research article semileptonic decays of to light tensor...

Research ArticleSemileptonic Decays of 119861(119861

119904) to Light Tensor Mesons

Reza Khosravi and S Sadeghi

Department of Physics Isfahan University of Technology Isfahan 84156-83111 Iran

Correspondence should be addressed to Reza Khosravi rezakhosravicciutacir

Received 23 June 2016 Revised 21 August 2016 Accepted 6 September 2016

Academic Editor Juan Jose Sanz-Cillero

Copyright copy 2016 R Khosravi and S Sadeghi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

The semileptonic 119861119904(119861) rarr 119870

lowast

2(119886

2 119891

2)ℓ] ℓ = 120591 120583 transitions are investigated in the frame work of the three-point QCD sum rules

Considering the quark condensate contributions the relevant form factors of these transitions are estimated The branching ratiosof these channel modes are also calculated at different values of the continuum thresholds of the tensor mesons and compared withthe obtained data for other approaches

1 Introduction

Investigation of the119861mesondecays into tensormesons is use-ful in several aspects such as CP asymmetries isospin sym-metries and the longitudinal and transverse polarizationfractions A large isospin violation has already been experi-mentally detected in 119861 rarr 120596119870

lowast

2(1430) mode [1] Also the

decay mode 119861 rarr 120601119870lowast

2(1430) is mainly dominated by the

longitudinal polarization [2 3] in contrast with 119861 rarr 120601119870lowast

where the transverse polarization is comparable with thelongitudinal one [4] Therefore nonleptonic and semilep-tonic decays of 119861 meson can play an important role in thestudy of the particle physics

In the flavor 119878119880(3) symmetry the light 119901-wave tensormesons with 119869119875 = 2+ containing isovector mesons 119886

2(1320)

isodoublet states 119870lowast

2(1430) and two isosinglet mesons

1198912(1270) and 1198911015840

2(1525) are building the ground state nonet

which has been experimentally established [5 6] The quarkcontent 119902119902 for the isovector and isodoublet tensor resonancesis obviousThe isoscalar tensor states119891

2(1270) and1198911015840

2(1525)

have mixing wave functions where mixing angle should besmall [7 8] Therefore 119891

2(1270) is primarily a (119906119906 + 119889119889)radic2

state while 1198911015840

2(1525) is dominantly 119904119904 [9]

As a nonperturbative method the QCD sum rules isa well established technique in the hadron physics sinceit is based on the fundamental QCD Lagrangian [10] Thesemileptonic decays of 119861 to the light mesons involving 120587

119870(119870lowast

119870lowast

0) and119886

1have been studied via the three-pointQCD

sum rules (3PSR) for instance 119861 rarr 120587ℓ] [11] 119861 rarr 119870ℓ+

ℓminus

119861 rarr 119870lowast

ℓ+

ℓminus[12ndash14] 119861

119904rarr 119870

lowast

0ℓ] [15] 119861

119904rarr (119870

lowast

0 119891

0)ℓ

+

ℓminus

[16] and 119861 rarr 1198861ℓ+

ℓminus [17] The determination of the form

factor value1198791(0) = 035 plusmn 005 relevant for the119861 rarr 119870

lowast

120574 and119861 rarr 119870

lowast

ℓ+

ℓminus [14 18] decays allowed prediction of the ratio

Γ(119861 rarr 119870lowast

120574)Γ(119887 rarr 119904120574) = 017 plusmn 005 which agrees with theexperimental measurements [19ndash21] The obtained results ofthe decay 119861 rarr 120587ℓ] [11] and simulations on the lattice [22ndash24]are in a reasonable agreement

In this work we investigate 119861(119861119904) rarr 119870

lowast

2(119886

2 119891

2)ℓ] decays

within the 3PSR method For analysis of these decays theform factors and their branching ratio values are calculatedSo far the form factors of the semileptonic decays 119861(119861

119904) rarr

119870lowast

2(119886

2 119891

2)ℓ] have been studied via different approaches such

as the LCSR [25] the perturbative QCD (PQCD) [5] thelarge energy effective theory (LEET) [26ndash28] and the ISGWIImodel [29] A comparison of our results for the form factorvalues in 1199022 = 0 and branching ratio data with predictionsobtained from other approaches especially the LCSR is alsomade

The plan of the present paper is as follows the 3PSRapproach for calculation of the relevant form factors of119861(119861

119904) rarr 119870

lowast

2(119886

2 119891

2)ℓ] decays is presented in Section 2 In

the final section the value of the form factors in 1199022 = 0 andthe branching ratio of the considered decays are reported Fora better analysis the form factors and differential branching

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 2352041 10 pageshttpdxdoiorg10115520162352041

2 Advances in High Energy Physics

Bs(p)

b u

vl

0

y x

s

Klowast2 (p

998400)

Figure 1 Schematic picture of the spectator mechanism for the119861119904rarr 119870

lowast

2ℓ] decay

ratios related to these semileptonic decays are plotted withrespect to the momentum transfer squared 1199022

2 Theoretical Framework

In order to study 119861(119861119904) rarr 119870

lowast

2(119886

2 119891

2)ℓ] decays we focus

on the exclusive decay 119861119904rarr 119870

lowast

2via the 3PSR The 119861

119904rarr

119870lowast

2ℓ] decay governed by the tree level 119887 rarr 119906 transition

(see Figure 1) In the framework of the 3PSR the first step isappropriate definition of correlation function In this workthe correlation function should be taken as

Π120572120573120583

(1199012

11990110158402

1199022

) = 119894∬119890119894(1199011015840119909minus119901119910)

⟨0 |

T 119895119870lowast

2

120572120573(119909) 119895

120583(0) 119895

119861119904(119910) | 0⟩ 119889

4

1199091198894

119910

(1)

where 119901 and 1199011015840 are four-momentum of the initial and finalmesons respectively 1199022 is the squared momentum transferand T is the time ordering operator 119895

120583= 119906120574

120583(1 minus 120574

5)119887 is

the transition current 119895119861119904 and 119895119870lowast

2

120572120573are also the interpolating

currents of 119861119904and the tensor meson 119870lowast

2 respectively With

considering all quantum numbers their interpolating cur-rents can be written as follows [33]

119895119861119904(119910) = 119887 (119910) 120574

5119904 (119910)

119895119870lowast

2

120572120573(119909)

=

119894

2

[119904 (119909) 120574120572

harr

119863120573(119909) 119906 (119909) + 119904 (119909) 120574

120573

harr

119863120572(119909) 119906 (119909)]

(2)

whereharr

119863120583(119909) is the four-derivative vector with respect to 119909

acting at the same time on the left and right It is given as

harr

119863120583(119909) =

1

2

[120583(119909) minus

120583(119909)]

120583(119909) =

120597120583(119909) minus 119894

119892

2

120582119886A119886

120583(119909)

120583(119909) =

120597120583(119909) + 119894

119892

2

120582119886A119886

120583(119909)

(3)

where 120582119886 and A119886

120583(119909) are the Gell-Mann matrices and the

external gluon fields respectively It should be noted thatthe second current in (2) interpolates a spin 2 particle formassless quarks In the general case to describe a spin 2 stateone has to use a current such that the trace of 119895119870

lowast

2

120572120573vanishes

The correlation function is a complex function of whichthe imaginary part comprises the computations of the phe-nomenology and real part comprises the computations of thetheoretical part (QCD) By linking these two parts via thedispersion relation the physical quantities are calculated Inthe phenomenological part of the QCD sum rules approachthe correlation function in (1) is calculated by inserting twocomplete sets of intermediate states with the same quantumnumbers as 119861

119904and 119870lowast

2 After performing four integrals over

119909 and 119910 it will be

Π120572120573120583

= minus

⟨0 | 119895119870lowast

2

120572120573| 119870

lowast

2(119901

1015840

)⟩ ⟨119870lowast

2(119901

1015840

) | 119895120583| 119861

119904(119901)⟩ ⟨119861

119904(119901) | 119895

119861119904| 0⟩

(1199012minus 119898

2

119861119904

) (11990110158402minus 119898

2

119870lowast

2

)

+ higher states (4)

In (4) the vacuum to initial and final meson state matrixelements is defined as

⟨0 | 119895119870lowast

2

120572120573| 119870

lowast

2(119901

1015840

120576)⟩ = 119891119870lowast

2

1198982

119870lowast

2

120576120572120573

⟨0 | 119895119861119904| 119861

119904(119901)⟩ = minus119894

119891119861119904

1198982

119861119904

(119898119887+ 119898

119904)

(5)

where 119891119870lowast

2

and 119891119861119904

are the leptonic decay constants of 119870lowast

2

and 119861119904mesons respectively 120576

120572120573is polarization tensor of

119870lowast

2 The transition current gives a contribution to these

matrix elements and it can be parametrized in terms ofsome form factors using the Lorentz invariance and parity

conservation The correspondence between a vector mesonand a tensor meson allows us to get these parametrizationsin a comparative way (for more information see [5]) Theparametrization of 119861 rarr 119879 form factors is analogous to the119861 rarr 119881 case except that 120576 is replaced by 120576

119879 as follows

119888119881⟨119870

lowast

2(119901

1015840

120576) | 119906120574120583(1 minus 120574

5) 119887 | 119861

119904(119901)⟩

= minus119894120576lowast

119879120583(119898

119861119904

+ 119898119870lowast

2

)1198601(119902

2

)

+ 119894 (119901 + 1199011015840

)120583

(120576lowast

119879sdot 119902)

1198602(119902

2

)

119898119861119904

+ 119898119870lowast

2

Advances in High Energy Physics 3

+ 119894119902120583(120576

lowast

119879sdot 119902)

2119898119870lowast

2

1199022

(1198603(119902

2

) minus 1198600(119902

2

))

+ 120598120583]120588120590120576

lowast]119879119901120588

1199011015840120590

2119881 (1199022

)

119898119861119904

+ 119898119870lowast

2

(6)

with 1198603(119902

2

)

=

119898119861119904

+ 119898119870lowast

2

2119898119870lowast

2

1198601(119902

2

) minus

119898119861119904

minus 119898119870lowast

2

2119898119870lowast

2

1198602(119902

2

)

1198600(0) = 119860

3(0)

(7)

where 119902 = 119901 minus 1199011015840 119875 = 119901 + 119901

1015840 and 120576lowast119879120583= (119901

120582119898

119861119904

)120576120583120582 The

factor 119888119881accounts for the flavor content of particles 119888

119881= radic2

for 1198862 119891

2and 119888

119881= 1 for 119870lowast

2[34] Inserting (5) and (6) in (4)

and performing summation over the polarization tensor as

120576120583]120576120572120573 =

1

2

119879120583120572119879]120573 +

1

2

119879120583120573119879]120572 minus

1

3

119879120583]119879120572120573 (8)

where 119879120583] = minus119892120583] +119901

1015840

1205831199011015840

]1198982

119870lowast

2

the final representation of thephysical side is obtained as

Π120572120573120583

=

119891119861119904

119898119861119904

(119898119887+ 119898

119904)

sdot

119891119870lowast

2

1198982

119870lowast

2

(1199012minus 119898

2

119861119904

) (11990110158402minus 119898

2

119870lowast

2

)

1198811015840

(1199022

) 119894120598120573120583120588120590

119901120572119901120588

1199011015840120590

+ 1198601015840

0(119902

2

) 1199011205721199011205731199011015840

120583+ 119860

1015840

1(119902

2

) 119892120573120583119901120572

+ 1198601015840

2(119902

2

) 119901120572119901120573119901120583 + higher states

(9)

For simplicity in calculations the following redefinitions havebeen used in (9)

1198811015840

(1199022

) =

119881 (1199022

)

119898119861119904

+ 119898119870lowast

2

1198601015840

0(119902

2

) = minus

119898119870lowast

2

(1198603(119902

2

) minus 1198600(119902

2

))

1199022

1198601015840

1(119902

2

) = minus

(119898119861119904

+ 119898119870lowast

2

)

2

1198601(119902

2

)

1198601015840

2(119902

2

) =

1198602(119902

2

)

2 (119898119861119904

+ 119898119870lowast

2

)

(10)

Now the QCD part of the correlation function is calculatedby expanding it in terms of the OPE at large negative value of1199022 as follows

Π120572120573120583

= 119862(0)

120572120573120583119868 + 119862

(3)

120572120573120583⟨0 | ΨΨ | 0⟩

+ 119862(4)

120572120573120583⟨0 | 119866

119886

120588]119866120588]119886| 0⟩

+ 119862(5)

120572120573120583⟨0 | Ψ120590

120588]119879119886

119866120588]119886Ψ | 0⟩ + sdot sdot sdot

(11)

where 119862(119894)

120572120573120583are the Wilson coefficients 119868 is the unit operator

Ψ is the local fermion field operator and 119866119886

120588] is the gluonstrength tensor In (11) the first term is contribution of theperturbative and the other terms are contribution of thenonperturbative part

To compute the portion of the perturbative part(Figure 1) using the Feynman rules for the bare loop weobtain

119862(0)

120572120573120583= minus

119894

4

∬119890119894(1199011015840119909minus119901119910)

Tr [119878119904(119909 minus 119910) 120574

120572

harr

119863120573(119909)

sdot 119878119906(minus119909) 120574

120583(1 minus 120574

5) 119878

119887(119910) 120574

5] + Tr [120572

larrrarr 120573] 1198894

119909 1198894

119910

(12)

taking the partial derivative with respect to 119909 of the quark freepropagators and performing the Fourier transformation andusing the Cutkosky rules that is 1(1199012 minus 1198982

) rarr minus2119894120587120575(1199012

minus

1198982

) imaginary part of 119862(0)

120572120573120583is calculated as

Im [119862(0)

120572120573120583] =

1

8120587

int120575 (1198962

minus 1198982

119904) 120575 ((119901 + 119896)

2

minus 1198982

119887)

sdot 120575 ((1199011015840

+ 119896)

2

minus 1198982

119906) (2119896 + 119901

1015840

)120573

Tr [(119896 + 119898119904)

sdot 120574120572(119901

1015840

+ 119896 + 119898119906) 120574

120583(1 minus 120574

5) (119901 + 119896 + 119898119887

) 1205745] + 120572

larrrarr 120573 1198894

119896

(13)

where 119896 is four-momentum of the spectator quark 119904 To solvethe integral in (13) wewill have to deal with the integrals suchas 119868

0 119868

120572 119868

120572120573 and 119868

120572120573120583with respect to 119896 For example 119868

120572120573120583can

be as

119868120572120573120583

(119904 1199041015840

1199022

) = int [119896120572119896120573119896120583] 120575 (119896

2

minus 1198982

119904)

sdot 120575 ((119901 + 119896)2

minus 1198982

119887) 120575 ((119901

1015840

+ 119896)

2

minus 1198982

119906) 119889

4

119896

(14)

where 119904 = 1199012 and 1199041015840 = 11990110158402 1198680 119868

120572 119868

120572120573 and 119868

120572120573120583can be taken as

an appropriate tensor structure as follows

1198680=

1

4radic120582 (119904 1199041015840 119902

2)

119868120572= 119861

1[119901

120572] + 119861

2[119901

1015840

120572]


b u

s s

(a)

b u

s s

(b)

Figure 2 The diagrams of the effective contributions of the condensate terms

119868120572120573= 119863

1[119892

120572120573] + 119863

2[119901

120572119901120573] + 119863

3[119901

1205721199011015840

120573+ 119901

1015840

120572119901120573]

+ 1198634[119901

1015840

1205721199011015840

120573]

119868120572120573120583

= 1198641[119892

120572120573119901120583+ 119892

120572120583119901120573+ 119892

120573120583119901120572]

+ 1198642[119892

1205721205731199011015840

120583+ 119892

1205721205831199011015840

120573+ 119892

1205731205831199011015840

120572]

+ 1198643[119901

120572119901120573119901120583]

+ 1198644[119901

1015840

120572119901120573119901120583+ 119901

1205721199011015840

120573119901120583+ 119901

1205721199011205731199011015840

120583]

+ 1198645[119901

1015840

1205721199011015840

120573119901120583+ 119901

1015840

1205721199011205731199011015840

120583+ 119901

1205721199011015840

1205731199011015840

120583]

+ 1198646[119901

1015840

1205721199011015840

1205731199011015840

120583]

(15)

The quantities 120582(119904 1199041015840 1199022) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4)

and 119864119903(119903 = 1 6) are indicated in Appendix Using the

relations in (15) Im[119862(0)

120572120573120583] can be calculated for each structure

corresponding to (9) as follows

Im [119862(0)

120572120573120583] = 120588

119881(119894120598

120573120583120588120590119901120572119901120588

1199011015840120590

) + 1205880(119901

1205721199011205731199011015840

120583)

+ 1205881(119892

120573120583119901120572) + 120588

2(119901

120572119901120573119901120583)

(16)

where the spectral densities 120588119894(119894 = 119881 0 1 2) are found as

120588119881(119904 119904

1015840

1199022

) = 241198611

radic120582 [1198611(119898

119904minus 119898

119887)

+ 1198612(119898

119904minus 119898

119906) + 119898

1199041198680]

1205880(119904 119904

1015840

1199022

) = 12 [1198632(119898

119904minus 119898

119887) + 119863

3(119898

119904minus 119898

119906)

+ 21198611119898

119904minus 2119864

4(119898

119887minus 119898

119904)]

1205881(119904 119904

1015840

1199022

) = 31198611[2119898

2

119904(119898

119887+ 119898

119906minus 119898

119904)

minus 119898119904(2119898

119887119898

119906+ 119906) + Δ (119898

119904minus 119898

119906) + Δ

1015840

(119898119904minus 119898

119887)]

+ 61198631(119898

119904minus 119898

119906) minus 24119864

1(119898

119887minus 119898

119904)

1205882(119904 119904

1015840

1199022

) = 24 [1198632119898

119904+ 119864

3(119898

119904minus 119898

119887)]

(17)

Using the dispersion relation the perturbative part contribu-tion of the correlation function can be calculated as follows

119862(0)

119894= ∬

120588119894(119904 119904

1015840

1199022

)

(119904 minus 1199012) (119904

1015840minus 119901

10158402)

1198891199041015840

119889119904 (18)

For calculation of the nonperturbative contributions(condensate terms) we consider the condensate terms ofdimensions 3 4 and 5 related to the contributions ofthe quark-quark gluon-gluon and quark-gluon condensaterespectivelyThey aremore important than the other terms inthe OPE In the 3PSR when the light quark is a spectator thegluon-gluon condensate contributions can be easily ignored[35] On the other hand the quark condensate contributionsof the light quark which is a nonspectator are zero afterapplying the double Borel transformation with respect toboth variables 1199012 and 11990110158402 because only one variable appearsin the denominator Therefore only two important diagramsof dimensions 3 4 and 5 remain from the nonperturbativepart contributions The diagrams of these contributions cor-responding to 119862(3)

120572120573120583and 119862(5)

120572120573120583are depicted in Figure 2 After

some calculations the nonperturbative part of the correlationfunction is obtained as follows

119862(3)

119881+ 119862

(5)

119881= minus

2120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

0+ 119862

(5)

0= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

1+ 119862

(5)

1=

120581

(1199012minus 119898

2

119887) (119901

10158402minus 119898

2

119906)

+

120581 [(119898119887+ 119898

119906)2

minus 1199022

]

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

2+ 119862

(5)

2= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

(19)

where 120581 = ((1198982

119904minus119898

2

02)16)⟨0 | 119904119904 | 0⟩1198982

0= (08plusmn02)GeV2

[35] and ⟨0 | 119904119904 | 0⟩ = (08 plusmn 02)⟨0 | 119906119906 | 0⟩ ⟨0 | 119906119906 | 0⟩ =⟨0 | 119889119889 | 0⟩ = minus(0240 plusmn 0010GeV)3 that is we choose the


Table 1 The values of the meson masses [30] and decay constants [31 32] in GeV

Meson 119861119904

119861 119870lowast

21198862

1198912

Mass 5366 5279 1425 1318 1275

Decay constant 0222 plusmn 0012 0186 plusmn 0014 0118 plusmn 0005 0107 plusmn 0006 0102 plusmn 0006

value of the condensates at a fixed renormalization scale ofabout 1 GeV [36 37]

The next step is to apply the Borel transformations withrespect to 1199012(1199012 rarr 119872

2

1) and 11990110158402(11990110158402 rarr 119872

2

2) on the phe-

nomenological as well as the perturbative and nonperturbat-ive parts of the correlation functions and equate these tworepresentations of the correlations The following sum rulesfor the form factors are derived

1198811015840

(1199022

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119881(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119881+ 119862

(5)

119881] 119889119904

1015840

119889119904

1198601015840

119899(119902

2

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119899(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119899+ 119862

(5)

119899] 119889119904

1015840

119889119904

(20)

where 119899 = 0 2 and 1199040and 1199041015840

0are the continuum thresholds

in the initial and final channels respectively The lower limitin the integration over 119904 is 119904

119871= 119898

2

119887+ (119898

2

119887(119898

2

119887minus 119902

2

))1199041015840 Also

transformation is defined as follows

[

1

(1199012minus 119898

2

119887)119898

(11990110158402minus 119898

2

119906)119899]

=

(minus1)119898+119899

Γ (119899) Γ (119898)

119890minus1198982

1198871198722

1119890minus1198982

1199061198722

2

(1198722

1)119898minus1

(1198722

2)119899minus1

(21)

where1198722

1and1198722

2are Borel mass parameters

In (20) to subtract the contributions of the higher statesand the continuum the quark-hadron duality assumption isalso used that is it is assumed that

120588higherstates

(119904 1199041015840

) = 120588 (119904 1199041015840

) 120579 (119904 minus 1199040) 120579 (119904

1015840

minus 1199041015840

0) (22)

We would like to provide the same results for 119861 rarr 1198862ℓ]

and 119861 rarr 1198912ℓ] decays With a little bit of change in the above

expressions such as 119904 harr 119889(119906) and 119898119870lowast

2

harr 1198981198862

(1198981198912

) we caneasily find similar results in (20) for the form factors of thenew transitions

3 Numerical Analysis

In this section we numerically analyze the sum rules for theform factors 119881(1199022) 119860

0(119902

2

) 1198601(119902

2

) and 1198602(119902

2

) as well asbranching ratio values of the transitions 119861(119861

119904) rarr 119879 where

119879 can be one of the tensor mesons 119870lowast

2 119886

2 or 119891

2 The values

of the meson masses and leptonic decay constants are chosenas presented in Table 1 Also119898

119887= 4820GeV119898

119904= 0150GeV

[38]119898120591= 1776GeV and119898

120583= 0105GeV [30]

From the 3PSR it is clear that the form factors alsocontain the continuum thresholds 119904

0and 1199041015840

0and the Borel

parameters 1198722

1and 119872

2

2as the main input These are not

physical quantities hence the form factors should be inde-pendent of these parameters The continuum thresholds1199040and 119904

1015840

0 are not completely arbitrary but these are in

correlation with the energy of the first exiting state withthe same quantum numbers as the considered interpolatingcurrents The value of the continuum threshold 119904

119861(119861119904)

0=

35GeV2 [39] is calculated from the 3PSR The values of thecontinuum threshold 1199041015840

0for the tensor mesons 119870lowast

2 119886

2 and

1198912are taken to be 119904119870

lowast

2

0= 313GeV2 1199041198862

0= 270GeV2 and

1199041198912

0= 253GeV2 respectively [9] In this work the variations

of 1199041198790(119879 = 119886

2 119870

lowast

2 119891

2) are considered to be plusmn02 In these

regions the dependence of the form factors on the continuumthreshold values is very small For instance we have shownthe variations of the form factor 119860119861

119904rarr119870lowast

2

1(119902

2

) for differentvalues of 119904119870

lowast

2

0in Figure 3 As can be seen these plots are very

close to each otherWe search for the intervals of the Borel parameters so

that our results are almost insensitive to their variations Onemore condition for the intervals of these parameters is the factthat the aforementioned intervals must suppress the higherstates continuum and contributions of the highest-orderoperators In other words the sum rules for the form factorsmust converge As a result we get 8GeV2

le 1198722

1le 12GeV2

and 4GeV2

le 1198722

2le 8GeV2 To show how the form factors

depend on the Borel mass parameters as examples we depictthe variations of the form factors 119881 119860

0 119860

1 and 119860

2for

119861119904rarr 119870

lowast

2ℓ] at 1199022 = 0 with respect to the variations of the

1198722

1and1198722

2parameters in their working regions in Figure 4

From these figures it is revealed that the form factors weaklydepend on these parameters in their working regions

In the Borel transform scheme the ratio of the nonpertur-bative to perturbative part of the form factor119881119861


2 is about119881

nonminusper(0)119881

per(0) ≃ 13 This value confirms that the

higher order corrections are small constituting a few percentand can easily be neglected Our calculation shows that thesame suppression is observed for all other form factors

The sum rules for the form factors are truncated at about0 le 119902

2

le 11GeV2The dependence of the form factors1198811198600

6 Advances in High Energy PhysicsA

B119904rarr

Klowast 2

1

sKlowast

20 = 313

sKlowast

20 = 323

sKlowast

20 = 333

minus02

minus01

0

01

02

03

2 4 6 80 10

q2

Figure 3The form factor of119860119861119904rarr119870lowast

2

1on 1199022 for different values of 119904119870

lowast

2

0

1198601 and 119860

2on 1199022 for 119861 rarr 119879 transitions is shown in Figure 5

However it is necessary to obtain the behavior of the formfactors with respect to 1199022 in the full physical region 0 le 1199022 le(119898

119861(119861119904)minus 119898

119879)2 in order to calculate the decay width of the

119861 rarr 119879 transitions So to extend our results we look for aparametrization of the form factors in such a way that in theregion 0 le 1199022 le (119898

119861(119861119904)minus119898

119879)2 this parametrization coincides

with the sum rules predictions Our numerical calculationsshow that the sufficient parametrization of the form factorswith respect to 1199022 is as follows

119891 (1199022

) =

119891 (0)

1 minus 119886 (1199022119898

2

119861(119861119904)) + 119887 (119902

2119898

2

119861(119861119904))

2 (23)

The values of the parameters 119891(0) 119886 and 119887 for the transitionform factors of 119861 rarr 119879 are given in Table 2

In Table 3 our results for the form factors of 119861 rarr 119879ℓ]decays in 1199022 = 0 are comparedwith those of other approachessuch as the LCSR the PQCD the LEET and the ISGW IImodel Our results are in good agreement with those of theLCSR PQCD and LEET in all cases

At the end of this section we would like to present thedifferential decay widths of the process under considerationUsing the parametrization of these transitions in terms ofthe form factors the differential decay width for 119861 rarr 119879ℓ]transition is obtained as

119889Γ (119861 rarr 119879ℓ])119889119902

2

=

1003816100381610038161003816119866119865119881119906119887

1003816100381610038161003816

2

radic120582 (1198982

119861 119898

2

119879 119902

2)

2561198983

11986112058731199022

(1 minus

1198982

ℓ

1199022)

2

sdot (119883119871+ 119883

++ 119883

minus)

(24)

Table 2 Parameter values appearing in the fit functions of the 119861 rarr119879ℓ] decays

Form factor 119891(0) 119886 119887

119881119861119904rarr119870

lowast

2 013 219 083119860

119861119904rarr119870lowast

2

1010 136 009

119881119861rarr1198862 013 210 075

119860119861rarr1198862

1011 145 023

119881119861rarr1198912 012 201 060

119860119861rarr1198912

1010 140 016

119860119861119904rarr119870

lowast

2

0023 377 421

119860119861119904rarr119870

lowast

2

2005 021 minus299

119860119861rarr1198862

0026 371 403

119860119861rarr1198862

2009 063 046

119860119861rarr1198912

0024 370 402

119860119861rarr1198912

2009 046 029

where119898ℓrepresents themess of the charged leptonTheother

parameters are defined as

119883119871=

1

9

120582

1198982

119879119898

2

119861

[(21199022

+ 1198982

ℓ) ℎ

2

0(119902

2

) + 31205821198982

ℓ119860

2

0(119902

2

)]

119883plusmn=

21199022

3

(21199022

+ 1198982

ℓ)

120582

81198982

119879119898

2

119861

[(119898119861+ 119898

119879) 119860

1(119902

2

)

∓

radic120582

119898119861+ 119898

119879

119881(1199022

)]

2

ℎ0(119902

2

) =

1

2119898119879

[(1198982

119861minus 119898

2

119879minus 119902

2

) (119898119861+ 119898

119879) 119860

1(119902

2

)

minus

120582

119898119861+ 119898

119879

1198602(119902

2

)]

(25)

Integrating (24) over 1199022 in the whole physical region andusing 119881

119906119887= (389 plusmn 044) times 10

minus3 [30] the branchingratios of the 119861 rarr 119879ℓ] are obtained The differentialbranching ratios of the 119861 rarr 119879ℓ] decays on 1199022 are shownin Figure 6 The branching ratio values of these decays arealso obtained as presented in Table 4 Furthermore this tablecontains the results estimated via the PQCD Considering theuncertainties our estimations for the branching ratio valuesof the119861 rarr 119879ℓ] decays are in consistent agreement with thoseof the PQCD

It should be noted that the uncertainties in the branchingratio values come from the form factors the CKMparameterand the meson and leptonmasses which are about 30 of thecentral values

In summary we considered 119861119904(119861) rarr 119870

lowast

2(119886

2 119891

2)ℓ]

channels and computed the relevant form factors consideringthe contribution of the quark condensate corrections Ourresults are in good agreement with those of the LCSRPQCD and LEET in all cases We also evaluated the totaldecays widths and the branching ratios of these decaysOur branching ratio values of these decays are in consistentagreement with those of the PQCD


Table 3 Comparison of the form factor values of 119861 rarr 119879ℓ] decays in 1199022 = 0 in different approaches

Form factor This work LCSR [25] PQCD [5] LEET [26ndash28] ISGW II [29]119881

119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005

minus004018 plusmn 003 032

119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)

Figure 4 The form factor of 119861119904rarr 119870

lowast

2on1198722

1and1198722

2

Table 4 Comparison of the branching ratio values of 119861 rarr 119879ℓ]decays with those of the PQCD (in units of 10minus4)

This work PQCD [5]Br (119861 rarr 119886

2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix

In this appendix the explicit expressions of the coefficients120582(119904 119904

1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)

Figure 5 The SR predictions for the form factors of the 119861(119861119904) rarr 119879ℓ] transitions on 1199022

a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05

Figure 6 The differential branching ratios of the semileptonic 119861 rarr 119879ℓ] decays on 1199022


1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Partial support of the Isfahan University of TechnologyResearch Council is appreciated

References

[1] B Aubert Y Karyotakis J P Lees et al ldquoObservation of 119861meson decays to 120596119870lowast and improved measurements for 120596120588 and120596119891

0rdquoPhysical ReviewD vol 79 no 5 Article ID052005 9 pages

2009[2] B Aubert M Bona Y Karyotakis et al ldquoObservation and

polarization measurements of 119861plusmn

rarr 120601119870plusmn

1and 119861plusmn

rarr 120601119870lowastplusmn

2rdquo

Physical Review Letters vol 101 no 16 Article ID 161801 2008[3] B Aubert M Bona Y Karyotakis et al ldquoTime-dependent and

time-integrated angular analysis of 119861 rarr 1205931198700

1198781205870 and 120593119870plusmn

120587∓rdquo

Physical Review D vol 78 no 9 Article ID 092008 27 pages2008

[4] E Barberio R Bernhard S Blyth et al ldquoAverages ofb-hadron and c-hadron properties at the end of 2007rdquohttpsarxivorgabs08081297

[5] W Wang ldquo119861 to tensor meson form factors in the perturbativeQCD approachrdquo Physical Review D vol 83 no 1 Article ID014008 13 pages 2011

[6] S V Dombrowski ldquoExperimental status of scalar and tensormesonsrdquo Nuclear Physics BmdashProceedings Supplements vol 56no 1-2 pp 125ndash135 1997

[7] C Amsler M Doser M Antonelli et al ldquoReview of particlephysicsrdquo Physics Letters B vol 667 no 1ndash5 pp 1ndash6 2008

[8] DM LiH Yu andQX Shen ldquoProperties of the tensormesonsf2(1270) and f 1015840

2(1525)rdquo Journal of Physics G Nuclear and Particle

Physics vol 27 no 4 pp 807ndash813 2001[9] H-Y Cheng Y Koike and K-C Yang ldquoTwo-parton light-cone

distribution amplitudes of tensor mesonsrdquo Physical Review Dvol 82 no 5 Article ID 054019 7 pages 2010

[10] K Azizi H Sundu and S Sahin ldquoSemileptonic 119861119904

rarr

119863lowast

1199042(2573)ℓ]

ℓtransition inQCDrdquoTheEuropean Physical Journal

C vol 75 article 197 pp 1ndash13 2015[11] P Ball ldquoSemileptonic decays119863 rarr 120587(120588)119890] and119861 rarr 120587(120588)119890] from

QCDsumrulesrdquoPhysical ReviewD vol 48 no 7 pp 3190ndash32031993

[12] P Colangelo F De Fazio P Santorelli and E Scrimieri ldquoQCDsum rule analysis of the decays119861 rarr 119870ℓ

+

ℓminus and 119861 rarr 119870

ℓ+

ℓminusrdquo

Physical Review D vol 53 no 7 pp 3672ndash3686 1996[13] P Colangelo F De Fazio P Santorelli and E Scrimieri ldquoRare

119861 rarr 119870()]] decays at B factoriesrdquo Physics Letters B vol 395no 3-4 pp 339ndash344 1997

[14] P Colangelo C A Dominguez G Nardulli and N PaverldquoRadiative 119861 rarr 119870 lowast 120574 transition in QCDrdquo Physics Letters Bvol 317 no 1-2 pp 183ndash189 1993

[15] M-Z Yang ldquoSemileptonic decay of 119861 and 119863 rarr 119870lowast

0(1430)ℓ]

from QCD sum rulerdquo Physical Review D vol 73 no 3 ArticleID 034027 9 pages 2006

[16] N Gharamany and R Khosravi ldquoAnalysis of the rare semilep-tonic decays of 119861

119904to 119891

0(980) and 119870

0(1430) scalar mesons in

QCD sum rulesrdquo Physical Review D vol 80 no 1 Article ID016009 2009

[17] R Khosravi ldquoForm factors and branching ratios of the FCNC119861 rarr 119886

1ℓ+

ℓminus decaysrdquo The European Physical Journal C vol 75

article 220 2015


[18] P Ball ldquoThe decay 119861 rarr 119870lowast

120574 from QCD sum rulesrdquo httpsarxivorgabshep-ph9308244

[19] T E Coan V Fadeyev Y Maravin et al ldquoStudy of exclusiveradiative 119861 meson decaysrdquo Physical Review Letters vol 84 no23 pp 5283ndash5287 2000

[20] S AhmedM S Alam S B Athar et al ldquo119887 rarr 119904120574 branching frac-tion and CP asymmetryrdquo httpsarxivorgabshep-ex9908022

[21] R Barate D Buskulic D Decamp et al ldquoA measurement of theinclusive 119887 rarr 119904120574 branching ratiordquo Physics Letters B vol 429 no1-2 pp 169ndash187 1998

[22] A Abada C R Allton Ph Boucaud et al ldquoSemi-leptonicdecays of heavy flavours on a fine grained latticerdquo NuclearPhysics B vol 416 no 2 pp 675ndash695 1994

[23] C R Allton M Crisafulli V Lubicz et al ldquoLattice calculationofD- and B-meson semileptonic decays using the clover actionat 120573 = 60 onAPErdquo Physics Letters B vol 345 no 4 pp 513ndash5231995

[24] D R Burford H D Duong J M Flynn et al ldquoForm factors for119861 rarr 120587119897]119897 and 119861 rarr 119870 lowast 120574 decays on the latticerdquo Nuclear PhysicsB vol 447 no 2-3 pp 425ndash437 1995

[25] K C Yang ldquoB to light tensor meson form factors derived fromlight-cone sum rulesrdquo Physics Letters B vol 695 no 5 pp 444ndash448 2011

[26] J Charles A Le Yaouanc L Oliver O Pene and J-C RaynalldquoHeavy-to-light form factors in the final hadron large energylimit covariant quark model approachrdquo Physics Letters B vol451 no 1-2 pp 187ndash194 1999

[27] D Ebert R N Faustov and V O Galkin ldquoForm factors ofheavy-to-light B decays at large recoilrdquo Physical Review D vol64 no 9 Article ID 094022 2001

[28] A Datta Y Gao A V Gritsan D London M Nagashimaand A Szynkman ldquoStudy of polarization in 119861 rarr 119881119879 decaysrdquoPhysical Review D vol 77 no 11 Article ID 114025 2008

[29] D Scora andN Isgur ldquoSemileptonic meson decays in the quarkmodel an updaterdquo Physical Review D vol 52 no 5 pp 2783ndash2812 1995

[30] J Beringer J-F Arguin R M Barnett et al ldquoReview of particlephysicsrdquo Physical Review D vol 86 no 1 Article ID 0100012012

[31] E Gamiz C T H Davies G P Lepage J Shigemitsu andM Wingate ldquoNeutral B meson mixing in unquenched latticeQCDrdquo Physical Review D vol 80 no 1 Article ID 014503 2009

[32] M J Baker J Bordes C A Dominguez J Penarrocha and KSchilcher ldquoBmeson decay constants 119891

119861119888 119891

119861119904and f

119861from QCD

sum rulesrdquo Journal of High Energy Physics vol 2014 no 7 article32 2014

[33] T M Aliev and M A Shifman ldquoOld tensor mesons in QCDsum rulesrdquo Physics Letters B vol 112 no 4-5 pp 401ndash405 1982

[34] P Ball and R Zwicky ldquoNew results on 119861 rarr 120587119870 120578 decay formfactors from light-cone sum rulesrdquo Physical Review D vol 71no 1 Article ID 014015 2005

[35] P Colangelo and A Khodjamirian ldquoQCD sum rules a modernperspectiverdquo in At the Frontier of Particle Physics Handbookof QCD M Shifman Ed vol 3 p 1495 World ScientificSingapore 2001

[36] J Bordes C A Dominguez P Moodley J Penarrocha andK Schilcher ldquoChiral corrections to the SU(2) x SU(2) Gell-Mann-Oakes-Renner relationrdquo Journal of High Energy Physicsvol 2010 p 64 2010

[37] J Bordes C A Dominguez P Moodley J Penarrocha andK Schilcher ldquoCorrections to the SU(3)times SU(3) Gell-Mann-Oakes-Renner relation and chiral couplings 119871119903

8and119867119903

2rdquo Journal

of High Energy Physics vol 2012 artice 102 2012[38] M-Q Huang ldquoExclusive semileptonic 119861

119904decays to excited 119863

119904

mesons search of 119863sJ(2317) and 119863sJ(2460)rdquo Physical Review Dvol 69 no 11 Article ID 114015 10 pages 2004

[39] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics Theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979

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


Bs(p)

b u

vl

0

y x

s

Klowast2 (p

998400)

Figure 1 Schematic picture of the spectator mechanism for the119861119904rarr 119870

lowast

2ℓ] decay

ratios related to these semileptonic decays are plotted withrespect to the momentum transfer squared 1199022

2 Theoretical Framework

In order to study 119861(119861119904) rarr 119870

lowast

2(119886

2 119891

2)ℓ] decays we focus

on the exclusive decay 119861119904rarr 119870

lowast

2via the 3PSR The 119861

119904rarr

119870lowast

2ℓ] decay governed by the tree level 119887 rarr 119906 transition

(see Figure 1) In the framework of the 3PSR the first step isappropriate definition of correlation function In this workthe correlation function should be taken as

Π120572120573120583

(1199012

11990110158402

1199022

) = 119894∬119890119894(1199011015840119909minus119901119910)

⟨0 |

T 119895119870lowast

2

120572120573(119909) 119895

120583(0) 119895

119861119904(119910) | 0⟩ 119889

4

1199091198894

119910

(1)

where 119901 and 1199011015840 are four-momentum of the initial and finalmesons respectively 1199022 is the squared momentum transferand T is the time ordering operator 119895

120583= 119906120574

120583(1 minus 120574

5)119887 is

the transition current 119895119861119904 and 119895119870lowast

2

120572120573are also the interpolating

currents of 119861119904and the tensor meson 119870lowast

2 respectively With

considering all quantum numbers their interpolating cur-rents can be written as follows [33]

119895119861119904(119910) = 119887 (119910) 120574

5119904 (119910)

119895119870lowast

2

120572120573(119909)

=

119894

2

[119904 (119909) 120574120572

harr

119863120573(119909) 119906 (119909) + 119904 (119909) 120574

120573

harr

119863120572(119909) 119906 (119909)]

(2)

whereharr

119863120583(119909) is the four-derivative vector with respect to 119909

acting at the same time on the left and right It is given as

harr

119863120583(119909) =

1

2

[120583(119909) minus

120583(119909)]

120583(119909) =

120597120583(119909) minus 119894

119892

2

120582119886A119886

120583(119909)

120583(119909) =

120597120583(119909) + 119894

119892

2

120582119886A119886

120583(119909)

(3)

where 120582119886 and A119886

120583(119909) are the Gell-Mann matrices and the

external gluon fields respectively It should be noted thatthe second current in (2) interpolates a spin 2 particle formassless quarks In the general case to describe a spin 2 stateone has to use a current such that the trace of 119895119870

lowast

2

120572120573vanishes

The correlation function is a complex function of whichthe imaginary part comprises the computations of the phe-nomenology and real part comprises the computations of thetheoretical part (QCD) By linking these two parts via thedispersion relation the physical quantities are calculated Inthe phenomenological part of the QCD sum rules approachthe correlation function in (1) is calculated by inserting twocomplete sets of intermediate states with the same quantumnumbers as 119861

119904and 119870lowast

2 After performing four integrals over

119909 and 119910 it will be

Π120572120573120583

= minus

⟨0 | 119895119870lowast

2

120572120573| 119870

lowast

2(119901

1015840

)⟩ ⟨119870lowast

2(119901

1015840

) | 119895120583| 119861

119904(119901)⟩ ⟨119861

119904(119901) | 119895

119861119904| 0⟩

(1199012minus 119898

2

119861119904

) (11990110158402minus 119898

2

119870lowast

2

)

+ higher states (4)

In (4) the vacuum to initial and final meson state matrixelements is defined as

⟨0 | 119895119870lowast

2

120572120573| 119870

lowast

2(119901

1015840

120576)⟩ = 119891119870lowast

2

1198982

119870lowast

2

120576120572120573

⟨0 | 119895119861119904| 119861

119904(119901)⟩ = minus119894

119891119861119904

1198982

119861119904

(119898119887+ 119898

119904)

(5)

where 119891119870lowast

2

and 119891119861119904

are the leptonic decay constants of 119870lowast

2

and 119861119904mesons respectively 120576

120572120573is polarization tensor of

119870lowast

2 The transition current gives a contribution to these

matrix elements and it can be parametrized in terms ofsome form factors using the Lorentz invariance and parity

conservation The correspondence between a vector mesonand a tensor meson allows us to get these parametrizationsin a comparative way (for more information see [5]) Theparametrization of 119861 rarr 119879 form factors is analogous to the119861 rarr 119881 case except that 120576 is replaced by 120576

119879 as follows

119888119881⟨119870

lowast

2(119901

1015840

120576) | 119906120574120583(1 minus 120574

5) 119887 | 119861

119904(119901)⟩

= minus119894120576lowast

119879120583(119898

119861119904

+ 119898119870lowast

2

)1198601(119902

2

)

+ 119894 (119901 + 1199011015840

)120583

(120576lowast

119879sdot 119902)

1198602(119902

2

)

119898119861119904

+ 119898119870lowast

2


+ 119894119902120583(120576

lowast

119879sdot 119902)

2119898119870lowast

2

1199022

(1198603(119902

2

) minus 1198600(119902

2

))

+ 120598120583]120588120590120576

lowast]119879119901120588

1199011015840120590

2119881 (1199022

)

119898119861119904

+ 119898119870lowast

2

(6)

with 1198603(119902

2

)

=

119898119861119904

+ 119898119870lowast

2

2119898119870lowast

2

1198601(119902

2

) minus

119898119861119904


2

2119898119870lowast

2

1198602(119902

2

)

1198600(0) = 119860

3(0)

(7)

where 119902 = 119901 minus 1199011015840 119875 = 119901 + 119901

1015840 and 120576lowast119879120583= (119901

120582119898

119861119904

)120576120583120582 The


119881= radic2

for 1198862 119891

2and 119888

119881= 1 for 119870lowast



120576120583]120576120572120573 =

1

2

119879120583120572119879]120573 +

1

2

119879120583120573119879]120572 minus

1

3

119879120583]119879120572120573 (8)

where 119879120583] = minus119892120583] +119901

1015840

1205831199011015840

]1198982

119870lowast

2


Π120572120573120583

=

119891119861119904

119898119861119904

(119898119887+ 119898

119904)

sdot

119891119870lowast

2

1198982

119870lowast

2

(1199012minus 119898

2

119861119904

) (11990110158402minus 119898

2

119870lowast

2

)

1198811015840

(1199022

) 119894120598120573120583120588120590

119901120572119901120588

1199011015840120590

+ 1198601015840

0(119902

2

) 1199011205721199011205731199011015840

120583+ 119860

1015840

1(119902

2

) 119892120573120583119901120572

+ 1198601015840

2(119902

2

) 119901120572119901120573119901120583 + higher states

(9)


1198811015840

(1199022

) =

119881 (1199022

)

119898119861119904

+ 119898119870lowast

2

1198601015840

0(119902

2

) = minus

119898119870lowast

2

(1198603(119902

2

) minus 1198600(119902

2

))

1199022

1198601015840

1(119902

2

) = minus

(119898119861119904

+ 119898119870lowast

2

)

2

1198601(119902

2

)

1198601015840

2(119902

2

) =

1198602(119902

2

)

2 (119898119861119904

+ 119898119870lowast

2

)

(10)


Π120572120573120583

= 119862(0)

120572120573120583119868 + 119862

(3)

120572120573120583⟨0 | ΨΨ | 0⟩

+ 119862(4)

120572120573120583⟨0 | 119866

119886

120588]119866120588]119886| 0⟩

+ 119862(5)

120572120573120583⟨0 | Ψ120590

120588]119879119886

119866120588]119886Ψ | 0⟩ + sdot sdot sdot

(11)

where 119862(119894)





119862(0)

120572120573120583= minus

119894

4

∬119890119894(1199011015840119909minus119901119910)

Tr [119878119904(119909 minus 119910) 120574

120572

harr

119863120573(119909)

sdot 119878119906(minus119909) 120574

120583(1 minus 120574

5) 119878

119887(119910) 120574

5] + Tr [120572

larrrarr 120573] 1198894

119909 1198894

119910

(12)


) rarr minus2119894120587120575(1199012

minus

1198982



Im [119862(0)

120572120573120583] =

1

8120587

int120575 (1198962

minus 1198982

119904) 120575 ((119901 + 119896)

2

minus 1198982

119887)

sdot 120575 ((1199011015840

+ 119896)

2

minus 1198982

119906) (2119896 + 119901

1015840

)120573

Tr [(119896 + 119898119904)

sdot 120574120572(119901

1015840

+ 119896 + 119898119906) 120574

120583(1 minus 120574

5) (119901 + 119896 + 119898119887

) 1205745] + 120572

larrrarr 120573 1198894

119896

(13)


0 119868

120572 119868

120572120573 and 119868


120572120573120583can

be as

119868120572120573120583

(119904 1199041015840

1199022

) = int [119896120572119896120573119896120583] 120575 (119896

2

minus 1198982

119904)

sdot 120575 ((119901 + 119896)2

minus 1198982

119887) 120575 ((119901

1015840

+ 119896)

2

minus 1198982

119906) 119889

4

119896

(14)

where 119904 = 1199012 and 1199041015840 = 11990110158402 1198680 119868

120572 119868

120572120573 and 119868

120572120573120583can be taken as


1198680=

1

4radic120582 (119904 1199041015840 119902

2)

119868120572= 119861

1[119901

120572] + 119861

2[119901

1015840

120572]


b u

s s

(a)

b u

s s

(b)


119868120572120573= 119863

1[119892

120572120573] + 119863

2[119901

120572119901120573] + 119863

3[119901

1205721199011015840

120573+ 119901

1015840

120572119901120573]

+ 1198634[119901

1015840

1205721199011015840

120573]

119868120572120573120583

= 1198641[119892

120572120573119901120583+ 119892

120572120583119901120573+ 119892

120573120583119901120572]

+ 1198642[119892

1205721205731199011015840

120583+ 119892

1205721205831199011015840

120573+ 119892

1205731205831199011015840

120572]

+ 1198643[119901

120572119901120573119901120583]

+ 1198644[119901

1015840

120572119901120573119901120583+ 119901

1205721199011015840

120573119901120583+ 119901

1205721199011205731199011015840

120583]

+ 1198645[119901

1015840

1205721199011015840

120573119901120583+ 119901

1015840

1205721199011205731199011015840

120583+ 119901

1205721199011015840

1205731199011015840

120583]

+ 1198646[119901

1015840

1205721199011015840

1205731199011015840

120583]

(15)

The quantities 120582(119904 1199041015840 1199022) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4)





Im [119862(0)

120572120573120583] = 120588

119881(119894120598

120573120583120588120590119901120572119901120588

1199011015840120590

) + 1205880(119901

1205721199011205731199011015840

120583)

+ 1205881(119892

120573120583119901120572) + 120588

2(119901

120572119901120573119901120583)

(16)


120588119881(119904 119904

1015840

1199022

) = 241198611

radic120582 [1198611(119898

119904minus 119898

119887)

+ 1198612(119898

119904minus 119898

119906) + 119898

1199041198680]

1205880(119904 119904

1015840

1199022

) = 12 [1198632(119898

119904minus 119898

119887) + 119863

3(119898

119904minus 119898

119906)

+ 21198611119898

119904minus 2119864

4(119898

119887minus 119898

119904)]

1205881(119904 119904

1015840

1199022

) = 31198611[2119898

2

119904(119898

119887+ 119898

119906minus 119898

119904)

minus 119898119904(2119898

119887119898

119906+ 119906) + Δ (119898

119904minus 119898

119906) + Δ

1015840

(119898119904minus 119898

119887)]

+ 61198631(119898

119904minus 119898

119906) minus 24119864

1(119898

119887minus 119898

119904)

1205882(119904 119904

1015840

1199022

) = 24 [1198632119898

119904+ 119864

3(119898

119904minus 119898

119887)]

(17)


119862(0)

119894= ∬

120588119894(119904 119904

1015840

1199022

)

(119904 minus 1199012) (119904

1015840minus 119901

10158402)

1198891199041015840

119889119904 (18)


120572120573120583and 119862(5)



119862(3)

119881+ 119862

(5)

119881= minus

2120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

0+ 119862

(5)

0= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

1+ 119862

(5)

1=

120581

(1199012minus 119898

2

119887) (119901

10158402minus 119898

2

119906)

+

120581 [(119898119887+ 119898

119906)2

minus 1199022

]

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

2+ 119862

(5)

2= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

(19)

where 120581 = ((1198982

119904minus119898

2

02)16)⟨0 | 119904119904 | 0⟩1198982

0= (08plusmn02)GeV2




Meson 119861119904

119861 119870lowast

21198862

1198912

Mass 5366 5279 1425 1318 1275




2

1) and 11990110158402(11990110158402 rarr 119872

2

2) on the phe-


1198811015840

(1199022

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119881(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119881+ 119862

(5)

119881] 119889119904

1015840

119889119904

1198601015840

119899(119902

2

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119899(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119899+ 119862

(5)

119899] 119889119904

1015840

119889119904

(20)

where 119899 = 0 2 and 1199040and 1199041015840



119871= 119898

2

119887+ (119898

2

119887(119898

2

119887minus 119902

2

))1199041015840 Also


[

1

(1199012minus 119898

2

119887)119898

(11990110158402minus 119898

2

119906)119899]

=

(minus1)119898+119899

Γ (119899) Γ (119898)

119890minus1198982

1198871198722

1119890minus1198982

1199061198722

2

(1198722

1)119898minus1

(1198722

2)119899minus1

(21)

where1198722

1and1198722



120588higherstates

(119904 1199041015840

) = 120588 (119904 1199041015840

) 120579 (119904 minus 1199040) 120579 (119904

1015840

minus 1199041015840

0) (22)




2

harr 1198981198862

(1198981198912




0(119902

2

) 1198601(119902

2

) and 1198602(119902

2


119904) rarr 119879 where


2 119886

2 or 119891

2 The values


119887= 4820GeV119898

119904= 0150GeV

[38]119898120591= 1776GeV and119898

120583= 0105GeV [30]


0and 1199041015840

0and the Borel

parameters 1198722

1and 119872

2



1015840



119861(119861119904)

0=



2 119886

2 and

1198912are taken to be 119904119870

lowast

2

0= 313GeV2 1199041198862

0= 270GeV2 and

1199041198912


of 1199041198790(119879 = 119886

2 119870

lowast

2 119891




2

1(119902

2


lowast

2




le 1198722

1le 12GeV2

and 4GeV2

le 1198722



0 119860

1 and 119860

2for

119861119904rarr 119870

lowast


1198722

1and1198722





2 is about119881





2



B119904rarr

Klowast 2

1

sKlowast

20 = 313

sKlowast

20 = 323

sKlowast

20 = 333

minus02

minus01

0

01

02

03

2 4 6 80 10

q2


2


lowast

2

0

1198601 and 119860



119861(119861119904)minus 119898



119861(119861119904)minus119898



119891 (1199022

) =

119891 (0)

1 minus 119886 (1199022119898

2

119861(119861119904)) + 119887 (119902

2119898

2

119861(119861119904))

2 (23)




119889Γ (119861 rarr 119879ℓ])119889119902

2

=

1003816100381610038161003816119866119865119881119906119887

1003816100381610038161003816

2

radic120582 (1198982

119861 119898

2

119879 119902

2)

2561198983

11986112058731199022

(1 minus

1198982

ℓ

1199022)

2

sdot (119883119871+ 119883

++ 119883

minus)

(24)


Form factor 119891(0) 119886 119887

119881119861119904rarr119870

lowast

2 013 219 083119860

119861119904rarr119870lowast

2

1010 136 009

119881119861rarr1198862 013 210 075

119860119861rarr1198862

1011 145 023

119881119861rarr1198912 012 201 060

119860119861rarr1198912

1010 140 016

119860119861119904rarr119870

lowast

2

0023 377 421

119860119861119904rarr119870

lowast

2

2005 021 minus299

119860119861rarr1198862

0026 371 403

119860119861rarr1198862

2009 063 046

119860119861rarr1198912

0024 370 402

119860119861rarr1198912

2009 046 029



119883119871=

1

9

120582

1198982

119879119898

2

119861

[(21199022

+ 1198982

ℓ) ℎ

2

0(119902

2

) + 31205821198982

ℓ119860

2

0(119902

2

)]

119883plusmn=

21199022

3

(21199022

+ 1198982

ℓ)

120582

81198982

119879119898

2

119861

[(119898119861+ 119898

119879) 119860

1(119902

2

)

∓

radic120582

119898119861+ 119898

119879

119881(1199022

)]

2

ℎ0(119902

2

) =

1

2119898119879

[(1198982

119861minus 119898

2

119879minus 119902

2

) (119898119861+ 119898

119879) 119860

1(119902

2

)

minus

120582

119898119861+ 119898

119879

1198602(119902

2

)]

(25)


119906119887= (389 plusmn 044) times 10




lowast

2(119886

2 119891

2)ℓ]





119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005


119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)


lowast

2on1198722

1and1198722

2



2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix


1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




+ 119894119902120583(120576

lowast

119879sdot 119902)

2119898119870lowast

2

1199022

(1198603(119902

2

) minus 1198600(119902

2

))

+ 120598120583]120588120590120576

lowast]119879119901120588

1199011015840120590

2119881 (1199022

)

119898119861119904

+ 119898119870lowast

2

(6)

with 1198603(119902

2

)

=

119898119861119904

+ 119898119870lowast

2

2119898119870lowast

2

1198601(119902

2

) minus

119898119861119904


2

2119898119870lowast

2

1198602(119902

2

)

1198600(0) = 119860

3(0)

(7)

where 119902 = 119901 minus 1199011015840 119875 = 119901 + 119901

1015840 and 120576lowast119879120583= (119901

120582119898

119861119904

)120576120583120582 The


119881= radic2

for 1198862 119891

2and 119888

119881= 1 for 119870lowast



120576120583]120576120572120573 =

1

2

119879120583120572119879]120573 +

1

2

119879120583120573119879]120572 minus

1

3

119879120583]119879120572120573 (8)

where 119879120583] = minus119892120583] +119901

1015840

1205831199011015840

]1198982

119870lowast

2


Π120572120573120583

=

119891119861119904

119898119861119904

(119898119887+ 119898

119904)

sdot

119891119870lowast

2

1198982

119870lowast

2

(1199012minus 119898

2

119861119904

) (11990110158402minus 119898

2

119870lowast

2

)

1198811015840

(1199022

) 119894120598120573120583120588120590

119901120572119901120588

1199011015840120590

+ 1198601015840

0(119902

2

) 1199011205721199011205731199011015840

120583+ 119860

1015840

1(119902

2

) 119892120573120583119901120572

+ 1198601015840

2(119902

2

) 119901120572119901120573119901120583 + higher states

(9)


1198811015840

(1199022

) =

119881 (1199022

)

119898119861119904

+ 119898119870lowast

2

1198601015840

0(119902

2

) = minus

119898119870lowast

2

(1198603(119902

2

) minus 1198600(119902

2

))

1199022

1198601015840

1(119902

2

) = minus

(119898119861119904

+ 119898119870lowast

2

)

2

1198601(119902

2

)

1198601015840

2(119902

2

) =

1198602(119902

2

)

2 (119898119861119904

+ 119898119870lowast

2

)

(10)


Π120572120573120583

= 119862(0)

120572120573120583119868 + 119862

(3)

120572120573120583⟨0 | ΨΨ | 0⟩

+ 119862(4)

120572120573120583⟨0 | 119866

119886

120588]119866120588]119886| 0⟩

+ 119862(5)

120572120573120583⟨0 | Ψ120590

120588]119879119886

119866120588]119886Ψ | 0⟩ + sdot sdot sdot

(11)

where 119862(119894)





119862(0)

120572120573120583= minus

119894

4

∬119890119894(1199011015840119909minus119901119910)

Tr [119878119904(119909 minus 119910) 120574

120572

harr

119863120573(119909)

sdot 119878119906(minus119909) 120574

120583(1 minus 120574

5) 119878

119887(119910) 120574

5] + Tr [120572

larrrarr 120573] 1198894

119909 1198894

119910

(12)


) rarr minus2119894120587120575(1199012

minus

1198982



Im [119862(0)

120572120573120583] =

1

8120587

int120575 (1198962

minus 1198982

119904) 120575 ((119901 + 119896)

2

minus 1198982

119887)

sdot 120575 ((1199011015840

+ 119896)

2

minus 1198982

119906) (2119896 + 119901

1015840

)120573

Tr [(119896 + 119898119904)

sdot 120574120572(119901

1015840

+ 119896 + 119898119906) 120574

120583(1 minus 120574

5) (119901 + 119896 + 119898119887

) 1205745] + 120572

larrrarr 120573 1198894

119896

(13)


0 119868

120572 119868

120572120573 and 119868


120572120573120583can

be as

119868120572120573120583

(119904 1199041015840

1199022

) = int [119896120572119896120573119896120583] 120575 (119896

2

minus 1198982

119904)

sdot 120575 ((119901 + 119896)2

minus 1198982

119887) 120575 ((119901

1015840

+ 119896)

2

minus 1198982

119906) 119889

4

119896

(14)

where 119904 = 1199012 and 1199041015840 = 11990110158402 1198680 119868

120572 119868

120572120573 and 119868

120572120573120583can be taken as


1198680=

1

4radic120582 (119904 1199041015840 119902

2)

119868120572= 119861

1[119901

120572] + 119861

2[119901

1015840

120572]


b u

s s

(a)

b u

s s

(b)


119868120572120573= 119863

1[119892

120572120573] + 119863

2[119901

120572119901120573] + 119863

3[119901

1205721199011015840

120573+ 119901

1015840

120572119901120573]

+ 1198634[119901

1015840

1205721199011015840

120573]

119868120572120573120583

= 1198641[119892

120572120573119901120583+ 119892

120572120583119901120573+ 119892

120573120583119901120572]

+ 1198642[119892

1205721205731199011015840

120583+ 119892

1205721205831199011015840

120573+ 119892

1205731205831199011015840

120572]

+ 1198643[119901

120572119901120573119901120583]

+ 1198644[119901

1015840

120572119901120573119901120583+ 119901

1205721199011015840

120573119901120583+ 119901

1205721199011205731199011015840

120583]

+ 1198645[119901

1015840

1205721199011015840

120573119901120583+ 119901

1015840

1205721199011205731199011015840

120583+ 119901

1205721199011015840

1205731199011015840

120583]

+ 1198646[119901

1015840

1205721199011015840

1205731199011015840

120583]

(15)

The quantities 120582(119904 1199041015840 1199022) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4)





Im [119862(0)

120572120573120583] = 120588

119881(119894120598

120573120583120588120590119901120572119901120588

1199011015840120590

) + 1205880(119901

1205721199011205731199011015840

120583)

+ 1205881(119892

120573120583119901120572) + 120588

2(119901

120572119901120573119901120583)

(16)


120588119881(119904 119904

1015840

1199022

) = 241198611

radic120582 [1198611(119898

119904minus 119898

119887)

+ 1198612(119898

119904minus 119898

119906) + 119898

1199041198680]

1205880(119904 119904

1015840

1199022

) = 12 [1198632(119898

119904minus 119898

119887) + 119863

3(119898

119904minus 119898

119906)

+ 21198611119898

119904minus 2119864

4(119898

119887minus 119898

119904)]

1205881(119904 119904

1015840

1199022

) = 31198611[2119898

2

119904(119898

119887+ 119898

119906minus 119898

119904)

minus 119898119904(2119898

119887119898

119906+ 119906) + Δ (119898

119904minus 119898

119906) + Δ

1015840

(119898119904minus 119898

119887)]

+ 61198631(119898

119904minus 119898

119906) minus 24119864

1(119898

119887minus 119898

119904)

1205882(119904 119904

1015840

1199022

) = 24 [1198632119898

119904+ 119864

3(119898

119904minus 119898

119887)]

(17)


119862(0)

119894= ∬

120588119894(119904 119904

1015840

1199022

)

(119904 minus 1199012) (119904

1015840minus 119901

10158402)

1198891199041015840

119889119904 (18)


120572120573120583and 119862(5)



119862(3)

119881+ 119862

(5)

119881= minus

2120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

0+ 119862

(5)

0= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

1+ 119862

(5)

1=

120581

(1199012minus 119898

2

119887) (119901

10158402minus 119898

2

119906)

+

120581 [(119898119887+ 119898

119906)2

minus 1199022

]

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

2+ 119862

(5)

2= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

(19)

where 120581 = ((1198982

119904minus119898

2

02)16)⟨0 | 119904119904 | 0⟩1198982

0= (08plusmn02)GeV2




Meson 119861119904

119861 119870lowast

21198862

1198912

Mass 5366 5279 1425 1318 1275




2

1) and 11990110158402(11990110158402 rarr 119872

2

2) on the phe-


1198811015840

(1199022

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119881(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119881+ 119862

(5)

119881] 119889119904

1015840

119889119904

1198601015840

119899(119902

2

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119899(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119899+ 119862

(5)

119899] 119889119904

1015840

119889119904

(20)

where 119899 = 0 2 and 1199040and 1199041015840



119871= 119898

2

119887+ (119898

2

119887(119898

2

119887minus 119902

2

))1199041015840 Also


[

1

(1199012minus 119898

2

119887)119898

(11990110158402minus 119898

2

119906)119899]

=

(minus1)119898+119899

Γ (119899) Γ (119898)

119890minus1198982

1198871198722

1119890minus1198982

1199061198722

2

(1198722

1)119898minus1

(1198722

2)119899minus1

(21)

where1198722

1and1198722



120588higherstates

(119904 1199041015840

) = 120588 (119904 1199041015840

) 120579 (119904 minus 1199040) 120579 (119904

1015840

minus 1199041015840

0) (22)




2

harr 1198981198862

(1198981198912




0(119902

2

) 1198601(119902

2

) and 1198602(119902

2


119904) rarr 119879 where


2 119886

2 or 119891

2 The values


119887= 4820GeV119898

119904= 0150GeV

[38]119898120591= 1776GeV and119898

120583= 0105GeV [30]


0and 1199041015840

0and the Borel

parameters 1198722

1and 119872

2



1015840



119861(119861119904)

0=



2 119886

2 and

1198912are taken to be 119904119870

lowast

2

0= 313GeV2 1199041198862

0= 270GeV2 and

1199041198912


of 1199041198790(119879 = 119886

2 119870

lowast

2 119891




2

1(119902

2


lowast

2




le 1198722

1le 12GeV2

and 4GeV2

le 1198722



0 119860

1 and 119860

2for

119861119904rarr 119870

lowast


1198722

1and1198722





2 is about119881





2



B119904rarr

Klowast 2

1

sKlowast

20 = 313

sKlowast

20 = 323

sKlowast

20 = 333

minus02

minus01

0

01

02

03

2 4 6 80 10

q2


2


lowast

2

0

1198601 and 119860



119861(119861119904)minus 119898



119861(119861119904)minus119898



119891 (1199022

) =

119891 (0)

1 minus 119886 (1199022119898

2

119861(119861119904)) + 119887 (119902

2119898

2

119861(119861119904))

2 (23)




119889Γ (119861 rarr 119879ℓ])119889119902

2

=

1003816100381610038161003816119866119865119881119906119887

1003816100381610038161003816

2

radic120582 (1198982

119861 119898

2

119879 119902

2)

2561198983

11986112058731199022

(1 minus

1198982

ℓ

1199022)

2

sdot (119883119871+ 119883

++ 119883

minus)

(24)


Form factor 119891(0) 119886 119887

119881119861119904rarr119870

lowast

2 013 219 083119860

119861119904rarr119870lowast

2

1010 136 009

119881119861rarr1198862 013 210 075

119860119861rarr1198862

1011 145 023

119881119861rarr1198912 012 201 060

119860119861rarr1198912

1010 140 016

119860119861119904rarr119870

lowast

2

0023 377 421

119860119861119904rarr119870

lowast

2

2005 021 minus299

119860119861rarr1198862

0026 371 403

119860119861rarr1198862

2009 063 046

119860119861rarr1198912

0024 370 402

119860119861rarr1198912

2009 046 029



119883119871=

1

9

120582

1198982

119879119898

2

119861

[(21199022

+ 1198982

ℓ) ℎ

2

0(119902

2

) + 31205821198982

ℓ119860

2

0(119902

2

)]

119883plusmn=

21199022

3

(21199022

+ 1198982

ℓ)

120582

81198982

119879119898

2

119861

[(119898119861+ 119898

119879) 119860

1(119902

2

)

∓

radic120582

119898119861+ 119898

119879

119881(1199022

)]

2

ℎ0(119902

2

) =

1

2119898119879

[(1198982

119861minus 119898

2

119879minus 119902

2

) (119898119861+ 119898

119879) 119860

1(119902

2

)

minus

120582

119898119861+ 119898

119879

1198602(119902

2

)]

(25)


119906119887= (389 plusmn 044) times 10




lowast

2(119886

2 119891

2)ℓ]





119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005


119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)


lowast

2on1198722

1and1198722

2



2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix


1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




b u

s s

(a)

b u

s s

(b)


119868120572120573= 119863

1[119892

120572120573] + 119863

2[119901

120572119901120573] + 119863

3[119901

1205721199011015840

120573+ 119901

1015840

120572119901120573]

+ 1198634[119901

1015840

1205721199011015840

120573]

119868120572120573120583

= 1198641[119892

120572120573119901120583+ 119892

120572120583119901120573+ 119892

120573120583119901120572]

+ 1198642[119892

1205721205731199011015840

120583+ 119892

1205721205831199011015840

120573+ 119892

1205731205831199011015840

120572]

+ 1198643[119901

120572119901120573119901120583]

+ 1198644[119901

1015840

120572119901120573119901120583+ 119901

1205721199011015840

120573119901120583+ 119901

1205721199011205731199011015840

120583]

+ 1198645[119901

1015840

1205721199011015840

120573119901120583+ 119901

1015840

1205721199011205731199011015840

120583+ 119901

1205721199011015840

1205731199011015840

120583]

+ 1198646[119901

1015840

1205721199011015840

1205731199011015840

120583]

(15)

The quantities 120582(119904 1199041015840 1199022) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4)





Im [119862(0)

120572120573120583] = 120588

119881(119894120598

120573120583120588120590119901120572119901120588

1199011015840120590

) + 1205880(119901

1205721199011205731199011015840

120583)

+ 1205881(119892

120573120583119901120572) + 120588

2(119901

120572119901120573119901120583)

(16)


120588119881(119904 119904

1015840

1199022

) = 241198611

radic120582 [1198611(119898

119904minus 119898

119887)

+ 1198612(119898

119904minus 119898

119906) + 119898

1199041198680]

1205880(119904 119904

1015840

1199022

) = 12 [1198632(119898

119904minus 119898

119887) + 119863

3(119898

119904minus 119898

119906)

+ 21198611119898

119904minus 2119864

4(119898

119887minus 119898

119904)]

1205881(119904 119904

1015840

1199022

) = 31198611[2119898

2

119904(119898

119887+ 119898

119906minus 119898

119904)

minus 119898119904(2119898

119887119898

119906+ 119906) + Δ (119898

119904minus 119898

119906) + Δ

1015840

(119898119904minus 119898

119887)]

+ 61198631(119898

119904minus 119898

119906) minus 24119864

1(119898

119887minus 119898

119904)

1205882(119904 119904

1015840

1199022

) = 24 [1198632119898

119904+ 119864

3(119898

119904minus 119898

119887)]

(17)


119862(0)

119894= ∬

120588119894(119904 119904

1015840

1199022

)

(119904 minus 1199012) (119904

1015840minus 119901

10158402)

1198891199041015840

119889119904 (18)


120572120573120583and 119862(5)



119862(3)

119881+ 119862

(5)

119881= minus

2120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

0+ 119862

(5)

0= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

1+ 119862

(5)

1=

120581

(1199012minus 119898

2

119887) (119901

10158402minus 119898

2

119906)

+

120581 [(119898119887+ 119898

119906)2

minus 1199022

]

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

119862(3)

2+ 119862

(5)

2= minus

4120581

(1199012minus 119898

2

119887)2

(11990110158402minus 119898

2

119906)

(19)

where 120581 = ((1198982

119904minus119898

2

02)16)⟨0 | 119904119904 | 0⟩1198982

0= (08plusmn02)GeV2




Meson 119861119904

119861 119870lowast

21198862

1198912

Mass 5366 5279 1425 1318 1275




2

1) and 11990110158402(11990110158402 rarr 119872

2

2) on the phe-


1198811015840

(1199022

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119881(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119881+ 119862

(5)

119881] 119889119904

1015840

119889119904

1198601015840

119899(119902

2

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119899(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119899+ 119862

(5)

119899] 119889119904

1015840

119889119904

(20)

where 119899 = 0 2 and 1199040and 1199041015840



119871= 119898

2

119887+ (119898

2

119887(119898

2

119887minus 119902

2

))1199041015840 Also


[

1

(1199012minus 119898

2

119887)119898

(11990110158402minus 119898

2

119906)119899]

=

(minus1)119898+119899

Γ (119899) Γ (119898)

119890minus1198982

1198871198722

1119890minus1198982

1199061198722

2

(1198722

1)119898minus1

(1198722

2)119899minus1

(21)

where1198722

1and1198722



120588higherstates

(119904 1199041015840

) = 120588 (119904 1199041015840

) 120579 (119904 minus 1199040) 120579 (119904

1015840

minus 1199041015840

0) (22)




2

harr 1198981198862

(1198981198912




0(119902

2

) 1198601(119902

2

) and 1198602(119902

2


119904) rarr 119879 where


2 119886

2 or 119891

2 The values


119887= 4820GeV119898

119904= 0150GeV

[38]119898120591= 1776GeV and119898

120583= 0105GeV [30]


0and 1199041015840

0and the Borel

parameters 1198722

1and 119872

2



1015840



119861(119861119904)

0=



2 119886

2 and

1198912are taken to be 119904119870

lowast

2

0= 313GeV2 1199041198862

0= 270GeV2 and

1199041198912


of 1199041198790(119879 = 119886

2 119870

lowast

2 119891




2

1(119902

2


lowast

2




le 1198722

1le 12GeV2

and 4GeV2

le 1198722



0 119860

1 and 119860

2for

119861119904rarr 119870

lowast


1198722

1and1198722





2 is about119881





2



B119904rarr

Klowast 2

1

sKlowast

20 = 313

sKlowast

20 = 323

sKlowast

20 = 333

minus02

minus01

0

01

02

03

2 4 6 80 10

q2


2


lowast

2

0

1198601 and 119860



119861(119861119904)minus 119898



119861(119861119904)minus119898



119891 (1199022

) =

119891 (0)

1 minus 119886 (1199022119898

2

119861(119861119904)) + 119887 (119902

2119898

2

119861(119861119904))

2 (23)




119889Γ (119861 rarr 119879ℓ])119889119902

2

=

1003816100381610038161003816119866119865119881119906119887

1003816100381610038161003816

2

radic120582 (1198982

119861 119898

2

119879 119902

2)

2561198983

11986112058731199022

(1 minus

1198982

ℓ

1199022)

2

sdot (119883119871+ 119883

++ 119883

minus)

(24)


Form factor 119891(0) 119886 119887

119881119861119904rarr119870

lowast

2 013 219 083119860

119861119904rarr119870lowast

2

1010 136 009

119881119861rarr1198862 013 210 075

119860119861rarr1198862

1011 145 023

119881119861rarr1198912 012 201 060

119860119861rarr1198912

1010 140 016

119860119861119904rarr119870

lowast

2

0023 377 421

119860119861119904rarr119870

lowast

2

2005 021 minus299

119860119861rarr1198862

0026 371 403

119860119861rarr1198862

2009 063 046

119860119861rarr1198912

0024 370 402

119860119861rarr1198912

2009 046 029



119883119871=

1

9

120582

1198982

119879119898

2

119861

[(21199022

+ 1198982

ℓ) ℎ

2

0(119902

2

) + 31205821198982

ℓ119860

2

0(119902

2

)]

119883plusmn=

21199022

3

(21199022

+ 1198982

ℓ)

120582

81198982

119879119898

2

119861

[(119898119861+ 119898

119879) 119860

1(119902

2

)

∓

radic120582

119898119861+ 119898

119879

119881(1199022

)]

2

ℎ0(119902

2

) =

1

2119898119879

[(1198982

119861minus 119898

2

119879minus 119902

2

) (119898119861+ 119898

119879) 119860

1(119902

2

)

minus

120582

119898119861+ 119898

119879

1198602(119902

2

)]

(25)


119906119887= (389 plusmn 044) times 10




lowast

2(119886

2 119891

2)ℓ]





119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005


119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)


lowast

2on1198722

1and1198722

2



2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix


1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Meson 119861119904

119861 119870lowast

21198862

1198912

Mass 5366 5279 1425 1318 1275




2

1) and 11990110158402(11990110158402 rarr 119872

2

2) on the phe-


1198811015840

(1199022

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119881(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119881+ 119862

(5)

119881] 119889119904

1015840

119889119904

1198601015840

119899(119902

2

) =

(119898119887+ 119898

119904) 119890

1198982

1198611199041198722

1119890

1198982

119870lowast

2

1198722

2

119891119861119904

119898119861119904

119891119870lowast

2

1198982

119870lowast

2

minus1

(2120587)2

sdot int

1199041015840

0

1198982

119904

int

1199040

119904119871

120588119899(119904 119904

1015840

1199022

) 119890minus119904119872

2

1119890minus11990410158401198722

2

+ [119862(3)

119899+ 119862

(5)

119899] 119889119904

1015840

119889119904

(20)

where 119899 = 0 2 and 1199040and 1199041015840



119871= 119898

2

119887+ (119898

2

119887(119898

2

119887minus 119902

2

))1199041015840 Also


[

1

(1199012minus 119898

2

119887)119898

(11990110158402minus 119898

2

119906)119899]

=

(minus1)119898+119899

Γ (119899) Γ (119898)

119890minus1198982

1198871198722

1119890minus1198982

1199061198722

2

(1198722

1)119898minus1

(1198722

2)119899minus1

(21)

where1198722

1and1198722



120588higherstates

(119904 1199041015840

) = 120588 (119904 1199041015840

) 120579 (119904 minus 1199040) 120579 (119904

1015840

minus 1199041015840

0) (22)




2

harr 1198981198862

(1198981198912




0(119902

2

) 1198601(119902

2

) and 1198602(119902

2


119904) rarr 119879 where


2 119886

2 or 119891

2 The values


119887= 4820GeV119898

119904= 0150GeV

[38]119898120591= 1776GeV and119898

120583= 0105GeV [30]


0and 1199041015840

0and the Borel

parameters 1198722

1and 119872

2



1015840



119861(119861119904)

0=



2 119886

2 and

1198912are taken to be 119904119870

lowast

2

0= 313GeV2 1199041198862

0= 270GeV2 and

1199041198912


of 1199041198790(119879 = 119886

2 119870

lowast

2 119891




2

1(119902

2


lowast

2




le 1198722

1le 12GeV2

and 4GeV2

le 1198722



0 119860

1 and 119860

2for

119861119904rarr 119870

lowast


1198722

1and1198722





2 is about119881





2



B119904rarr

Klowast 2

1

sKlowast

20 = 313

sKlowast

20 = 323

sKlowast

20 = 333

minus02

minus01

0

01

02

03

2 4 6 80 10

q2


2


lowast

2

0

1198601 and 119860



119861(119861119904)minus 119898



119861(119861119904)minus119898



119891 (1199022

) =

119891 (0)

1 minus 119886 (1199022119898

2

119861(119861119904)) + 119887 (119902

2119898

2

119861(119861119904))

2 (23)




119889Γ (119861 rarr 119879ℓ])119889119902

2

=

1003816100381610038161003816119866119865119881119906119887

1003816100381610038161003816

2

radic120582 (1198982

119861 119898

2

119879 119902

2)

2561198983

11986112058731199022

(1 minus

1198982

ℓ

1199022)

2

sdot (119883119871+ 119883

++ 119883

minus)

(24)


Form factor 119891(0) 119886 119887

119881119861119904rarr119870

lowast

2 013 219 083119860

119861119904rarr119870lowast

2

1010 136 009

119881119861rarr1198862 013 210 075

119860119861rarr1198862

1011 145 023

119881119861rarr1198912 012 201 060

119860119861rarr1198912

1010 140 016

119860119861119904rarr119870

lowast

2

0023 377 421

119860119861119904rarr119870

lowast

2

2005 021 minus299

119860119861rarr1198862

0026 371 403

119860119861rarr1198862

2009 063 046

119860119861rarr1198912

0024 370 402

119860119861rarr1198912

2009 046 029



119883119871=

1

9

120582

1198982

119879119898

2

119861

[(21199022

+ 1198982

ℓ) ℎ

2

0(119902

2

) + 31205821198982

ℓ119860

2

0(119902

2

)]

119883plusmn=

21199022

3

(21199022

+ 1198982

ℓ)

120582

81198982

119879119898

2

119861

[(119898119861+ 119898

119879) 119860

1(119902

2

)

∓

radic120582

119898119861+ 119898

119879

119881(1199022

)]

2

ℎ0(119902

2

) =

1

2119898119879

[(1198982

119861minus 119898

2

119879minus 119902

2

) (119898119861+ 119898

119879) 119860

1(119902

2

)

minus

120582

119898119861+ 119898

119879

1198602(119902

2

)]

(25)


119906119887= (389 plusmn 044) times 10




lowast

2(119886

2 119891

2)ℓ]





119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005


119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)


lowast

2on1198722

1and1198722

2



2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix


1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




B119904rarr

Klowast 2

1

sKlowast

20 = 313

sKlowast

20 = 323

sKlowast

20 = 333

minus02

minus01

0

01

02

03

2 4 6 80 10

q2


2


lowast

2

0

1198601 and 119860



119861(119861119904)minus 119898



119861(119861119904)minus119898



119891 (1199022

) =

119891 (0)

1 minus 119886 (1199022119898

2

119861(119861119904)) + 119887 (119902

2119898

2

119861(119861119904))

2 (23)




119889Γ (119861 rarr 119879ℓ])119889119902

2

=

1003816100381610038161003816119866119865119881119906119887

1003816100381610038161003816

2

radic120582 (1198982

119861 119898

2

119879 119902

2)

2561198983

11986112058731199022

(1 minus

1198982

ℓ

1199022)

2

sdot (119883119871+ 119883

++ 119883

minus)

(24)


Form factor 119891(0) 119886 119887

119881119861119904rarr119870

lowast

2 013 219 083119860

119861119904rarr119870lowast

2

1010 136 009

119881119861rarr1198862 013 210 075

119860119861rarr1198862

1011 145 023

119881119861rarr1198912 012 201 060

119860119861rarr1198912

1010 140 016

119860119861119904rarr119870

lowast

2

0023 377 421

119860119861119904rarr119870

lowast

2

2005 021 minus299

119860119861rarr1198862

0026 371 403

119860119861rarr1198862

2009 063 046

119860119861rarr1198912

0024 370 402

119860119861rarr1198912

2009 046 029



119883119871=

1

9

120582

1198982

119879119898

2

119861

[(21199022

+ 1198982

ℓ) ℎ

2

0(119902

2

) + 31205821198982

ℓ119860

2

0(119902

2

)]

119883plusmn=

21199022

3

(21199022

+ 1198982

ℓ)

120582

81198982

119879119898

2

119861

[(119898119861+ 119898

119879) 119860

1(119902

2

)

∓

radic120582

119898119861+ 119898

119879

119881(1199022

)]

2

ℎ0(119902

2

) =

1

2119898119879

[(1198982

119861minus 119898

2

119879minus 119902

2

) (119898119861+ 119898

119879) 119860

1(119902

2

)

minus

120582

119898119861+ 119898

119879

1198602(119902

2

)]

(25)


119906119887= (389 plusmn 044) times 10




lowast

2(119886

2 119891

2)ℓ]





119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005


119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)


lowast

2on1198722

1and1198722

2



2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix


1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119861119904rarr119870lowast

2013 plusmn 003 015 plusmn 002 018

+005

minus004mdash mdash

119860119861119904rarr119870

lowast

2

0023 plusmn 006 022 plusmn 004 015

+004

minus003mdash mdash

119860119861119904rarr119870

lowast

2

1010 plusmn 002 012 plusmn 002 011

+003

minus002mdash mdash

119860119861119904rarr119870

lowast

2

2005 plusmn 001 005 plusmn 002 007

+002

minus002mdash mdash

119881119861rarr1198862

013 plusmn 003 018 plusmn 002 018+005


119860119861rarr1198862

0026 plusmn 007 021 plusmn 004 018

+006


119860119861rarr1198862

1011 plusmn 004 014 plusmn 002 011

+003


119860119861rarr1198862

2009 plusmn 002 009 plusmn 002 006

+002


119881119861rarr1198912

012 plusmn 004 018 plusmn 002 012+003


119860119861rarr1198912

0024 plusmn 006 020 plusmn 004 013

+004


119860119861rarr1198912

1010 plusmn 002 014 plusmn 002 008

+002


119860119861rarr1198912

2009 plusmn 002 010 plusmn 002 004

+001


minus02

minus01

0

01

02

03

04

98 11 1210

M21

minus02

minus01

0

01

02

03

04

5 6 7 84

M22

VA0

A1

A2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Bsrarr

Klowast 2

)


lowast

2on1198722

1and1198722

2



2120583]) 082 plusmn 025 116

+081

minus057

Br (119861119904rarr 119870

lowast

2120583]) 065 plusmn 020 073

+048

minus033

Br (119861 rarr 1198912120583]) 077 plusmn 023 069

+048

minus034

Br (119861 rarr 1198862120591]) 051 plusmn 017 041

+029

minus020

Br (119861119904rarr 119870

lowast

2120591]) 035 plusmn 011 025

+017

minus012

Br (119861 rarr 1198912120591]) 053 plusmn 018 025

+018

minus013

Appendix


1015840

1199022

) 119861119897(119897 = 1 2) 119863

119895(119895 = 1 4) and 119864

119903(119903 =

1 6) are given

120582 (119904 1199041015840

1199022

) = 1199042

+ 11990410158402

+ (1199022

)

2

minus 21199041199022

minus 21199041015840

1199022

minus 21199041199041015840

1198611=

1198680

120582 (119904 1199041015840 119902

2)

[21199041015840

Δ minus Δ1015840

119906]

1198612=

1198680

120582 (119904 1199041015840 119902

2)

[2119904Δ1015840

minus Δ119906]

1198631= minus

1198680

2120582 (119904 1199041015840 119902

2)

[41199041199041015840

1198982

119904minus 119904Δ

10158402

minus 1199041015840

Δ2

minus 1199062

1198982

119904

+ 119906ΔΔ1015840

]

1198632= minus

1198680

1205822(119904 119904

1015840 119902

2)

[811990411990410158402

1198982

119904minus 2119904119904

1015840

Δ1015840

minus 611990410158402

Δ2

minus 21199062

1199041015840

1198982

119904+ 6119904

1015840

119906ΔΔ1015840

minus 1199062

Δ10158402

]

1198633=

1198680

1205822(119904 119904

1015840 119902

2)

[41199041199041015840

1199061198982

119904+ 4119904119904

1015840

ΔΔ1015840

minus 3119904119906Δ10158402

minus 3119906Δ2

1199041015840

minus 1199063

1198982

3+ 2119906

2

ΔΔ1015840

]


minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

104 6 820

q2

VA0

A1

A2

minus1

0

1

2

2 4 6 8 100

q2

VA0

A1

A2

Form

fact

ors (

Bsrarr

Klowast 2

)

Form

fact

ors (

Brarr

a 2)

Form

fact

ors (

Brarr

f2)


a2Klowast

2f2

a2Klowast

2f2

q22 4 6 8 10 12 14 16

0

02

04

06

08

1

6 84 12 14 1610

q2

0

02

04

06

08

dBr(Brarr

T120591)dq2middot1

05

dBr(Brarr

T120583)dq2middot1

05



1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1198634=

1198680

1205822(119904 119904

1015840 119902

2)

[minus61199041015840

119906ΔΔ1015840

+ 61199042

Δ10158402

minus 81199042

1199041015840

1198982

119904

+ 21199062

1199041198982

119904+ 119906

2

Δ2

+ 21199041199041015840

Δ2

]

1198641=

1198680

21205822(119904 119904

1015840 119902

2)

[811990410158402

1198982

119904Δ119904 minus 2119904

1015840

1198982

119904Δ119906

2

minus 41199061198982

119904Δ1015840

1199041199041015840

+ 1199063

1198982

119904Δ1015840

minus 211990410158402

Δ3

+ 31199041015840

119906Δ2

Δ1015840

minus 2Δ10158402

Δ1199041199041015840

minus Δ10158402

Δ1199062

+ 119906119904Δ3

]

1198642=

1198680

21205822(119904 119904

1015840 119902

2)

[81199042

1198982

119904Δ1015840

1199041015840

minus 21199042

Δ10158403

minus 41199061198982

119904Δ119904119904

1015840

minus 2Δ2

Δ1015840

1199041199041015840

+ 3119906119904Δ10158402

Δ minus 21199041198982

119904Δ1015840

1199062

+ 1199041015840

119906Δ3

+ 1199063

1198982

119904Δ minus Δ

2

Δ1015840

1199062

]

1198643= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199041198982

119904Δ119904

10158403

minus 2411990411990410158402

1199061198982

119904Δ1015840

minus 1211990411990410158402

Δ10158402

Δ + 6119904119906Δ10158403

1199041015840

minus 2011990410158403

Δ3

+ 3011990410158402

119906Δ2

Δ1015840

minus 1211990410158402

1198982

119904Δ119906

2

minus 121199041015840

Δ10158402

Δ1199062

+ 61199041015840

1199063

1198982

119904Δ1015840

+ 1199063

Δ10158403

]

1198644= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ1015840

11990410158402

minus 41199042

Δ10158403

1199041015840

minus 1211990411990410158402

Δ2

Δ1015840

minus 2411990411990410158402

1199061198982

119904Δ + 3119906

3

Δ10158402

Δ

+ 18119904119906Δ10158402

Δ1199041015840

minus 4119904Δ10158403

1199062

+ 1011990410158402

119906Δ3

+ 61199041015840

1199063

1198982

119904Δ

minus 121199041015840

Δ2

Δ1015840

1199062

minus 21198982

119904Δ1015840

1199064

+ 41199041199041015840

1199062

1198982

119904Δ1015840

]

1198645= minus

1198680

1205823(119904 119904

1015840 119902

2)

[161199042

1198982

119904Δ119904

10158402

minus 241199042

1199041015840

1199061198982

119904Δ1015840

minus 121199042

1199041015840

Δ10158402

Δ + 101199061199042

Δ10158403

minus 411990411990410158402

Δ3

+ 41199041199041015840

1199062

1198982

119904Δ

+ 18119904119906Δ2

Δ1015840

1199041015840

+ 61199041199063

1198982

119904Δ1015840

minus 12119904Δ2

Δ1199062

minus 41199041015840

Δ3

1199062

minus 21198982

119904Δ119906

4

+ 31199063

Δ2

Δ1015840

]

1198646= minus

1198680

1205823(119904 119904

1015840 119902

2)

[481199043

1198982

119904Δ1015840

1199041015840

minus 201199043

Δ10158403

minus 121199042

Δ2

Δ1015840

1199041015840

minus 241199042

1199041015840

1199061198982

119904Δ minus 12119904

2

1198982

119904Δ1015840

1199062

+ 301199061199042

Δ10158402

Δ + 6119904119906Δ3

1199041015840

minus 12119904Δ2

Δ1015840

1199062

+ 61199041199063

1198982

119904Δ

+ 1199063

Δ3

]

Δ = 119904 + 1198982

119904minus 119898

2

119887 Δ

1015840

= 1199041015840

+ 1198982

119904minus 119898

2

119906 119906 = 119904 + 119904

1015840

minus 1199022

(A1)

Competing Interests


Acknowledgments


References






1and 119861plusmn


2rdquo



1198781205870 and 120593119870plusmn

120587∓rdquo











rarr

119863lowast

1199042(2573)ℓ]





+


ℓ+

ℓminusrdquo





0(1430)ℓ]



119904to 119891

0(980) and 119870




1ℓ+


article 220 2015


















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




















119861119888 119891

119861119904and f

119861from QCD







8and119867119903

2rdquo Journal



119904








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 semileptonic decays of to light tensor...

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