862

119894incorporating the hard contributions above typical

scale of 119905 process-dependent rescattering subamplitudes 119879accounting for the heavy quark decay and wave functions Φof all participating hadrons which is expressed as

int119889119896119862119894(119905) 119879 (119905 119896)Φ (119896) 119890

minus119878

(4)

where 119896 is the momentum of valence quarks and 119890minus119878 is theSudakov factor

23 Kinematic Variables The light cone kinematic variablesin Υ(119899119878) rest frame are defined as follows

119901Υ= 119901

1=

1198981

radic2

(1 1 0)

119901119861lowast

119888

= 1199012= (119901

+

2 119901

minus

2 0)

1199013= (119901

minus

3 119901

+

3 0)

119901plusmn

119894=

(119864119894plusmn 119901)

radic2

119896119894= 119909

119894119901119894+ (0 0

119896119894perp)

120598

1=

1199011

1198981

minus

1198981

1199011sdot 119899

+

119899+


120598

2=

1199012

1198982

minus

1198982

1199012sdot 119899

minus

119899minus

120598perp

12= (0 0 1)

119899+= (1 0 0)

119899minus= (0 1 0)

119904 = 21199012sdot 119901

3

119905 = 21199011sdot 119901

2= 2119898

11198642

119906 = 21199011sdot 119901

3= 2119898

11198643

119901 =

radic[1198982

1minus (119898

2+ 119898

3)

2

] [1198982

1minus (119898

2minus 119898

3)

2

]

21198981

(5)




120598perp



119894sdot 119901

119894= 0


119894 119898

119894 and 120598

119894corresponds







⟨0

10038161003816100381610038161003816119887119894(119911) 119887

119895(0)

10038161003816100381610038161003816Υ (119901

1 120598

1)⟩ =

119891Υ

4


120598

1[119898

1Φ

VΥ(119896

1) minus 1199011Φ

119905

Υ(119896

1)]

119895119894

⟨0

10038161003816100381610038161003816119887119894(119911) 119887

119895(0)

10038161003816100381610038161003816Υ (119901

1 120598

perp

1)⟩ =

119891Υ

4


120598perp

1[119898

1Φ

119881

Υ(119896

1) minus 1199011Φ

119879

Υ(119896

1)]

119895119894

⟨119861lowast

119888(119901

2 120598

2)

10038161003816100381610038161003816119888119894(119911) 119887

119895(0)

100381610038161003816100381610038160⟩ =

119891119861lowast

119888

4

sdot int

1

0

11988911989631198901198941198962sdot119911

120598

2[119898

2Φ

V119861lowast

119888

(1198962) + 1199012Φ

119905

119861lowast

119888

(1198962)]

119895119894

⟨119861lowast

119888(119901

2 120598

perp

2)

10038161003816100381610038161003816119888119894(119911) 119887

119895(0)

100381610038161003816100381610038160⟩ =

119891119861lowast

119888

4

sdot int

1

0

11988911989621198901198941198962sdot119911

120598perp

2[119898

2Φ

119881

119861lowast

119888

(1198962) + 1199012Φ

119879

119861lowast

119888

(1198962)]

119895119894

⟨119875 (1199013)

10038161003816100381610038161003816119906119894(0) 119902

119895(119911)

100381610038161003816100381610038160⟩ =

119894119891119875

4

sdot int 11988911989631198901198941198963sdot119911

1205745[1199013Φ

119886

119875(119896

3) + 120583

119875Φ

119901

119875(119896

3)

+ 120583119875(119899minus119899+ minus 1)Φ

119905

119875(119896

3)]

119895119894

(6)

where 119891Υ 119891

119861lowast

119888


119888 and


Υ(119899119878)≃ 2119898

119887and 119898

119861lowast

119888

≃

119898119887+ 119898





119888mesons could


1206011119878(119896) sim 119890

minus119896

2

21205732

1206012119878(119896) sim 119890

minus119896

2

21205732

(2119896

2

minus 31205732

)

1206013119878(119896) sim 119890

minus119896

2

21205732

(4119896

4

minus 20119896

2

1205732

+ 151205734

)

(7)


perp|119899119878⟩ sim 120573


119896

2

997888rarr

1

4

sum

119894

119896

2

119894perp+ 119898

2

119902119894

119909119894

(8)

where 119909119894and 119898

119902119894



written as [17]

120601V119879Υ(1119878)

(119909) = 119860119909119909 expminus119898

2

119887

81205732

1119909119909

120601119905

Υ(1119878)(119909) = 119861119905

2 expminus119898

2

119887

81205732

1119909119909

120601119881

Υ(1119878)(119909) = 119862 (1 + 119905

2

) expminus119898

2

119887

81205732

1119909119909

120601V119905119881119879Υ(2119878)

(119909) = 119863120601V119905119881119879Υ(1119878)

(119909) 1 +

1198982

119887

21205732

1119909119909

120601V119905119881119879Υ(3119878)

(119909) = 119864120601V119905119881119879Υ(1119878)

(119909)(1 minus

1198982

119887

21205732

1119909119909

)

2

+ 6

120601V119879119861lowast

119888

(119909) = 119865119909119909 expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909


120601119905

119861lowast

119888

(119909) = 1198661199052 expminus

1199091198982

119888+ 119909119898

2

119887

81205732

2119909119909

120601119881

119861lowast

119888

(119909) = 119867(1 minus 1199052

) expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909

(9)


119894≃ 120585

119894120572119904(120585

119894) with 120585

119894= 119898

1198942 and QCD


perp


int

1

0

119889119909120601119894

Υ(119899119878)(119909) = int

1

0

119889119909120601119894

119861lowast

119888

(119909) = 1

for 119894 = V 119905 119881 119879(10)





119888meson are



120601119886

119875(119909) = 6119909119909sum

119894=0

11988611989411986232

119894(119905) (11)


int

1

0

120601119886

119875(119909) 119889119909 = 1 (12)

where 11986232


11986232

0(119905) = 1

11986232

1(119905) = 3119905

11986232

2(119905) =

3

2

(51199052

minus 1)

(13)


119894 note that 119886



119894= 0 for 119894 = 1 3 5






lowast


A (Υ (119899119878) 997888rarr 119861lowast

119888119875) = A

119871(120598

1 120598

2) +A

119873(120598

perp

1 120598

perp

2)

+ 119894A119879120598120583V120572120573120598

120583

1120598V2119901120572

1119901

120573

2

(14)


A0= minus119862Asum

119895

A119895

119871(120598

1 120598

2)

A= radic2119862Asum

119895

A119895

119873(120598

perp

1 120598

perp

2)


119901sum

119895

A119895

119879

119862A = 119894119881119888119887119881lowast

119906119902

119866119865

radic2

119862119865

119873119888

120587119891Υ119891119861lowast

119888

119891119875

(15)



on 119860119895





119894are




lowast


B119903 =1

12120587

119901

1198982

ΥΓΥ

1003816100381610038161003816A

0

1003816100381610038161003816

2

+1003816100381610038161003816A

1003816100381610038161003816

2

+1003816100381610038161003816A

perp

1003816100381610038161003816

2

(16)


119894and








lowast





060402 08 10000

1

2

3

4

120601Blowast119888(x)

120601Υ(3S)(x)120601Υ(2S)(x)

120601Υ(1S)(x)

(a)

00 02 04 06 08 100

1

2

3

4

5

6

120601tBlowast119888(x)

120601tΥ(3S)(x)120601tΥ(2S)(x)

120601tΥ(1S)(x)

(b)

000

1

2

3

4

04 06 08 1002

120601VBlowast119888(x)

120601VΥ(3S)(x)120601VΥ(2S)(x)

120601VΥ(1S)(x)

(c)


119888mesons

u k3

bb

p3

p2p1

120587

d(k3)

G Blowastc

b(k1) c(k2)

Υ

)(

(a)

bb

u

120587

G Blowastc

b c

d

Υ

(b)

bb

u

G Blowastc

b c

d

120587

Υ

(c)

G

b c

120587

bb

u

Blowastc

d

Υ

(d)







119906119904119881

lowast

119906119889|2

sim

1205822



lowast

119888119875







6906

3811 014 017

2477

049 038 00806050402 0701 0803

120572s120587

0

20

40

60

80(

)


(a)

550

021 013 01606050402 0701 0803

120572s120587

3988

328 088 039 0070

20

40

60

80

()


(b)

011 013


06050402 0701 0803120572s120587

0

20

40

60

80

()

50674531

247 074 033 019 005

Υ

(c)


119888120587 decay (a) Υ(2119878) rarr 119861

lowast

119888120587 decay (b) and Υ(3119878) rarr 119861

lowast

119888120587 decay (c) from




+0023

minus0024[1] 120582 = 022537 plusmn 000061 [1]


119887= 478 plusmn 006 GeV [1] 119891

120587= 13041 plusmn 020MeV [1]

119898119888= 167 plusmn 007 GeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119898119861lowast

119888


119888

= 422 plusmn 13MeV [15]c119886119870

1(1 GeV) = minus006 plusmn 003 [16] 119891

Υ(1119878)= 6764 plusmn 107MeV [17]

119886119870

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(2119878)= 4730 plusmn 237MeV [17]

119886120587

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(3119878)= 4095 plusmn 294MeV [17]


119888




119888


119888


lowast

119888119875 decays


119888119875 decays


119888120587 Υ(2119878) rarr 119861

lowast

119888120587 Υ(3119878) rarr 119861

lowast

119888120587

1010

timesB119903 435+029+019+044+017


+013+026+040+009


+012+009+048+007



119888119870 Υ(2119878) rarr 119861

lowast

119888119870 Υ(3119878) rarr 119861

lowast

119888119870

1011

timesB119903 345+023+013+038+013


+011+007+036+007


+009+008+040+005





119888120587 decay



gt

119898Υ(2119878)

gt 119898Υ(1119878)


Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)


lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for


Υ(119899119878)in (16) has


Υ(119899119878)Γ


value 1198912

Υ(1119878)Γ





4 Summary


119888120587 and119861lowast



119904120587 le 03


lowast

119888120587 (119861lowast



Appendix


119888119875 Decays




A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)


119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120598

2=

1199012

1198982

minus

1198982

1199012sdot 119899

minus

119899minus

120598perp

12= (0 0 1)

119899+= (1 0 0)

119899minus= (0 1 0)

119904 = 21199012sdot 119901

3

119905 = 21199011sdot 119901

2= 2119898

11198642

119906 = 21199011sdot 119901

3= 2119898

11198643

119901 =

radic[1198982

1minus (119898

2+ 119898

3)

2

] [1198982

1minus (119898

2minus 119898

3)

2

]

21198981

(5)




120598perp



119894sdot 119901

119894= 0


119894 119898

119894 and 120598

119894corresponds







⟨0

10038161003816100381610038161003816119887119894(119911) 119887

119895(0)

10038161003816100381610038161003816Υ (119901

1 120598

1)⟩ =

119891Υ

4


120598

1[119898

1Φ

VΥ(119896

1) minus 1199011Φ

119905

Υ(119896

1)]

119895119894

⟨0

10038161003816100381610038161003816119887119894(119911) 119887

119895(0)

10038161003816100381610038161003816Υ (119901

1 120598

perp

1)⟩ =

119891Υ

4


120598perp

1[119898

1Φ

119881

Υ(119896

1) minus 1199011Φ

119879

Υ(119896

1)]

119895119894

⟨119861lowast

119888(119901

2 120598

2)

10038161003816100381610038161003816119888119894(119911) 119887

119895(0)

100381610038161003816100381610038160⟩ =

119891119861lowast

119888

4

sdot int

1

0

11988911989631198901198941198962sdot119911

120598

2[119898

2Φ

V119861lowast

119888

(1198962) + 1199012Φ

119905

119861lowast

119888

(1198962)]

119895119894

⟨119861lowast

119888(119901

2 120598

perp

2)

10038161003816100381610038161003816119888119894(119911) 119887

119895(0)

100381610038161003816100381610038160⟩ =

119891119861lowast

119888

4

sdot int

1

0

11988911989621198901198941198962sdot119911

120598perp

2[119898

2Φ

119881

119861lowast

119888

(1198962) + 1199012Φ

119879

119861lowast

119888

(1198962)]

119895119894

⟨119875 (1199013)

10038161003816100381610038161003816119906119894(0) 119902

119895(119911)

100381610038161003816100381610038160⟩ =

119894119891119875

4

sdot int 11988911989631198901198941198963sdot119911

1205745[1199013Φ

119886

119875(119896

3) + 120583

119875Φ

119901

119875(119896

3)

+ 120583119875(119899minus119899+ minus 1)Φ

119905

119875(119896

3)]

119895119894

(6)

where 119891Υ 119891

119861lowast

119888


119888 and


Υ(119899119878)≃ 2119898

119887and 119898

119861lowast

119888

≃

119898119887+ 119898





119888mesons could


1206011119878(119896) sim 119890

minus119896

2

21205732

1206012119878(119896) sim 119890

minus119896

2

21205732

(2119896

2

minus 31205732

)

1206013119878(119896) sim 119890

minus119896

2

21205732

(4119896

4

minus 20119896

2

1205732

+ 151205734

)

(7)


perp|119899119878⟩ sim 120573


119896

2

997888rarr

1

4

sum

119894

119896

2

119894perp+ 119898

2

119902119894

119909119894

(8)

where 119909119894and 119898

119902119894



written as [17]

120601V119879Υ(1119878)

(119909) = 119860119909119909 expminus119898

2

119887

81205732

1119909119909

120601119905

Υ(1119878)(119909) = 119861119905

2 expminus119898

2

119887

81205732

1119909119909

120601119881

Υ(1119878)(119909) = 119862 (1 + 119905

2

) expminus119898

2

119887

81205732

1119909119909

120601V119905119881119879Υ(2119878)

(119909) = 119863120601V119905119881119879Υ(1119878)

(119909) 1 +

1198982

119887

21205732

1119909119909

120601V119905119881119879Υ(3119878)

(119909) = 119864120601V119905119881119879Υ(1119878)

(119909)(1 minus

1198982

119887

21205732

1119909119909

)

2

+ 6

120601V119879119861lowast

119888

(119909) = 119865119909119909 expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909


120601119905

119861lowast

119888

(119909) = 1198661199052 expminus

1199091198982

119888+ 119909119898

2

119887

81205732

2119909119909

120601119881

119861lowast

119888

(119909) = 119867(1 minus 1199052

) expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909

(9)


119894≃ 120585

119894120572119904(120585

119894) with 120585

119894= 119898

1198942 and QCD


perp


int

1

0

119889119909120601119894

Υ(119899119878)(119909) = int

1

0

119889119909120601119894

119861lowast

119888

(119909) = 1

for 119894 = V 119905 119881 119879(10)





119888meson are



120601119886

119875(119909) = 6119909119909sum

119894=0

11988611989411986232

119894(119905) (11)


int

1

0

120601119886

119875(119909) 119889119909 = 1 (12)

where 11986232


11986232

0(119905) = 1

11986232

1(119905) = 3119905

11986232

2(119905) =

3

2

(51199052

minus 1)

(13)


119894 note that 119886



119894= 0 for 119894 = 1 3 5






lowast


A (Υ (119899119878) 997888rarr 119861lowast

119888119875) = A

119871(120598

1 120598

2) +A

119873(120598

perp

1 120598

perp

2)

+ 119894A119879120598120583V120572120573120598

120583

1120598V2119901120572

1119901

120573

2

(14)


A0= minus119862Asum

119895

A119895

119871(120598

1 120598

2)

A= radic2119862Asum

119895

A119895

119873(120598

perp

1 120598

perp

2)


119901sum

119895

A119895

119879

119862A = 119894119881119888119887119881lowast

119906119902

119866119865

radic2

119862119865

119873119888

120587119891Υ119891119861lowast

119888

119891119875

(15)



on 119860119895





119894are




lowast


B119903 =1

12120587

119901

1198982

ΥΓΥ

1003816100381610038161003816A

0

1003816100381610038161003816

2

+1003816100381610038161003816A

1003816100381610038161003816

2

+1003816100381610038161003816A

perp

1003816100381610038161003816

2

(16)


119894and








lowast





060402 08 10000

1

2

3

4

120601Blowast119888(x)

120601Υ(3S)(x)120601Υ(2S)(x)

120601Υ(1S)(x)

(a)

00 02 04 06 08 100

1

2

3

4

5

6

120601tBlowast119888(x)

120601tΥ(3S)(x)120601tΥ(2S)(x)

120601tΥ(1S)(x)

(b)

000

1

2

3

4

04 06 08 1002

120601VBlowast119888(x)

120601VΥ(3S)(x)120601VΥ(2S)(x)

120601VΥ(1S)(x)

(c)


119888mesons

u k3

bb

p3

p2p1

120587

d(k3)

G Blowastc

b(k1) c(k2)

Υ

)(

(a)

bb

u

120587

G Blowastc

b c

d

Υ

(b)

bb

u

G Blowastc

b c

d

120587

Υ

(c)

G

b c

120587

bb

u

Blowastc

d

Υ

(d)







119906119904119881

lowast

119906119889|2

sim

1205822



lowast

119888119875







6906

3811 014 017

2477

049 038 00806050402 0701 0803

120572s120587

0

20

40

60

80(

)


(a)

550

021 013 01606050402 0701 0803

120572s120587

3988

328 088 039 0070

20

40

60

80

()


(b)

011 013


06050402 0701 0803120572s120587

0

20

40

60

80

()

50674531

247 074 033 019 005

Υ

(c)


119888120587 decay (a) Υ(2119878) rarr 119861

lowast

119888120587 decay (b) and Υ(3119878) rarr 119861

lowast

119888120587 decay (c) from




+0023

minus0024[1] 120582 = 022537 plusmn 000061 [1]


119887= 478 plusmn 006 GeV [1] 119891

120587= 13041 plusmn 020MeV [1]

119898119888= 167 plusmn 007 GeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119898119861lowast

119888


119888

= 422 plusmn 13MeV [15]c119886119870

1(1 GeV) = minus006 plusmn 003 [16] 119891

Υ(1119878)= 6764 plusmn 107MeV [17]

119886119870

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(2119878)= 4730 plusmn 237MeV [17]

119886120587

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(3119878)= 4095 plusmn 294MeV [17]


119888




119888


119888


lowast

119888119875 decays


119888119875 decays


119888120587 Υ(2119878) rarr 119861

lowast

119888120587 Υ(3119878) rarr 119861

lowast

119888120587

1010

timesB119903 435+029+019+044+017


+013+026+040+009


+012+009+048+007



119888119870 Υ(2119878) rarr 119861

lowast

119888119870 Υ(3119878) rarr 119861

lowast

119888119870

1011

timesB119903 345+023+013+038+013


+011+007+036+007


+009+008+040+005





119888120587 decay



gt

119898Υ(2119878)

gt 119898Υ(1119878)


Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)


lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for


Υ(119899119878)in (16) has


Υ(119899119878)Γ


value 1198912

Υ(1119878)Γ





4 Summary


119888120587 and119861lowast



119904120587 le 03


lowast

119888120587 (119861lowast



Appendix


119888119875 Decays




A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)


119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120601119905

119861lowast

119888

(119909) = 1198661199052 expminus

1199091198982

119888+ 119909119898

2

119887

81205732

2119909119909

120601119881

119861lowast

119888

(119909) = 119867(1 minus 1199052

) expminus119909119898

2

119888+ 119909119898

2

119887

81205732

2119909119909

(9)


119894≃ 120585

119894120572119904(120585

119894) with 120585

119894= 119898

1198942 and QCD


perp


int

1

0

119889119909120601119894

Υ(119899119878)(119909) = int

1

0

119889119909120601119894

119861lowast

119888

(119909) = 1

for 119894 = V 119905 119881 119879(10)





119888meson are



120601119886

119875(119909) = 6119909119909sum

119894=0

11988611989411986232

119894(119905) (11)


int

1

0

120601119886

119875(119909) 119889119909 = 1 (12)

where 11986232


11986232

0(119905) = 1

11986232

1(119905) = 3119905

11986232

2(119905) =

3

2

(51199052

minus 1)

(13)


119894 note that 119886



119894= 0 for 119894 = 1 3 5






lowast


A (Υ (119899119878) 997888rarr 119861lowast

119888119875) = A

119871(120598

1 120598

2) +A

119873(120598

perp

1 120598

perp

2)

+ 119894A119879120598120583V120572120573120598

120583

1120598V2119901120572

1119901

120573

2

(14)


A0= minus119862Asum

119895

A119895

119871(120598

1 120598

2)

A= radic2119862Asum

119895

A119895

119873(120598

perp

1 120598

perp

2)


119901sum

119895

A119895

119879

119862A = 119894119881119888119887119881lowast

119906119902

119866119865

radic2

119862119865

119873119888

120587119891Υ119891119861lowast

119888

119891119875

(15)



on 119860119895





119894are




lowast


B119903 =1

12120587

119901

1198982

ΥΓΥ

1003816100381610038161003816A

0

1003816100381610038161003816

2

+1003816100381610038161003816A

1003816100381610038161003816

2

+1003816100381610038161003816A

perp

1003816100381610038161003816

2

(16)


119894and








lowast





060402 08 10000

1

2

3

4

120601Blowast119888(x)

120601Υ(3S)(x)120601Υ(2S)(x)

120601Υ(1S)(x)

(a)

00 02 04 06 08 100

1

2

3

4

5

6

120601tBlowast119888(x)

120601tΥ(3S)(x)120601tΥ(2S)(x)

120601tΥ(1S)(x)

(b)

000

1

2

3

4

04 06 08 1002

120601VBlowast119888(x)

120601VΥ(3S)(x)120601VΥ(2S)(x)

120601VΥ(1S)(x)

(c)


119888mesons

u k3

bb

p3

p2p1

120587

d(k3)

G Blowastc

b(k1) c(k2)

Υ

)(

(a)

bb

u

120587

G Blowastc

b c

d

Υ

(b)

bb

u

G Blowastc

b c

d

120587

Υ

(c)

G

b c

120587

bb

u

Blowastc

d

Υ

(d)







119906119904119881

lowast

119906119889|2

sim

1205822



lowast

119888119875







6906

3811 014 017

2477

049 038 00806050402 0701 0803

120572s120587

0

20

40

60

80(

)


(a)

550

021 013 01606050402 0701 0803

120572s120587

3988

328 088 039 0070

20

40

60

80

()


(b)

011 013


06050402 0701 0803120572s120587

0

20

40

60

80

()

50674531

247 074 033 019 005

Υ

(c)


119888120587 decay (a) Υ(2119878) rarr 119861

lowast

119888120587 decay (b) and Υ(3119878) rarr 119861

lowast

119888120587 decay (c) from




+0023

minus0024[1] 120582 = 022537 plusmn 000061 [1]


119887= 478 plusmn 006 GeV [1] 119891

120587= 13041 plusmn 020MeV [1]

119898119888= 167 plusmn 007 GeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119898119861lowast

119888


119888

= 422 plusmn 13MeV [15]c119886119870

1(1 GeV) = minus006 plusmn 003 [16] 119891

Υ(1119878)= 6764 plusmn 107MeV [17]

119886119870

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(2119878)= 4730 plusmn 237MeV [17]

119886120587

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(3119878)= 4095 plusmn 294MeV [17]


119888




119888


119888


lowast

119888119875 decays


119888119875 decays


119888120587 Υ(2119878) rarr 119861

lowast

119888120587 Υ(3119878) rarr 119861

lowast

119888120587

1010

timesB119903 435+029+019+044+017


+013+026+040+009


+012+009+048+007



119888119870 Υ(2119878) rarr 119861

lowast

119888119870 Υ(3119878) rarr 119861

lowast

119888119870

1011

timesB119903 345+023+013+038+013


+011+007+036+007


+009+008+040+005





119888120587 decay



gt

119898Υ(2119878)

gt 119898Υ(1119878)


Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)


lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for


Υ(119899119878)in (16) has


Υ(119899119878)Γ


value 1198912

Υ(1119878)Γ





4 Summary


119888120587 and119861lowast



119904120587 le 03


lowast

119888120587 (119861lowast



Appendix


119888119875 Decays




A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)


119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




060402 08 10000

1

2

3

4

120601Blowast119888(x)

120601Υ(3S)(x)120601Υ(2S)(x)

120601Υ(1S)(x)

(a)

00 02 04 06 08 100

1

2

3

4

5

6

120601tBlowast119888(x)

120601tΥ(3S)(x)120601tΥ(2S)(x)

120601tΥ(1S)(x)

(b)

000

1

2

3

4

04 06 08 1002

120601VBlowast119888(x)

120601VΥ(3S)(x)120601VΥ(2S)(x)

120601VΥ(1S)(x)

(c)


119888mesons

u k3

bb

p3

p2p1

120587

d(k3)

G Blowastc

b(k1) c(k2)

Υ

)(

(a)

bb

u

120587

G Blowastc

b c

d

Υ

(b)

bb

u

G Blowastc

b c

d

120587

Υ

(c)

G

b c

120587

bb

u

Blowastc

d

Υ

(d)







119906119904119881

lowast

119906119889|2

sim

1205822



lowast

119888119875







6906

3811 014 017

2477

049 038 00806050402 0701 0803

120572s120587

0

20

40

60

80(

)


(a)

550

021 013 01606050402 0701 0803

120572s120587

3988

328 088 039 0070

20

40

60

80

()


(b)

011 013


06050402 0701 0803120572s120587

0

20

40

60

80

()

50674531

247 074 033 019 005

Υ

(c)


119888120587 decay (a) Υ(2119878) rarr 119861

lowast

119888120587 decay (b) and Υ(3119878) rarr 119861

lowast

119888120587 decay (c) from




+0023

minus0024[1] 120582 = 022537 plusmn 000061 [1]


119887= 478 plusmn 006 GeV [1] 119891

120587= 13041 plusmn 020MeV [1]

119898119888= 167 plusmn 007 GeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119898119861lowast

119888


119888

= 422 plusmn 13MeV [15]c119886119870

1(1 GeV) = minus006 plusmn 003 [16] 119891

Υ(1119878)= 6764 plusmn 107MeV [17]

119886119870

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(2119878)= 4730 plusmn 237MeV [17]

119886120587

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(3119878)= 4095 plusmn 294MeV [17]


119888




119888


119888


lowast

119888119875 decays


119888119875 decays


119888120587 Υ(2119878) rarr 119861

lowast

119888120587 Υ(3119878) rarr 119861

lowast

119888120587

1010

timesB119903 435+029+019+044+017


+013+026+040+009


+012+009+048+007



119888119870 Υ(2119878) rarr 119861

lowast

119888119870 Υ(3119878) rarr 119861

lowast

119888119870

1011

timesB119903 345+023+013+038+013


+011+007+036+007


+009+008+040+005





119888120587 decay



gt

119898Υ(2119878)

gt 119898Υ(1119878)


Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)


lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for


Υ(119899119878)in (16) has


Υ(119899119878)Γ


value 1198912

Υ(1119878)Γ





4 Summary


119888120587 and119861lowast



119904120587 le 03


lowast

119888120587 (119861lowast



Appendix


119888119875 Decays




A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)


119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




6906

3811 014 017

2477

049 038 00806050402 0701 0803

120572s120587

0

20

40

60

80(

)


(a)

550

021 013 01606050402 0701 0803

120572s120587

3988

328 088 039 0070

20

40

60

80

()


(b)

011 013


06050402 0701 0803120572s120587

0

20

40

60

80

()

50674531

247 074 033 019 005

Υ

(c)


119888120587 decay (a) Υ(2119878) rarr 119861

lowast

119888120587 decay (b) and Υ(3119878) rarr 119861

lowast

119888120587 decay (c) from




+0023

minus0024[1] 120582 = 022537 plusmn 000061 [1]


119887= 478 plusmn 006 GeV [1] 119891

120587= 13041 plusmn 020MeV [1]

119898119888= 167 plusmn 007 GeV [1] 119891

119870= 1562 plusmn 07MeV [1]

119898119861lowast

119888


119888

= 422 plusmn 13MeV [15]c119886119870

1(1 GeV) = minus006 plusmn 003 [16] 119891

Υ(1119878)= 6764 plusmn 107MeV [17]

119886119870

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(2119878)= 4730 plusmn 237MeV [17]

119886120587

2(1 GeV) = 025 plusmn 015 [16] 119891

Υ(3119878)= 4095 plusmn 294MeV [17]


119888




119888


119888


lowast

119888119875 decays


119888119875 decays


119888120587 Υ(2119878) rarr 119861

lowast

119888120587 Υ(3119878) rarr 119861

lowast

119888120587

1010

timesB119903 435+029+019+044+017


+013+026+040+009


+012+009+048+007



119888119870 Υ(2119878) rarr 119861

lowast

119888119870 Υ(3119878) rarr 119861

lowast

119888119870

1011

timesB119903 345+023+013+038+013


+011+007+036+007


+009+008+040+005





119888120587 decay



gt

119898Υ(2119878)

gt 119898Υ(1119878)


Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)


lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for


Υ(119899119878)in (16) has


Υ(119899119878)Γ


value 1198912

Υ(1119878)Γ





4 Summary


119888120587 and119861lowast



119904120587 le 03


lowast

119888120587 (119861lowast



Appendix


119888119875 Decays




A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)


119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119888120587 decay



gt

119898Υ(2119878)

gt 119898Υ(1119878)


Υ(3119878)lt

ΓΥ(2119878)

lt ΓΥ(1119878)


lowast

119888119875) gt

B119903(Υ(2119878) rarr 119861lowast

119888119875) gt B119903(Υ(1119878) rarr 119861

lowast

119888119875) for


Υ(119899119878)in (16) has


Υ(119899119878)Γ


value 1198912

Υ(1119878)Γ





4 Summary


119888120587 and119861lowast



119904120587 le 03


lowast

119888120587 (119861lowast



Appendix


119888119875 Decays




A119886

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

VΥ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)


119888

(1199092) [119898

2

1119904 minus (4119898

2

11199012

+ 1198982

2119906) 119909

2]

+ 120601119905

119861lowast

119888

(1199092)119898

2119898

119887119906

A119886

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886)119898

1120601

119881

119861lowast

119888

(1199092)119898

2(119906 minus 119904119909

2)

+ 120601119879

119861lowast

119888

(1199092)119898

119887119904

A119886

119879= minus2119898

1int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119886 119887

1 119887

2) 119864

119886119887(119905

119886) 120601

119881

Υ(119909

1)

sdot 120572119904(119905

119886) 119886

1(119905

119886) 120601

119879

119861lowast

119888

(1199092)119898

119887+ 120601

119881

119861lowast

119888

(1199092)119898

21199092

A119887

119871= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

V119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

VΥ(119909

1) [119898

2

2119906 minus 119898

2

1(119904 minus 4119901

2

) 1199091]

+ 120601119905

Υ(119909

1)119898

1119898

119888119904

A119887

119873= int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887)119898

2120601

119881

Υ(119909

1)119898

1(119904 minus 119906119909

1)

+ 120601119879

Υ(119909

1)119898

119888119906

A119887

119879= minus2119898

2int

1

0

1198891199091int

1

0

1198891199092int

infin

0

1198871119889119887

1

sdot int

infin

0

1198872119889119887

2119867

119886119887(120572

119890 120573

119887 119887

2 119887

1) 119864

119886119887(119905

119887) 120601

119881

119861lowast

119888

(1199092)

sdot 120572119904(119905

119887) 119886

1(119905

119887) 120601

119881

Υ(119909

1)119898

11199091+ 120601

119879

Υ(119909

1)119898

119888

A119888

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119888) 119862

2(119905

119888) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199091minus 119909

3)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119906119909

1minus 119904119909

2minus 2119898

2

31199093)

sdot 120575 (1198871minus 119887

2)


A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




A119888

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

1minus 119909

3) + 119898

2

2119906 (119909

3minus 119909

2)

A119888

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119888 119887

2 119887

3) 119864

119888119889(119905

119888) 119862

2(119905

119888) 120572

119904(119905

119888)

sdot 120575 (1198871minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3) 119898

2

1(119909

3minus 119909

1)

+ 1198982

2(119909

2minus 119909

3)

A119889

119871=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 120601

119886

119875(119909

3)

sdot 120572119904(119905

119889) 119862

2(119905

119889) 120601

VΥ(119909

1) 120601

V119861lowast

119888

(1199092) 4119898

2

11199012

(1199093minus 119909

2)

+ 120601119905

Υ(119909

1) 120601

119905

119861lowast

119888

(1199092)119898

1119898

2(119904119909

2+ 2119898

2

31199093minus 119906119909

1)

sdot 120575 (1198871minus 119887

2)

A119889

119873=

1

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1119904 (119909

3minus 119909

1) + 119898

2

2119906 (119909

2minus 119909

3)

A119889

119879=

2

119873119888

int

1

0

1198891199091int

1

0

1198891199092int

1

0

1198891199093int

infin

0

1198891198871int

infin

0

1198872119889119887

2

sdot int

infin

0

1198873119889119887

3119867

119888119889(120572

119890 120573

119889 119887

2 119887

3) 119864

119888119889(119905

119889) 119862

2(119905

119889)

sdot 120572119904(119905

119889) 120575 (119887

1minus 119887

2) 120601

119879

Υ(119909

1) 120601

119879

119861lowast

119888

(1199092) 120601

119886

119875(119909

3)

sdot 1198982

1(119909

1minus 119909

3) minus 119898

2

2(119909

2minus 119909

3)

(A1)

where 119909119894= 1 minus 119909

119894 variable 119909





the scale of 119905 1198861= 119862

1+ 119862

2119873

119888


119867119886119887(120572

119890 120573 119887

119894 119887

119895) = 119870


119894)

sdot 120579 (119887119894minus 119887

119895)119870


119894) 119868


119895)

+ (119887119894larrrarr 119887

119895)

119867119888119889(120572

119890 120573 119887

2 119887

3) = 120579 (minus120573)119870


3)

+

120587

2

120579 (120573) [1198941198690(radic120573119887

3) minus 119884

0(radic120573119887

3)]

sdot 120579 (1198872minus 119887

3)119870


2) 119868


3)

+ (1198872larrrarr 119887

3)

(A2)

where 1198690and119884

0(119868

0and119870



119886) is the


119894corresponds


120572 = 1199092

1119898

2

1+ 119909

2

2119898

2

2minus 119909

11199092119905

120573119886= 119898

2

1minus 119898

2

119887+ 119909

2

2119898

2

2minus 119909

2119905

120573119887= 119898

2

2minus 119898

2

119888+ 119909

2

1119898

2

1minus 119909

1119905

120573119888= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

120573119889= 119909

2

1119898

2

1+ 119909

2

2119898

2

2+ 119909

2

3119898

2

3minus 119909

11199092119905 minus 119909

11199093119906

+ 11990921199093119904

(A3)


119894are defined


119905119886(119887)


1

1198871

1

1198872

) (A4)

119905119888(119889)


1003816100381610038161003816

1

1198872

1

1198873

) (A5)

119864119886119887(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) (A6)

119864119888119889(119905) = exp minus119878

Υ(119905) minus 119878

119861lowast

119888

(119905) minus 119878119875(119905) (A7)

119878Υ(119905) = 119904 (119909

1 119901

+

1

1

1198871

) + 2int

119905

11198871

119889120583

120583

120574119902 (A8)

119878119861lowast

119888

(119905) = 119904 (1199092 119901

+

2

1

1198872

) + 2int

119905

11198872

119889120583

120583

120574119902 (A9)


119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119878120587119870(119905) = 119904 (119909

3 119901

+

3

1

1198873

) + 119904 (1199093 119901

+

3

1

1198873

)

+ 2int

119905

11198873

119889120583

120583

120574119902

(A10)

where 120574119902= minus120572



Competing Interests


Acknowledgments


References







plusmn

rarr 119861plusmn




























119873119873= 5TeVrdquo









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



Research Article Decays with Perturbative QCD ApproachResearch Article , Decays with Perturbative...

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Transcript of Research Article Decays with Perturbative QCD ApproachResearch Article , Decays with Perturbative...