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Research ArticleSemileptonic Transition of Tensor 120594
1198882(1119875) to 119863
119904Meson
J Y Sungu1 H Sundu1 and K Azizi2
1 Department of Physics Kocaeli University 41380 Izmit Turkey2Department of Physics Dogus University Acıbadem-Kadıkoy 34722 Istanbul Turkey
Correspondence should be addressed to K Azizi kazizidogusedutr
Received 13 May 2014 Revised 17 July 2014 Accepted 17 July 2014 Published 13 October 2014
Academic Editor Kingman Cheung
Copyright copy 2014 J Y Sungu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3
Taking into account the two-gluon condensate corrections the transition form factors of the semileptonic 1205941198882
rarr 119863119904ℓ](ℓ = 119890 120583)
decay channel are calculated via three-point QCD sum rules These form factors are used to estimate the decay width of thetransition under consideration in both electron and muon channels The obtained results can be used both in direct search forsuch decay channels at charm factories and in analysis of the B
119888meson decay at LHC
1 Introduction
Charmonium physics lies in the boundary region betweenperturbative and nonperturbative QCD Hence the studyof the exclusive decay of charmonium such as 120594
119888119869states
provides essential tools to test the perturbative and nonper-turbative aspects of QCD Recently interest in charmoniumspectroscopy has been renewed with the discovery of numer-ous charmonium and charmonium-like states Comparedto other states we have very limited information aboutthe charmonium 120594
119888119869(119869 = 0 1 2) states and their decay
properties More experimental data and theoretical results onexclusive decay of 119875-wave charmonia are needed to betterunderstand the decay dynamics of these states The produc-tion and decaymechanisms of the 120594
119888119869states are actively being
studied Even though these states are not directly produced in119890+
119890minus collisions they are produced abundantly in 120595(3686) 1198641
transitions The large 120595(3686) data sample taken with BESIIIsupplies a good opportunity for a detailed study of 120594
119888119869states
[1]The 120594
1198882meson whose inclusive decay is the subject of
the present study is a tensor 119888119888 bound state with quantumnumbers 119869119875119862 = 2
++ This meson specially attracts the interestof experimentalists for testing the predictions of perturbativeQCD in the laboratory [2ndash4]The first observation of 120594
1198882was
reported in B-decay at CLEO experiment in 2001 [5] The
measurements of the same collaboration on the two-photondecay rates of the even-parity scalar 120594
119887(1198880) and tensor 120594
119887(1198882)
states [6] were motivation to investigate the properties ofthese mesons and their radiative and strong decay (for a listsee [7]) both theoretically and experimentally Compared totheir hadronic and radiative decay the semileptonic decay ofsuch states has not been studied more As the semileptonicchannels contribute significantly to the total decay widthmore theoretical and experimental studies on these transi-tions are needed In this respect we analyze the semileptonic1205941198882
rarr 119863119904ℓ] transition with (ℓ = 119890 120583) via three-point QCD
sum rules in the present work Taking into account the two-gluon condensate contribution we calculate the transitionform factors associated with this channel The fit functionof form factors is then used to estimate the correspondingdecay width in both 119890 and 120583 channels Our results can beused in direct search for the semileptonic decay of the 120594
1198882
meson at charm factories It is expected that the semileptonic119861119888rarr 120594
1198882119897] decay has considerable contributions to the total
decay width of the 119861119888meson [8] hence our results on the
semileptonic decay of 1205941198882state can also be used in analysis of
the 119861119888meson decay at LHC
The plan of this paper is as follows In the next sectionusing an appropriate three-point correlation function wederive QCD sum rules for the form factors defining thesemileptonic 120594
1198882rarr 119863
119904ℓ] transitions The last section
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 252795 7 pageshttpdxdoiorg1011552014252795
2 Advances in High Energy Physics
is devoted to the numerical analysis of the form factorsdetermination of their behavior in terms of transferredmomentum squared estimation of the decay width of thetransitions under consideration and concluding remarks
2 QCD Sum Rules for Transition Form Factor
QCD sum rule method has been a useful and successfulnonperturbative tool to describe physical parameters ofhadrons [9] In this model the hadronic parameters of theground-state hadrons are extracted via equating the followingtwo-alternative representations through a dispersion relationthe first is the operator product expansion (OPE) of the Borel-transformed correlation function of the two relevant currentsand the second is the expression of the same correlationfunction calculated in terms of hadronic degrees of freedom
One of the most efficient tools to do quantitative analysisof the semileptonic decay is based on their low-energyeffective Hamiltonian The effective Hamiltonian for the1205941198882
rarr 119863119904ℓ] decay which is based on the three-level 119888 rarr 119904
transition at quark level can be written as
Heff (119888 997888rarr 119904ℓ]ℓ) =
119866119865
radic2
119881119888119904119888120574120583(1 minus 120574
5) 119904119897120574
120583
(1 minus 1205745) ] (1)
where 119866119865is the Fermi coupling constant and 119881
119888119904is an
element of the Cabibbo-Kobayashi-Maskawa (CKM) matrixBy sandwiching the effective Hamiltonian between the initialand final states we obtain the following matrix elements forthe vector and axial-vector parts of the transition current119895tr120583= 119888120574
120583(1 minus 120574
5)119904 parameterized in terms of form factors
⟨119863119904(1199011015840
) | 119895tr119881120583
| 1205941198882(119901 120576)⟩ = ℎ (119902
2
) 120598120583]120579120578120598
]120582119875120582119875120579
119902120578
⟨119863119904(1199011015840
) | 119895tr119860120583
| 1205941198882(119901 120576)⟩
= 119894 119870 (1199022
) 120598120583]119875
]+ 120598120579120578119875120579
119875120578
[119875120583119887+(1199022
) + 119902120583119887minus(1199022
)]
(2)
where ℎ(1199022
) 119870(1199022
) 119887+(1199022
) and 119887minus(1199022
) are transition formfactors 120598
120579120578is the polarization tensor of the 120594
1198882meson 119875
120583=
(119901 + 1199011015840
)120583 and 119902
120583= (119901 minus 119901
1015840
)120583 Our main task in the following
is to calculate these transition form factors via QCD sum ruletechnique For this aimwe consider the following three-pointcorrelation function
Π120583120572120573
(119901 1199011015840
119902)
= 1198942
int1198894
119909119890minus119894119901sdot119909
times int1198894
1199101198901198941199011015840sdot119910
⟨0 | T | 119895119863119904(119910) 119895
tr119881(119860)120583
(0) 119895dagger1205941198882
120572120573(119909) | 0⟩
(3)
where T is the time-ordered operator To proceed we needto define the interpolating currents of the initial and finalmesonic states These interpolating fields can be written as
119895119863119904(119910) = 119888 (119910) 119894120574
5119904 (119910)
1198951205941198882
120572120573(119909) =
119894
2
[119888 (119909) 120574120572
harr
D120573 (119909) 119888 (119909) + 119888 (119909) 120574120573
harr
D120572 (119909) 119888 (119909)]
(4)
where the two-side derivativeharr
D120573(119909) is defined as
harr
D120573 (119909) =1
2
[D120573(119909) minus
D120573(119909)] (5)
with
D120573(119909) =
120597120573(119909) minus 119894
119892
2
120582119886
119860119886
120573(119909)
D120573(119909) =
120597120573(119909) + 119894
119892
2
120582119886
119860119886
120573(119909)
(6)
Here 120582119886 are the Gell-Mann matrices and 119860
119886
120573(119909) are the
external gluon fields Considering the Fock-Schwinger gauge(119909120573119860119886
120573(119909) = 0) these external fields are expressed in terms of
the gluon field strength tensor in the following way
119860119886
120573(119909) = int
1
0
119889120572120572119909120573119866119886
120573] (120572119909)
=
1
2
119909120573119866119886
120573] (0) +1
3
119909120578119909120573D120578119866119886
120573] (0) + sdot sdot sdot
(7)
In order to calculate the hadronic side of the afore-mentioned correlation function we will insert appropriatecomplete sets of intermediate states with the same quantumnumbers as the mentioned interpolating fields into (3) Afterperforming integrals over four 119909 and 119910 we get
ΠHAD120583120572120573
(119901 1199011015840
119902) = ⟨0 | 119895119863119904(0) | 119863
119904(1199011015840
)⟩
times ⟨119863119904(1199011015840
) | 119895tr119881(119860)120583
(0) | 1205941198882(119901 120576)⟩
times ⟨1205941198882(119901 120576) | 119895
dagger1205941198882
120572120573(0) | 0⟩
times ((11990110158402
minus 1198982
119863119904
) (1199012
minus 1198982
1205941198882
))
minus1
+ sdot sdot sdot
(8)
where sdot sdot sdot symbolizes the contribution of higher states andthe continuum To proceed we need to know the matrixelements ⟨0 | 119895
119863119904(0) | 119863
119904(1199011015840
)⟩ and ⟨1205941198882(119901 120576) | 119895
dagger1205941198882
120572120573(0) | 0⟩
which are defined in terms of the decay constantsmasses andpolarization tensor of the initial state
⟨0 | 119895119863119904(0) | 119863
119904(1199011015840
)⟩ = 119894
119891119863119904
1198982
119863119904
119898119888+ 119898
119904
⟨1205941198882(119901 120576) | 119895
dagger1205941198882
120572120573(0) | 0⟩ = 119891
1205941198882
1198983
1205941198882
120576lowast120582
120572120573
(9)
Advances in High Energy Physics 3
where 1198911205941198882
and 119891119863119904
are leptonic decay constants of 1205941198882and119863
119904
mesons respectively Combining all matrix elements given in(2) and (9) in (8) the final representation of the correlationfunction for the hadronic side is obtained as
ΠHAD120583120572120573
(119901 1199011015840
119902)
=
1198911205941198882
119891119863119904
1198981205941198882
1198982
119863119904
8 (119898119888+ 119898
119904) (119901
10158402minus 119898
2
119863119904
) (1199012minus 119898
2
1205941198882
)
times
2
3
[minusΔ119870(1199022
) minus Δ1015840
119887minus(1199022
)] 119902120583119892120573120572
+
2
3
[(minusΔ + 41198982
1205941198882
)119870 (1199022
) minus Δ1015840
119887+(1199022
)] 119875120583119892120573120572
minus 119894 (Δ minus 41198982
1205941198882
) ℎ (1199022
) 120576120582]120573120583119875120582119875120572119902]
+Δ119870(1199022
) 119902120572119892120573120583
+ other structures + sdot sdot sdot
(10)
where
Δ = 1198982
119863119904
+ 31198982
1205941198882
minus 1199022
Δ1015840
= 1198984
119863119904
minus 21198982
119863119904
(1198982
1205941198882
+ 1199022
) + (1198982
1205941198882
minus 1199022
)
2
(11)
and we have held only the structures which we are goingto choose in order to find the corresponding form factorsNote that for obtaining the above representation we haveperformed summation over the polarization tensor using
sum
120582
120576120582
120583]120576lowast120582
120572120573=
1
2
120578120583120572
120578]120573 +1
2
120578120583120573
120578]120572 minus1
3
120578120583]120578120572120573 (12)
where
120578120583] = minus119892
120583] +119901120583119901]
1198982
1205941198882
(13)
In OPE side the correlation function is calculated indeep Euclidean region (see for instance [10 11]) Placingthe explicit forms of the interpolating currents into thecorrelation function and contracting out all quark pairs viaWickrsquos theorem we obtain
ΠOPE120583120572120573
(119901 1199011015840
119902)
=
1198943
4
int1198894
119909
times int1198894
119910119890minus119894119901sdot119909
1198901198941199011015840sdot119910
times Tr [1205745119878119896119895
119904(119910) 120574
120583(1 minus 120574
5) 119878119895119894
119888(minus119909)
times 120574120572
harr
D120573 (119909) 119878119894119896
119888(119909 minus 119910)] + [120573 larrrarr 120572]
(14)
where 119878119888and 119878
119902are the heavy and light quark propagator
respectively They are given by [12]
119878119894119895
119888(119909)
=
119894
(2120587)4
times int1198894
119896119890minus119894119896sdot119909
times
120575119894119895
119896 minus 119898119888
minus
119892119904119866120572120573
119894119895
4
120590120572120573
(119896 + 119898119888) + (119896 + 119898
119888) 120590120572120573
(1198962minus 119898
2
119888)2
+
1205872
3
⟨
120572119904119866119866
120587
⟩120575119894119895119898119888
1198962
+ 119898119888119896
(1198962minus 119898
2
119888)4+ sdot sdot sdot
119878119894119895
119904(119909) = 119894
119909212058721199094120575119894119895minus
119898119904
412058721199092120575119894119895minus
⟨119904119904⟩
12
(1 minus 119894
119898119904
4119909) 120575
119894119895
minus
1199092
192
1198982
0⟨119904119904⟩ (1 minus 119894
119898119904
6119909) 120575
119894119895
minus 119894
119894119892119904
3212058721199092119866119894119895
120583] (119909120590120583]
+ 120590120583]
119909) + sdot sdot sdot
(15)
Despite being very small compared to the perturbativepart we include the contribution coming from the gluoncondensate terms as nonperturbative effects The correlationfunction in OPE side is also written as
ΠOPE120583120572120573
(119901 1199011015840
119902) = (Πpert1
(1199022
) + Πnon-pert1
(1199022
)) 119902120572119892120573120583
+ (Πpert2
(1199022
) + Πnon-pert2
(1199022
)) 119902120583119892120573120572
+ (Πpert3
(1199022
) + Πnon-pert3
(1199022
)) 119875120583119892120573120572
+ (Πpert4
(1199022
) + Πnon-pert4
(1199022
)) 120576120582]120573120583119875120582119875120572119902]
+ other structures(16)
whereΠpert119894
(1199022
)with 119894 = 1 2 3 4 are the perturbative parts ofthe coefficients of the selected structures They are expressedin terms of double dispersion integrals as
Πpert119894
(1199022
) = int119889119904int1198891199041015840
120588119894(119904 119904
1015840
1199022
)
(119904 minus 1199012) (1199041015840minus 11990110158402)
+ subtracted terms
(17)
where the spectral densities 120588119894(119904 119904
1015840
1199022
) are obtained bytaking the imaginary parts of the Π
pert119894
functions that is120588119894(119904 119904
1015840
1199022
) = (1120587) Im[Πpert119894
] Replacing the explicit expres-sions of the above propagators into (14) and performing
4 Advances in High Energy Physics
integrals over four 119909 and 119910 we find the spectral densitiescorresponding to four different Dirac structures as
1205881(119904 119904
1015840
1199022
) =
3
321205872int
1
0
119889119909int
1minus119909
0
119889119910 [119898119904(4119909 + 2119910 minus 3)
+119898119888(8119909 + 4119910 minus 5)]
1205882(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909
times int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904(4119909 + 2119910 minus 3)]
1205883(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904+ 2119898
119904119910]
1205884(119904 119904
1015840
1199022
) = 0
(18)
From a similar way we also calculate the functionsΠ
non-pert119894
(1199022
)The QCD sum rules for the form factors are obtained
by matching the coefficients of the same structures fromboth sides of the correlation function To suppress thecontributions of the higher states and continuum the doubleBorel transformation with respect to quantities 1199012 and 119901
10158402 isapplied to both sides of the obtained sum rules according tothe following rule
B1198722B11987210158402
1
(1199012minus 119898
2
1)119886
1
(11990110158402
minus 1198982
2)119887
997888rarr (minus1)119886+119887
(1198722
)
119886minus1
(11987210158402
)
119887minus1
Γ (119886) Γ (119887)
119890minus1198982
11198722
119890minus1198982
211987210158402
(19)
where 1198722 and 119872
10158402 are the Borel mass parameters Tofurther suppress the contributions of the higher state andcontinuum we perform continuum subtraction and use thequark-hadron duality assumption As a result we get thefollowing sum rules for the form factors
119870(1199022
)
=
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205881(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert1
119887minus(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205882(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert2
minus
Δ
Δ1015840119870(119902
2
)
119887+(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205883(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert3
+
minusΔ + 41198982
1205941198882
Δ1015840
119870(1199022
)
ℎ (1199022
)
= minus
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
(Δ minus 41198982
1205941198882
)
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205884(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert4
(20)
where 1199040and 119904
1015840
0are continuum thresholds in the initial and
final channels respectively The function 119871(119904 1199041015840
1199022
) is givenby
119871 (119904 1199041015840
1199022
) = 1199041015840
119910 (1 minus 119909 minus 119910) minus 1198982
119888(119909 + 119910)
+ 1199022
119909 (1 minus 119909 minus 119910) + 119904119909119910
(21)
Advances in High Energy Physics 5
8 9 10 11 12
8 9 10 11 12K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
Total contributionPerturbative contribution
M2(GeV2
)
Nonperturbative contribution
(a)K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
5 6 7 8
5 6 7 8
Total contributionPerturbative contributionNonperturbative contribution
M9984002
(GeV2)
(b)
Figure 1 (a) 119870(1199022
= 0) as a function of the Borel mass parameter 1198722 at 11987210158402
= 65GeV2 (b) 119870(1199022
= 0) as a function of the Borel massparameter11987210158402 at1198722
= 10GeV2
The functions B1198722B11987210158402Π
non-pert119894
are very lengthy hence wedo not present their explicit expressions here We shouldstress that the contributions of the light quark condensatesare eliminated by applying the double Borel transformationswith respect to the initial and final momenta hence in theB1198722B11987210158402Π
non-pert119894
functions we only consider the two-gluoncondensate contributions
3 Numerical Results and Discussion
In this section we present our numerical results for the formfactors of the semileptonic 120594
1198882rarr 119863
119904ℓ] transition whose
sum rules have been found in the previous section For thisaim we use the following input parameters 119898
119888= (1275 plusmn
0025)GeV 119898119904= 95 plusmn 5MeV [7] 119866
119865= 117 times 10
minus5 GeVminus2⟨1205721199041198662
120587⟩ = (0012 plusmn 0004)GeV4 ⟨119904119904(1GeV)⟩ = minus08(024 plusmn
001)3 GeV3 [13 14] 1198982
0(1GeV) = (08 plusmn 02)GeV2 [15 16]
119891119863119904
= 245 plusmn 157 plusmn 45MeV [17] and 1198911205941198882
= 00111 plusmn 00062
[18]The sum rules for the form factors also contain four
auxiliary parameters two Borelmass parameters1198722 and11987210158402
as well as two continuum thresholds 1199040and 119904
1015840
0 According
to the criteria of the method the physical quantities suchas form factors should be independent of these parametersHence we will look for regions such that the dependenceof form factors on these helping parameters is weak Thecontinuum thresholds 119904
0and 119904
1015840
0are not totally arbitrary but
they depend on the energy of the first excited states withthe same quantum numbers as the interpolating currents ofthe initial and final channels respectively Our numericalcalculations reveal that in the intervals 13GeV2 le 119904
0le
15GeV2 and 45GeV2 le 1199041015840
0le 6GeV2 the form factors
demonstrate weak dependence on continuum thresholdsThe working region for the Borel mass parameters is
determined with the requirements that not only the higherstates and continuum contributions are suppressed but alsothe contributions of the higher order operators are small thatis the sum rules are convergent From these restrictions theworking regions for the Borel parameters are found to be8GeV2 le 119872
2
le 12GeV2 and 5GeV2 le 11987210158402
le 8GeV2Note that the above regions for the continuum thresholds areobtained according to the standard criteria of the QCD sumrules that is the continuum thresholds are independent ofBorel mass parameters However some recent works (see forinstance [19]) show that the standard criteria do not lead tocorrect results and the continuum thresholds should be takenas functions of Borel masses Following [19] we will add anextra 15 systematic error to the uncertainties of the formfactors
As an example we depict the dependence of form factor119870(119902
2
) on 1198722 and 119872
10158402 in Figure 1 at 1199022
= 0 From thisfigure we observe that the form factor 119870(119902
2
) demonstratesa good stability with respect to the variations of Borel massparameters in their working regions It is also clear that theperturbative part exceeds the nonperturbative one substan-tially and constitutes approximately the whole contribution
Having determined the working regions for the auxiliaryparameters we proceed to find the behaviors of the formfactors in terms of 119902
2 The sum rules for the form factorsare truncated at some points below the perturbative cut soto extend our results to the full physical region we look fora parameterization of the form factors such that its resultscoincide with the results of sum rules at reliable region Our
6 Advances in High Energy Physics
minus25
minus20
minus15
minus10
minus05
minus25
minus20
minus15
minus10
minus05
00 05 10 15 20 25
00 05 10 15 20 25
QCD sum ruleFit function
K(q2)
q2(GeV2
)
Figure 2 119870(1199022
) as a function of 1199022 at 1198722
= 10GeV2 and 11987210158402
=
65GeV2
analysis shows that the form factors are well fitted to thefunction
119891 (1199022
) = 1198910exp[119888
1
1199022
1198982
fit+ 1198882(
1199022
1198982
fit)
2
] (22)
where the values of the parameters 1198910 1198881 1198882 and 119898
2
fitobtained using 119872
2
= 10GeV2 and 11987210158402
= 65GeV2 for1205941198882
rarr 119863119904ℓ] transition are presented in Table 1 The
errors appearing in the results belong to the uncertaintiesin the input parameters those coming from determinationof the working regions of the auxiliary parameters and thepreviously discussed systematic uncertainties As an examplewe depict the dependence of the form factor 119870(119902
2
) on 1199022 at
1198722
= 10GeV2 and11987210158402
= 65GeV2 in Figure 2 which showsa good fitting of the sum rules results to those obtained fromthe above fit function
Our final purpose in this section is to obtain the decaywidth of the 120594
1198882rarr 119863
119904ℓ] transition The differential decay
width for this transition is obtained as
119889Γ
1198891199022
=
1198662
1198651198812
119888119904
210321198987
1205941198882
12058731199026(1198982
ℓminus 1199022
)
2
Δ101584032
times
10038161003816100381610038161003816119887minus(1199022
)
10038161003816100381610038161003816
2
Δ1015840
1198982
ℓ1199024
+
10038161003816100381610038161003816119887+(1199022
)
10038161003816100381610038161003816
2
times Δ1015840
[(1198982
119863119904
minus 1198982
1205941198882
)
2
1198982
ℓ+ (119898
2
119863119904
minus 1198982
1205941198882
)
2
1199022
minus 2 (1198982
119863119904
+ 1198982
1205941198882
) 1199024
+ 1199026
]
Table 1 Parameters appearing in the fit function of the form factors
1198910
1198881
1198882
1198982
fit
119870(1199022
) minus(109 plusmn 038) 290 minus068 1265119887minus(1199022
) 205 plusmn 072GeVminus2 701 2186 1265119887+(1199022
) 242 plusmn 085GeVminus2 563 12951 1265ℎ(1199022
) minus(442 plusmn 155) times 10minus9 GeVminus2 minus191 111 1265
Table 2 Numerical results of decay width for different leptonchannels
Γ (GeV)1205941198882
rarr 119863119904120583]120583
(160 plusmn 067) times 10minus11
1205941198882
rarr 119863119904119890]119890
(162 plusmn 068) times 10minus11
+ 2Re [119870 (1199022
) 119887lowast
+(1199022
)]
times Δ1015840
[minus1199024
+ 1198982
119863119904
(1198982
ℓ+ 1199022
) minus 1198982
1205941198882
(1198982
ℓ+ 1199022
)]
minus 2Re [119887minus(1199022
) 119887lowast
+(1199022
)] Δ1015840
1198982
ℓ1199022
(1198982
119863119904
minus 1198982
1205941198882
)
+
10038161003816100381610038161003816119870 (119902
2
)
10038161003816100381610038161003816
2
times [1198984
119863119904
(1198982
ℓ+ 1199022
) + 1198984
1205941198882
(1198982
ℓ+ 1199022
)
+ 1199024
(1198982
ℓ+ 1199022
) minus 21198982
119863119904
times (1198982
1205941198882
+ 1199022
) (1198982
ℓ+ 1199022
) + 1198982
1205941198882
1199022
(1198982
ℓ+ 5119902
2
)]
+ 3
10038161003816100381610038161003816ℎ (119902
2
)
10038161003816100381610038161003816
2
Δ1015840
1198982
1205941198882
1199022
(1198982
ℓ+ 1199022
)
minus2Re [119870 (1199022
) 119887lowast
minus(1199022
)] Δ1015840
1198982
ℓ1199022
(23)
After performing integration over 1199022 in (23) in the interval
1198982
ℓle 119902
2
le (1198981205941198882
minus 119898119863119904
)2 we obtain the decay widths in
both 119890 and 120583 channels as presented in Table 2 Consideringthe developments in experimental side we hope that it willbe possible to study such decay channels in the experimentin near future Comparison of future data with theoreticalcalculations will help us get useful information on thestructure of 120594
1198882tensor meson as well as the perturbative and
nonperturbative aspects of QCDThe obtained results in thiswork can also be used in the analysis of the 119861
119888meson decay
at LHC as the 119861119888
rarr 1205941198882is expected to have a considerable
contribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Abilikim M N Achasov X C Ai et al ldquoMeasurement of120594119888119869decaying into 120578
1015840K+Kminusrdquo Physical Review D vol 89 ArticleID 074030 2014
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
2 Advances in High Energy Physics
is devoted to the numerical analysis of the form factorsdetermination of their behavior in terms of transferredmomentum squared estimation of the decay width of thetransitions under consideration and concluding remarks
2 QCD Sum Rules for Transition Form Factor
QCD sum rule method has been a useful and successfulnonperturbative tool to describe physical parameters ofhadrons [9] In this model the hadronic parameters of theground-state hadrons are extracted via equating the followingtwo-alternative representations through a dispersion relationthe first is the operator product expansion (OPE) of the Borel-transformed correlation function of the two relevant currentsand the second is the expression of the same correlationfunction calculated in terms of hadronic degrees of freedom
One of the most efficient tools to do quantitative analysisof the semileptonic decay is based on their low-energyeffective Hamiltonian The effective Hamiltonian for the1205941198882
rarr 119863119904ℓ] decay which is based on the three-level 119888 rarr 119904
transition at quark level can be written as
Heff (119888 997888rarr 119904ℓ]ℓ) =
119866119865
radic2
119881119888119904119888120574120583(1 minus 120574
5) 119904119897120574
120583
(1 minus 1205745) ] (1)
where 119866119865is the Fermi coupling constant and 119881
119888119904is an
element of the Cabibbo-Kobayashi-Maskawa (CKM) matrixBy sandwiching the effective Hamiltonian between the initialand final states we obtain the following matrix elements forthe vector and axial-vector parts of the transition current119895tr120583= 119888120574
120583(1 minus 120574
5)119904 parameterized in terms of form factors
⟨119863119904(1199011015840
) | 119895tr119881120583
| 1205941198882(119901 120576)⟩ = ℎ (119902
2
) 120598120583]120579120578120598
]120582119875120582119875120579
119902120578
⟨119863119904(1199011015840
) | 119895tr119860120583
| 1205941198882(119901 120576)⟩
= 119894 119870 (1199022
) 120598120583]119875
]+ 120598120579120578119875120579
119875120578
[119875120583119887+(1199022
) + 119902120583119887minus(1199022
)]
(2)
where ℎ(1199022
) 119870(1199022
) 119887+(1199022
) and 119887minus(1199022
) are transition formfactors 120598
120579120578is the polarization tensor of the 120594
1198882meson 119875
120583=
(119901 + 1199011015840
)120583 and 119902
120583= (119901 minus 119901
1015840
)120583 Our main task in the following
is to calculate these transition form factors via QCD sum ruletechnique For this aimwe consider the following three-pointcorrelation function
Π120583120572120573
(119901 1199011015840
119902)
= 1198942
int1198894
119909119890minus119894119901sdot119909
times int1198894
1199101198901198941199011015840sdot119910
⟨0 | T | 119895119863119904(119910) 119895
tr119881(119860)120583
(0) 119895dagger1205941198882
120572120573(119909) | 0⟩
(3)
where T is the time-ordered operator To proceed we needto define the interpolating currents of the initial and finalmesonic states These interpolating fields can be written as
119895119863119904(119910) = 119888 (119910) 119894120574
5119904 (119910)
1198951205941198882
120572120573(119909) =
119894
2
[119888 (119909) 120574120572
harr
D120573 (119909) 119888 (119909) + 119888 (119909) 120574120573
harr
D120572 (119909) 119888 (119909)]
(4)
where the two-side derivativeharr
D120573(119909) is defined as
harr
D120573 (119909) =1
2
[D120573(119909) minus
D120573(119909)] (5)
with
D120573(119909) =
120597120573(119909) minus 119894
119892
2
120582119886
119860119886
120573(119909)
D120573(119909) =
120597120573(119909) + 119894
119892
2
120582119886
119860119886
120573(119909)
(6)
Here 120582119886 are the Gell-Mann matrices and 119860
119886
120573(119909) are the
external gluon fields Considering the Fock-Schwinger gauge(119909120573119860119886
120573(119909) = 0) these external fields are expressed in terms of
the gluon field strength tensor in the following way
119860119886
120573(119909) = int
1
0
119889120572120572119909120573119866119886
120573] (120572119909)
=
1
2
119909120573119866119886
120573] (0) +1
3
119909120578119909120573D120578119866119886
120573] (0) + sdot sdot sdot
(7)
In order to calculate the hadronic side of the afore-mentioned correlation function we will insert appropriatecomplete sets of intermediate states with the same quantumnumbers as the mentioned interpolating fields into (3) Afterperforming integrals over four 119909 and 119910 we get
ΠHAD120583120572120573
(119901 1199011015840
119902) = ⟨0 | 119895119863119904(0) | 119863
119904(1199011015840
)⟩
times ⟨119863119904(1199011015840
) | 119895tr119881(119860)120583
(0) | 1205941198882(119901 120576)⟩
times ⟨1205941198882(119901 120576) | 119895
dagger1205941198882
120572120573(0) | 0⟩
times ((11990110158402
minus 1198982
119863119904
) (1199012
minus 1198982
1205941198882
))
minus1
+ sdot sdot sdot
(8)
where sdot sdot sdot symbolizes the contribution of higher states andthe continuum To proceed we need to know the matrixelements ⟨0 | 119895
119863119904(0) | 119863
119904(1199011015840
)⟩ and ⟨1205941198882(119901 120576) | 119895
dagger1205941198882
120572120573(0) | 0⟩
which are defined in terms of the decay constantsmasses andpolarization tensor of the initial state
⟨0 | 119895119863119904(0) | 119863
119904(1199011015840
)⟩ = 119894
119891119863119904
1198982
119863119904
119898119888+ 119898
119904
⟨1205941198882(119901 120576) | 119895
dagger1205941198882
120572120573(0) | 0⟩ = 119891
1205941198882
1198983
1205941198882
120576lowast120582
120572120573
(9)
Advances in High Energy Physics 3
where 1198911205941198882
and 119891119863119904
are leptonic decay constants of 1205941198882and119863
119904
mesons respectively Combining all matrix elements given in(2) and (9) in (8) the final representation of the correlationfunction for the hadronic side is obtained as
ΠHAD120583120572120573
(119901 1199011015840
119902)
=
1198911205941198882
119891119863119904
1198981205941198882
1198982
119863119904
8 (119898119888+ 119898
119904) (119901
10158402minus 119898
2
119863119904
) (1199012minus 119898
2
1205941198882
)
times
2
3
[minusΔ119870(1199022
) minus Δ1015840
119887minus(1199022
)] 119902120583119892120573120572
+
2
3
[(minusΔ + 41198982
1205941198882
)119870 (1199022
) minus Δ1015840
119887+(1199022
)] 119875120583119892120573120572
minus 119894 (Δ minus 41198982
1205941198882
) ℎ (1199022
) 120576120582]120573120583119875120582119875120572119902]
+Δ119870(1199022
) 119902120572119892120573120583
+ other structures + sdot sdot sdot
(10)
where
Δ = 1198982
119863119904
+ 31198982
1205941198882
minus 1199022
Δ1015840
= 1198984
119863119904
minus 21198982
119863119904
(1198982
1205941198882
+ 1199022
) + (1198982
1205941198882
minus 1199022
)
2
(11)
and we have held only the structures which we are goingto choose in order to find the corresponding form factorsNote that for obtaining the above representation we haveperformed summation over the polarization tensor using
sum
120582
120576120582
120583]120576lowast120582
120572120573=
1
2
120578120583120572
120578]120573 +1
2
120578120583120573
120578]120572 minus1
3
120578120583]120578120572120573 (12)
where
120578120583] = minus119892
120583] +119901120583119901]
1198982
1205941198882
(13)
In OPE side the correlation function is calculated indeep Euclidean region (see for instance [10 11]) Placingthe explicit forms of the interpolating currents into thecorrelation function and contracting out all quark pairs viaWickrsquos theorem we obtain
ΠOPE120583120572120573
(119901 1199011015840
119902)
=
1198943
4
int1198894
119909
times int1198894
119910119890minus119894119901sdot119909
1198901198941199011015840sdot119910
times Tr [1205745119878119896119895
119904(119910) 120574
120583(1 minus 120574
5) 119878119895119894
119888(minus119909)
times 120574120572
harr
D120573 (119909) 119878119894119896
119888(119909 minus 119910)] + [120573 larrrarr 120572]
(14)
where 119878119888and 119878
119902are the heavy and light quark propagator
respectively They are given by [12]
119878119894119895
119888(119909)
=
119894
(2120587)4
times int1198894
119896119890minus119894119896sdot119909
times
120575119894119895
119896 minus 119898119888
minus
119892119904119866120572120573
119894119895
4
120590120572120573
(119896 + 119898119888) + (119896 + 119898
119888) 120590120572120573
(1198962minus 119898
2
119888)2
+
1205872
3
⟨
120572119904119866119866
120587
⟩120575119894119895119898119888
1198962
+ 119898119888119896
(1198962minus 119898
2
119888)4+ sdot sdot sdot
119878119894119895
119904(119909) = 119894
119909212058721199094120575119894119895minus
119898119904
412058721199092120575119894119895minus
⟨119904119904⟩
12
(1 minus 119894
119898119904
4119909) 120575
119894119895
minus
1199092
192
1198982
0⟨119904119904⟩ (1 minus 119894
119898119904
6119909) 120575
119894119895
minus 119894
119894119892119904
3212058721199092119866119894119895
120583] (119909120590120583]
+ 120590120583]
119909) + sdot sdot sdot
(15)
Despite being very small compared to the perturbativepart we include the contribution coming from the gluoncondensate terms as nonperturbative effects The correlationfunction in OPE side is also written as
ΠOPE120583120572120573
(119901 1199011015840
119902) = (Πpert1
(1199022
) + Πnon-pert1
(1199022
)) 119902120572119892120573120583
+ (Πpert2
(1199022
) + Πnon-pert2
(1199022
)) 119902120583119892120573120572
+ (Πpert3
(1199022
) + Πnon-pert3
(1199022
)) 119875120583119892120573120572
+ (Πpert4
(1199022
) + Πnon-pert4
(1199022
)) 120576120582]120573120583119875120582119875120572119902]
+ other structures(16)
whereΠpert119894
(1199022
)with 119894 = 1 2 3 4 are the perturbative parts ofthe coefficients of the selected structures They are expressedin terms of double dispersion integrals as
Πpert119894
(1199022
) = int119889119904int1198891199041015840
120588119894(119904 119904
1015840
1199022
)
(119904 minus 1199012) (1199041015840minus 11990110158402)
+ subtracted terms
(17)
where the spectral densities 120588119894(119904 119904
1015840
1199022
) are obtained bytaking the imaginary parts of the Π
pert119894
functions that is120588119894(119904 119904
1015840
1199022
) = (1120587) Im[Πpert119894
] Replacing the explicit expres-sions of the above propagators into (14) and performing
4 Advances in High Energy Physics
integrals over four 119909 and 119910 we find the spectral densitiescorresponding to four different Dirac structures as
1205881(119904 119904
1015840
1199022
) =
3
321205872int
1
0
119889119909int
1minus119909
0
119889119910 [119898119904(4119909 + 2119910 minus 3)
+119898119888(8119909 + 4119910 minus 5)]
1205882(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909
times int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904(4119909 + 2119910 minus 3)]
1205883(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904+ 2119898
119904119910]
1205884(119904 119904
1015840
1199022
) = 0
(18)
From a similar way we also calculate the functionsΠ
non-pert119894
(1199022
)The QCD sum rules for the form factors are obtained
by matching the coefficients of the same structures fromboth sides of the correlation function To suppress thecontributions of the higher states and continuum the doubleBorel transformation with respect to quantities 1199012 and 119901
10158402 isapplied to both sides of the obtained sum rules according tothe following rule
B1198722B11987210158402
1
(1199012minus 119898
2
1)119886
1
(11990110158402
minus 1198982
2)119887
997888rarr (minus1)119886+119887
(1198722
)
119886minus1
(11987210158402
)
119887minus1
Γ (119886) Γ (119887)
119890minus1198982
11198722
119890minus1198982
211987210158402
(19)
where 1198722 and 119872
10158402 are the Borel mass parameters Tofurther suppress the contributions of the higher state andcontinuum we perform continuum subtraction and use thequark-hadron duality assumption As a result we get thefollowing sum rules for the form factors
119870(1199022
)
=
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205881(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert1
119887minus(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205882(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert2
minus
Δ
Δ1015840119870(119902
2
)
119887+(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205883(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert3
+
minusΔ + 41198982
1205941198882
Δ1015840
119870(1199022
)
ℎ (1199022
)
= minus
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
(Δ minus 41198982
1205941198882
)
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205884(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert4
(20)
where 1199040and 119904
1015840
0are continuum thresholds in the initial and
final channels respectively The function 119871(119904 1199041015840
1199022
) is givenby
119871 (119904 1199041015840
1199022
) = 1199041015840
119910 (1 minus 119909 minus 119910) minus 1198982
119888(119909 + 119910)
+ 1199022
119909 (1 minus 119909 minus 119910) + 119904119909119910
(21)
Advances in High Energy Physics 5
8 9 10 11 12
8 9 10 11 12K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
Total contributionPerturbative contribution
M2(GeV2
)
Nonperturbative contribution
(a)K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
5 6 7 8
5 6 7 8
Total contributionPerturbative contributionNonperturbative contribution
M9984002
(GeV2)
(b)
Figure 1 (a) 119870(1199022
= 0) as a function of the Borel mass parameter 1198722 at 11987210158402
= 65GeV2 (b) 119870(1199022
= 0) as a function of the Borel massparameter11987210158402 at1198722
= 10GeV2
The functions B1198722B11987210158402Π
non-pert119894
are very lengthy hence wedo not present their explicit expressions here We shouldstress that the contributions of the light quark condensatesare eliminated by applying the double Borel transformationswith respect to the initial and final momenta hence in theB1198722B11987210158402Π
non-pert119894
functions we only consider the two-gluoncondensate contributions
3 Numerical Results and Discussion
In this section we present our numerical results for the formfactors of the semileptonic 120594
1198882rarr 119863
119904ℓ] transition whose
sum rules have been found in the previous section For thisaim we use the following input parameters 119898
119888= (1275 plusmn
0025)GeV 119898119904= 95 plusmn 5MeV [7] 119866
119865= 117 times 10
minus5 GeVminus2⟨1205721199041198662
120587⟩ = (0012 plusmn 0004)GeV4 ⟨119904119904(1GeV)⟩ = minus08(024 plusmn
001)3 GeV3 [13 14] 1198982
0(1GeV) = (08 plusmn 02)GeV2 [15 16]
119891119863119904
= 245 plusmn 157 plusmn 45MeV [17] and 1198911205941198882
= 00111 plusmn 00062
[18]The sum rules for the form factors also contain four
auxiliary parameters two Borelmass parameters1198722 and11987210158402
as well as two continuum thresholds 1199040and 119904
1015840
0 According
to the criteria of the method the physical quantities suchas form factors should be independent of these parametersHence we will look for regions such that the dependenceof form factors on these helping parameters is weak Thecontinuum thresholds 119904
0and 119904
1015840
0are not totally arbitrary but
they depend on the energy of the first excited states withthe same quantum numbers as the interpolating currents ofthe initial and final channels respectively Our numericalcalculations reveal that in the intervals 13GeV2 le 119904
0le
15GeV2 and 45GeV2 le 1199041015840
0le 6GeV2 the form factors
demonstrate weak dependence on continuum thresholdsThe working region for the Borel mass parameters is
determined with the requirements that not only the higherstates and continuum contributions are suppressed but alsothe contributions of the higher order operators are small thatis the sum rules are convergent From these restrictions theworking regions for the Borel parameters are found to be8GeV2 le 119872
2
le 12GeV2 and 5GeV2 le 11987210158402
le 8GeV2Note that the above regions for the continuum thresholds areobtained according to the standard criteria of the QCD sumrules that is the continuum thresholds are independent ofBorel mass parameters However some recent works (see forinstance [19]) show that the standard criteria do not lead tocorrect results and the continuum thresholds should be takenas functions of Borel masses Following [19] we will add anextra 15 systematic error to the uncertainties of the formfactors
As an example we depict the dependence of form factor119870(119902
2
) on 1198722 and 119872
10158402 in Figure 1 at 1199022
= 0 From thisfigure we observe that the form factor 119870(119902
2
) demonstratesa good stability with respect to the variations of Borel massparameters in their working regions It is also clear that theperturbative part exceeds the nonperturbative one substan-tially and constitutes approximately the whole contribution
Having determined the working regions for the auxiliaryparameters we proceed to find the behaviors of the formfactors in terms of 119902
2 The sum rules for the form factorsare truncated at some points below the perturbative cut soto extend our results to the full physical region we look fora parameterization of the form factors such that its resultscoincide with the results of sum rules at reliable region Our
6 Advances in High Energy Physics
minus25
minus20
minus15
minus10
minus05
minus25
minus20
minus15
minus10
minus05
00 05 10 15 20 25
00 05 10 15 20 25
QCD sum ruleFit function
K(q2)
q2(GeV2
)
Figure 2 119870(1199022
) as a function of 1199022 at 1198722
= 10GeV2 and 11987210158402
=
65GeV2
analysis shows that the form factors are well fitted to thefunction
119891 (1199022
) = 1198910exp[119888
1
1199022
1198982
fit+ 1198882(
1199022
1198982
fit)
2
] (22)
where the values of the parameters 1198910 1198881 1198882 and 119898
2
fitobtained using 119872
2
= 10GeV2 and 11987210158402
= 65GeV2 for1205941198882
rarr 119863119904ℓ] transition are presented in Table 1 The
errors appearing in the results belong to the uncertaintiesin the input parameters those coming from determinationof the working regions of the auxiliary parameters and thepreviously discussed systematic uncertainties As an examplewe depict the dependence of the form factor 119870(119902
2
) on 1199022 at
1198722
= 10GeV2 and11987210158402
= 65GeV2 in Figure 2 which showsa good fitting of the sum rules results to those obtained fromthe above fit function
Our final purpose in this section is to obtain the decaywidth of the 120594
1198882rarr 119863
119904ℓ] transition The differential decay
width for this transition is obtained as
119889Γ
1198891199022
=
1198662
1198651198812
119888119904
210321198987
1205941198882
12058731199026(1198982
ℓminus 1199022
)
2
Δ101584032
times
10038161003816100381610038161003816119887minus(1199022
)
10038161003816100381610038161003816
2
Δ1015840
1198982
ℓ1199024
+
10038161003816100381610038161003816119887+(1199022
)
10038161003816100381610038161003816
2
times Δ1015840
[(1198982
119863119904
minus 1198982
1205941198882
)
2
1198982
ℓ+ (119898
2
119863119904
minus 1198982
1205941198882
)
2
1199022
minus 2 (1198982
119863119904
+ 1198982
1205941198882
) 1199024
+ 1199026
]
Table 1 Parameters appearing in the fit function of the form factors
1198910
1198881
1198882
1198982
fit
119870(1199022
) minus(109 plusmn 038) 290 minus068 1265119887minus(1199022
) 205 plusmn 072GeVminus2 701 2186 1265119887+(1199022
) 242 plusmn 085GeVminus2 563 12951 1265ℎ(1199022
) minus(442 plusmn 155) times 10minus9 GeVminus2 minus191 111 1265
Table 2 Numerical results of decay width for different leptonchannels
Γ (GeV)1205941198882
rarr 119863119904120583]120583
(160 plusmn 067) times 10minus11
1205941198882
rarr 119863119904119890]119890
(162 plusmn 068) times 10minus11
+ 2Re [119870 (1199022
) 119887lowast
+(1199022
)]
times Δ1015840
[minus1199024
+ 1198982
119863119904
(1198982
ℓ+ 1199022
) minus 1198982
1205941198882
(1198982
ℓ+ 1199022
)]
minus 2Re [119887minus(1199022
) 119887lowast
+(1199022
)] Δ1015840
1198982
ℓ1199022
(1198982
119863119904
minus 1198982
1205941198882
)
+
10038161003816100381610038161003816119870 (119902
2
)
10038161003816100381610038161003816
2
times [1198984
119863119904
(1198982
ℓ+ 1199022
) + 1198984
1205941198882
(1198982
ℓ+ 1199022
)
+ 1199024
(1198982
ℓ+ 1199022
) minus 21198982
119863119904
times (1198982
1205941198882
+ 1199022
) (1198982
ℓ+ 1199022
) + 1198982
1205941198882
1199022
(1198982
ℓ+ 5119902
2
)]
+ 3
10038161003816100381610038161003816ℎ (119902
2
)
10038161003816100381610038161003816
2
Δ1015840
1198982
1205941198882
1199022
(1198982
ℓ+ 1199022
)
minus2Re [119870 (1199022
) 119887lowast
minus(1199022
)] Δ1015840
1198982
ℓ1199022
(23)
After performing integration over 1199022 in (23) in the interval
1198982
ℓle 119902
2
le (1198981205941198882
minus 119898119863119904
)2 we obtain the decay widths in
both 119890 and 120583 channels as presented in Table 2 Consideringthe developments in experimental side we hope that it willbe possible to study such decay channels in the experimentin near future Comparison of future data with theoreticalcalculations will help us get useful information on thestructure of 120594
1198882tensor meson as well as the perturbative and
nonperturbative aspects of QCDThe obtained results in thiswork can also be used in the analysis of the 119861
119888meson decay
at LHC as the 119861119888
rarr 1205941198882is expected to have a considerable
contribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Abilikim M N Achasov X C Ai et al ldquoMeasurement of120594119888119869decaying into 120578
1015840K+Kminusrdquo Physical Review D vol 89 ArticleID 074030 2014
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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FluidsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in Condensed Matter Physics
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 3
where 1198911205941198882
and 119891119863119904
are leptonic decay constants of 1205941198882and119863
119904
mesons respectively Combining all matrix elements given in(2) and (9) in (8) the final representation of the correlationfunction for the hadronic side is obtained as
ΠHAD120583120572120573
(119901 1199011015840
119902)
=
1198911205941198882
119891119863119904
1198981205941198882
1198982
119863119904
8 (119898119888+ 119898
119904) (119901
10158402minus 119898
2
119863119904
) (1199012minus 119898
2
1205941198882
)
times
2
3
[minusΔ119870(1199022
) minus Δ1015840
119887minus(1199022
)] 119902120583119892120573120572
+
2
3
[(minusΔ + 41198982
1205941198882
)119870 (1199022
) minus Δ1015840
119887+(1199022
)] 119875120583119892120573120572
minus 119894 (Δ minus 41198982
1205941198882
) ℎ (1199022
) 120576120582]120573120583119875120582119875120572119902]
+Δ119870(1199022
) 119902120572119892120573120583
+ other structures + sdot sdot sdot
(10)
where
Δ = 1198982
119863119904
+ 31198982
1205941198882
minus 1199022
Δ1015840
= 1198984
119863119904
minus 21198982
119863119904
(1198982
1205941198882
+ 1199022
) + (1198982
1205941198882
minus 1199022
)
2
(11)
and we have held only the structures which we are goingto choose in order to find the corresponding form factorsNote that for obtaining the above representation we haveperformed summation over the polarization tensor using
sum
120582
120576120582
120583]120576lowast120582
120572120573=
1
2
120578120583120572
120578]120573 +1
2
120578120583120573
120578]120572 minus1
3
120578120583]120578120572120573 (12)
where
120578120583] = minus119892
120583] +119901120583119901]
1198982
1205941198882
(13)
In OPE side the correlation function is calculated indeep Euclidean region (see for instance [10 11]) Placingthe explicit forms of the interpolating currents into thecorrelation function and contracting out all quark pairs viaWickrsquos theorem we obtain
ΠOPE120583120572120573
(119901 1199011015840
119902)
=
1198943
4
int1198894
119909
times int1198894
119910119890minus119894119901sdot119909
1198901198941199011015840sdot119910
times Tr [1205745119878119896119895
119904(119910) 120574
120583(1 minus 120574
5) 119878119895119894
119888(minus119909)
times 120574120572
harr
D120573 (119909) 119878119894119896
119888(119909 minus 119910)] + [120573 larrrarr 120572]
(14)
where 119878119888and 119878
119902are the heavy and light quark propagator
respectively They are given by [12]
119878119894119895
119888(119909)
=
119894
(2120587)4
times int1198894
119896119890minus119894119896sdot119909
times
120575119894119895
119896 minus 119898119888
minus
119892119904119866120572120573
119894119895
4
120590120572120573
(119896 + 119898119888) + (119896 + 119898
119888) 120590120572120573
(1198962minus 119898
2
119888)2
+
1205872
3
⟨
120572119904119866119866
120587
⟩120575119894119895119898119888
1198962
+ 119898119888119896
(1198962minus 119898
2
119888)4+ sdot sdot sdot
119878119894119895
119904(119909) = 119894
119909212058721199094120575119894119895minus
119898119904
412058721199092120575119894119895minus
⟨119904119904⟩
12
(1 minus 119894
119898119904
4119909) 120575
119894119895
minus
1199092
192
1198982
0⟨119904119904⟩ (1 minus 119894
119898119904
6119909) 120575
119894119895
minus 119894
119894119892119904
3212058721199092119866119894119895
120583] (119909120590120583]
+ 120590120583]
119909) + sdot sdot sdot
(15)
Despite being very small compared to the perturbativepart we include the contribution coming from the gluoncondensate terms as nonperturbative effects The correlationfunction in OPE side is also written as
ΠOPE120583120572120573
(119901 1199011015840
119902) = (Πpert1
(1199022
) + Πnon-pert1
(1199022
)) 119902120572119892120573120583
+ (Πpert2
(1199022
) + Πnon-pert2
(1199022
)) 119902120583119892120573120572
+ (Πpert3
(1199022
) + Πnon-pert3
(1199022
)) 119875120583119892120573120572
+ (Πpert4
(1199022
) + Πnon-pert4
(1199022
)) 120576120582]120573120583119875120582119875120572119902]
+ other structures(16)
whereΠpert119894
(1199022
)with 119894 = 1 2 3 4 are the perturbative parts ofthe coefficients of the selected structures They are expressedin terms of double dispersion integrals as
Πpert119894
(1199022
) = int119889119904int1198891199041015840
120588119894(119904 119904
1015840
1199022
)
(119904 minus 1199012) (1199041015840minus 11990110158402)
+ subtracted terms
(17)
where the spectral densities 120588119894(119904 119904
1015840
1199022
) are obtained bytaking the imaginary parts of the Π
pert119894
functions that is120588119894(119904 119904
1015840
1199022
) = (1120587) Im[Πpert119894
] Replacing the explicit expres-sions of the above propagators into (14) and performing
4 Advances in High Energy Physics
integrals over four 119909 and 119910 we find the spectral densitiescorresponding to four different Dirac structures as
1205881(119904 119904
1015840
1199022
) =
3
321205872int
1
0
119889119909int
1minus119909
0
119889119910 [119898119904(4119909 + 2119910 minus 3)
+119898119888(8119909 + 4119910 minus 5)]
1205882(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909
times int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904(4119909 + 2119910 minus 3)]
1205883(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904+ 2119898
119904119910]
1205884(119904 119904
1015840
1199022
) = 0
(18)
From a similar way we also calculate the functionsΠ
non-pert119894
(1199022
)The QCD sum rules for the form factors are obtained
by matching the coefficients of the same structures fromboth sides of the correlation function To suppress thecontributions of the higher states and continuum the doubleBorel transformation with respect to quantities 1199012 and 119901
10158402 isapplied to both sides of the obtained sum rules according tothe following rule
B1198722B11987210158402
1
(1199012minus 119898
2
1)119886
1
(11990110158402
minus 1198982
2)119887
997888rarr (minus1)119886+119887
(1198722
)
119886minus1
(11987210158402
)
119887minus1
Γ (119886) Γ (119887)
119890minus1198982
11198722
119890minus1198982
211987210158402
(19)
where 1198722 and 119872
10158402 are the Borel mass parameters Tofurther suppress the contributions of the higher state andcontinuum we perform continuum subtraction and use thequark-hadron duality assumption As a result we get thefollowing sum rules for the form factors
119870(1199022
)
=
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205881(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert1
119887minus(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205882(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert2
minus
Δ
Δ1015840119870(119902
2
)
119887+(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205883(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert3
+
minusΔ + 41198982
1205941198882
Δ1015840
119870(1199022
)
ℎ (1199022
)
= minus
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
(Δ minus 41198982
1205941198882
)
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205884(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert4
(20)
where 1199040and 119904
1015840
0are continuum thresholds in the initial and
final channels respectively The function 119871(119904 1199041015840
1199022
) is givenby
119871 (119904 1199041015840
1199022
) = 1199041015840
119910 (1 minus 119909 minus 119910) minus 1198982
119888(119909 + 119910)
+ 1199022
119909 (1 minus 119909 minus 119910) + 119904119909119910
(21)
Advances in High Energy Physics 5
8 9 10 11 12
8 9 10 11 12K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
Total contributionPerturbative contribution
M2(GeV2
)
Nonperturbative contribution
(a)K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
5 6 7 8
5 6 7 8
Total contributionPerturbative contributionNonperturbative contribution
M9984002
(GeV2)
(b)
Figure 1 (a) 119870(1199022
= 0) as a function of the Borel mass parameter 1198722 at 11987210158402
= 65GeV2 (b) 119870(1199022
= 0) as a function of the Borel massparameter11987210158402 at1198722
= 10GeV2
The functions B1198722B11987210158402Π
non-pert119894
are very lengthy hence wedo not present their explicit expressions here We shouldstress that the contributions of the light quark condensatesare eliminated by applying the double Borel transformationswith respect to the initial and final momenta hence in theB1198722B11987210158402Π
non-pert119894
functions we only consider the two-gluoncondensate contributions
3 Numerical Results and Discussion
In this section we present our numerical results for the formfactors of the semileptonic 120594
1198882rarr 119863
119904ℓ] transition whose
sum rules have been found in the previous section For thisaim we use the following input parameters 119898
119888= (1275 plusmn
0025)GeV 119898119904= 95 plusmn 5MeV [7] 119866
119865= 117 times 10
minus5 GeVminus2⟨1205721199041198662
120587⟩ = (0012 plusmn 0004)GeV4 ⟨119904119904(1GeV)⟩ = minus08(024 plusmn
001)3 GeV3 [13 14] 1198982
0(1GeV) = (08 plusmn 02)GeV2 [15 16]
119891119863119904
= 245 plusmn 157 plusmn 45MeV [17] and 1198911205941198882
= 00111 plusmn 00062
[18]The sum rules for the form factors also contain four
auxiliary parameters two Borelmass parameters1198722 and11987210158402
as well as two continuum thresholds 1199040and 119904
1015840
0 According
to the criteria of the method the physical quantities suchas form factors should be independent of these parametersHence we will look for regions such that the dependenceof form factors on these helping parameters is weak Thecontinuum thresholds 119904
0and 119904
1015840
0are not totally arbitrary but
they depend on the energy of the first excited states withthe same quantum numbers as the interpolating currents ofthe initial and final channels respectively Our numericalcalculations reveal that in the intervals 13GeV2 le 119904
0le
15GeV2 and 45GeV2 le 1199041015840
0le 6GeV2 the form factors
demonstrate weak dependence on continuum thresholdsThe working region for the Borel mass parameters is
determined with the requirements that not only the higherstates and continuum contributions are suppressed but alsothe contributions of the higher order operators are small thatis the sum rules are convergent From these restrictions theworking regions for the Borel parameters are found to be8GeV2 le 119872
2
le 12GeV2 and 5GeV2 le 11987210158402
le 8GeV2Note that the above regions for the continuum thresholds areobtained according to the standard criteria of the QCD sumrules that is the continuum thresholds are independent ofBorel mass parameters However some recent works (see forinstance [19]) show that the standard criteria do not lead tocorrect results and the continuum thresholds should be takenas functions of Borel masses Following [19] we will add anextra 15 systematic error to the uncertainties of the formfactors
As an example we depict the dependence of form factor119870(119902
2
) on 1198722 and 119872
10158402 in Figure 1 at 1199022
= 0 From thisfigure we observe that the form factor 119870(119902
2
) demonstratesa good stability with respect to the variations of Borel massparameters in their working regions It is also clear that theperturbative part exceeds the nonperturbative one substan-tially and constitutes approximately the whole contribution
Having determined the working regions for the auxiliaryparameters we proceed to find the behaviors of the formfactors in terms of 119902
2 The sum rules for the form factorsare truncated at some points below the perturbative cut soto extend our results to the full physical region we look fora parameterization of the form factors such that its resultscoincide with the results of sum rules at reliable region Our
6 Advances in High Energy Physics
minus25
minus20
minus15
minus10
minus05
minus25
minus20
minus15
minus10
minus05
00 05 10 15 20 25
00 05 10 15 20 25
QCD sum ruleFit function
K(q2)
q2(GeV2
)
Figure 2 119870(1199022
) as a function of 1199022 at 1198722
= 10GeV2 and 11987210158402
=
65GeV2
analysis shows that the form factors are well fitted to thefunction
119891 (1199022
) = 1198910exp[119888
1
1199022
1198982
fit+ 1198882(
1199022
1198982
fit)
2
] (22)
where the values of the parameters 1198910 1198881 1198882 and 119898
2
fitobtained using 119872
2
= 10GeV2 and 11987210158402
= 65GeV2 for1205941198882
rarr 119863119904ℓ] transition are presented in Table 1 The
errors appearing in the results belong to the uncertaintiesin the input parameters those coming from determinationof the working regions of the auxiliary parameters and thepreviously discussed systematic uncertainties As an examplewe depict the dependence of the form factor 119870(119902
2
) on 1199022 at
1198722
= 10GeV2 and11987210158402
= 65GeV2 in Figure 2 which showsa good fitting of the sum rules results to those obtained fromthe above fit function
Our final purpose in this section is to obtain the decaywidth of the 120594
1198882rarr 119863
119904ℓ] transition The differential decay
width for this transition is obtained as
119889Γ
1198891199022
=
1198662
1198651198812
119888119904
210321198987
1205941198882
12058731199026(1198982
ℓminus 1199022
)
2
Δ101584032
times
10038161003816100381610038161003816119887minus(1199022
)
10038161003816100381610038161003816
2
Δ1015840
1198982
ℓ1199024
+
10038161003816100381610038161003816119887+(1199022
)
10038161003816100381610038161003816
2
times Δ1015840
[(1198982
119863119904
minus 1198982
1205941198882
)
2
1198982
ℓ+ (119898
2
119863119904
minus 1198982
1205941198882
)
2
1199022
minus 2 (1198982
119863119904
+ 1198982
1205941198882
) 1199024
+ 1199026
]
Table 1 Parameters appearing in the fit function of the form factors
1198910
1198881
1198882
1198982
fit
119870(1199022
) minus(109 plusmn 038) 290 minus068 1265119887minus(1199022
) 205 plusmn 072GeVminus2 701 2186 1265119887+(1199022
) 242 plusmn 085GeVminus2 563 12951 1265ℎ(1199022
) minus(442 plusmn 155) times 10minus9 GeVminus2 minus191 111 1265
Table 2 Numerical results of decay width for different leptonchannels
Γ (GeV)1205941198882
rarr 119863119904120583]120583
(160 plusmn 067) times 10minus11
1205941198882
rarr 119863119904119890]119890
(162 plusmn 068) times 10minus11
+ 2Re [119870 (1199022
) 119887lowast
+(1199022
)]
times Δ1015840
[minus1199024
+ 1198982
119863119904
(1198982
ℓ+ 1199022
) minus 1198982
1205941198882
(1198982
ℓ+ 1199022
)]
minus 2Re [119887minus(1199022
) 119887lowast
+(1199022
)] Δ1015840
1198982
ℓ1199022
(1198982
119863119904
minus 1198982
1205941198882
)
+
10038161003816100381610038161003816119870 (119902
2
)
10038161003816100381610038161003816
2
times [1198984
119863119904
(1198982
ℓ+ 1199022
) + 1198984
1205941198882
(1198982
ℓ+ 1199022
)
+ 1199024
(1198982
ℓ+ 1199022
) minus 21198982
119863119904
times (1198982
1205941198882
+ 1199022
) (1198982
ℓ+ 1199022
) + 1198982
1205941198882
1199022
(1198982
ℓ+ 5119902
2
)]
+ 3
10038161003816100381610038161003816ℎ (119902
2
)
10038161003816100381610038161003816
2
Δ1015840
1198982
1205941198882
1199022
(1198982
ℓ+ 1199022
)
minus2Re [119870 (1199022
) 119887lowast
minus(1199022
)] Δ1015840
1198982
ℓ1199022
(23)
After performing integration over 1199022 in (23) in the interval
1198982
ℓle 119902
2
le (1198981205941198882
minus 119898119863119904
)2 we obtain the decay widths in
both 119890 and 120583 channels as presented in Table 2 Consideringthe developments in experimental side we hope that it willbe possible to study such decay channels in the experimentin near future Comparison of future data with theoreticalcalculations will help us get useful information on thestructure of 120594
1198882tensor meson as well as the perturbative and
nonperturbative aspects of QCDThe obtained results in thiswork can also be used in the analysis of the 119861
119888meson decay
at LHC as the 119861119888
rarr 1205941198882is expected to have a considerable
contribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Abilikim M N Achasov X C Ai et al ldquoMeasurement of120594119888119869decaying into 120578
1015840K+Kminusrdquo Physical Review D vol 89 ArticleID 074030 2014
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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FluidsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in Condensed Matter Physics
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AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
integrals over four 119909 and 119910 we find the spectral densitiescorresponding to four different Dirac structures as
1205881(119904 119904
1015840
1199022
) =
3
321205872int
1
0
119889119909int
1minus119909
0
119889119910 [119898119904(4119909 + 2119910 minus 3)
+119898119888(8119909 + 4119910 minus 5)]
1205882(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909
times int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904(4119909 + 2119910 minus 3)]
1205883(119904 119904
1015840
1199022
) =
3
161205872int
1
0
119889119909int
1minus119909
0
119889119910 [minus119898119888minus 119898
119904+ 2119898
119904119910]
1205884(119904 119904
1015840
1199022
) = 0
(18)
From a similar way we also calculate the functionsΠ
non-pert119894
(1199022
)The QCD sum rules for the form factors are obtained
by matching the coefficients of the same structures fromboth sides of the correlation function To suppress thecontributions of the higher states and continuum the doubleBorel transformation with respect to quantities 1199012 and 119901
10158402 isapplied to both sides of the obtained sum rules according tothe following rule
B1198722B11987210158402
1
(1199012minus 119898
2
1)119886
1
(11990110158402
minus 1198982
2)119887
997888rarr (minus1)119886+119887
(1198722
)
119886minus1
(11987210158402
)
119887minus1
Γ (119886) Γ (119887)
119890minus1198982
11198722
119890minus1198982
211987210158402
(19)
where 1198722 and 119872
10158402 are the Borel mass parameters Tofurther suppress the contributions of the higher state andcontinuum we perform continuum subtraction and use thequark-hadron duality assumption As a result we get thefollowing sum rules for the form factors
119870(1199022
)
=
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205881(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert1
119887minus(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205882(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert2
minus
Δ
Δ1015840119870(119902
2
)
119887+(1199022
)
= minus
12 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
Δ10158401198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205883(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert3
+
minusΔ + 41198982
1205941198882
Δ1015840
119870(1199022
)
ℎ (1199022
)
= minus
8 (119898119888+ 119898
119904)
1198911205941198882
119891119863119904
1198982
119863119904
1198981205941198882
(Δ minus 41198982
1205941198882
)
1198901198982
12059411988821198722
1198901198982
11986311990411987210158402
times int
1199040
41198982
119888
119889119904
times int
1199041015840
0
(119898119888+119898119904)2
1198891199041015840
119890minus119904119872
2
119890minus119904101584011987210158402
1205884(119904 119904
1015840
1199022
)
times 120579 [119871 (119904 1199041015840
1199022
)] +B1198722B11987210158402Π
non-pert4
(20)
where 1199040and 119904
1015840
0are continuum thresholds in the initial and
final channels respectively The function 119871(119904 1199041015840
1199022
) is givenby
119871 (119904 1199041015840
1199022
) = 1199041015840
119910 (1 minus 119909 minus 119910) minus 1198982
119888(119909 + 119910)
+ 1199022
119909 (1 minus 119909 minus 119910) + 119904119909119910
(21)
Advances in High Energy Physics 5
8 9 10 11 12
8 9 10 11 12K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
Total contributionPerturbative contribution
M2(GeV2
)
Nonperturbative contribution
(a)K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
5 6 7 8
5 6 7 8
Total contributionPerturbative contributionNonperturbative contribution
M9984002
(GeV2)
(b)
Figure 1 (a) 119870(1199022
= 0) as a function of the Borel mass parameter 1198722 at 11987210158402
= 65GeV2 (b) 119870(1199022
= 0) as a function of the Borel massparameter11987210158402 at1198722
= 10GeV2
The functions B1198722B11987210158402Π
non-pert119894
are very lengthy hence wedo not present their explicit expressions here We shouldstress that the contributions of the light quark condensatesare eliminated by applying the double Borel transformationswith respect to the initial and final momenta hence in theB1198722B11987210158402Π
non-pert119894
functions we only consider the two-gluoncondensate contributions
3 Numerical Results and Discussion
In this section we present our numerical results for the formfactors of the semileptonic 120594
1198882rarr 119863
119904ℓ] transition whose
sum rules have been found in the previous section For thisaim we use the following input parameters 119898
119888= (1275 plusmn
0025)GeV 119898119904= 95 plusmn 5MeV [7] 119866
119865= 117 times 10
minus5 GeVminus2⟨1205721199041198662
120587⟩ = (0012 plusmn 0004)GeV4 ⟨119904119904(1GeV)⟩ = minus08(024 plusmn
001)3 GeV3 [13 14] 1198982
0(1GeV) = (08 plusmn 02)GeV2 [15 16]
119891119863119904
= 245 plusmn 157 plusmn 45MeV [17] and 1198911205941198882
= 00111 plusmn 00062
[18]The sum rules for the form factors also contain four
auxiliary parameters two Borelmass parameters1198722 and11987210158402
as well as two continuum thresholds 1199040and 119904
1015840
0 According
to the criteria of the method the physical quantities suchas form factors should be independent of these parametersHence we will look for regions such that the dependenceof form factors on these helping parameters is weak Thecontinuum thresholds 119904
0and 119904
1015840
0are not totally arbitrary but
they depend on the energy of the first excited states withthe same quantum numbers as the interpolating currents ofthe initial and final channels respectively Our numericalcalculations reveal that in the intervals 13GeV2 le 119904
0le
15GeV2 and 45GeV2 le 1199041015840
0le 6GeV2 the form factors
demonstrate weak dependence on continuum thresholdsThe working region for the Borel mass parameters is
determined with the requirements that not only the higherstates and continuum contributions are suppressed but alsothe contributions of the higher order operators are small thatis the sum rules are convergent From these restrictions theworking regions for the Borel parameters are found to be8GeV2 le 119872
2
le 12GeV2 and 5GeV2 le 11987210158402
le 8GeV2Note that the above regions for the continuum thresholds areobtained according to the standard criteria of the QCD sumrules that is the continuum thresholds are independent ofBorel mass parameters However some recent works (see forinstance [19]) show that the standard criteria do not lead tocorrect results and the continuum thresholds should be takenas functions of Borel masses Following [19] we will add anextra 15 systematic error to the uncertainties of the formfactors
As an example we depict the dependence of form factor119870(119902
2
) on 1198722 and 119872
10158402 in Figure 1 at 1199022
= 0 From thisfigure we observe that the form factor 119870(119902
2
) demonstratesa good stability with respect to the variations of Borel massparameters in their working regions It is also clear that theperturbative part exceeds the nonperturbative one substan-tially and constitutes approximately the whole contribution
Having determined the working regions for the auxiliaryparameters we proceed to find the behaviors of the formfactors in terms of 119902
2 The sum rules for the form factorsare truncated at some points below the perturbative cut soto extend our results to the full physical region we look fora parameterization of the form factors such that its resultscoincide with the results of sum rules at reliable region Our
6 Advances in High Energy Physics
minus25
minus20
minus15
minus10
minus05
minus25
minus20
minus15
minus10
minus05
00 05 10 15 20 25
00 05 10 15 20 25
QCD sum ruleFit function
K(q2)
q2(GeV2
)
Figure 2 119870(1199022
) as a function of 1199022 at 1198722
= 10GeV2 and 11987210158402
=
65GeV2
analysis shows that the form factors are well fitted to thefunction
119891 (1199022
) = 1198910exp[119888
1
1199022
1198982
fit+ 1198882(
1199022
1198982
fit)
2
] (22)
where the values of the parameters 1198910 1198881 1198882 and 119898
2
fitobtained using 119872
2
= 10GeV2 and 11987210158402
= 65GeV2 for1205941198882
rarr 119863119904ℓ] transition are presented in Table 1 The
errors appearing in the results belong to the uncertaintiesin the input parameters those coming from determinationof the working regions of the auxiliary parameters and thepreviously discussed systematic uncertainties As an examplewe depict the dependence of the form factor 119870(119902
2
) on 1199022 at
1198722
= 10GeV2 and11987210158402
= 65GeV2 in Figure 2 which showsa good fitting of the sum rules results to those obtained fromthe above fit function
Our final purpose in this section is to obtain the decaywidth of the 120594
1198882rarr 119863
119904ℓ] transition The differential decay
width for this transition is obtained as
119889Γ
1198891199022
=
1198662
1198651198812
119888119904
210321198987
1205941198882
12058731199026(1198982
ℓminus 1199022
)
2
Δ101584032
times
10038161003816100381610038161003816119887minus(1199022
)
10038161003816100381610038161003816
2
Δ1015840
1198982
ℓ1199024
+
10038161003816100381610038161003816119887+(1199022
)
10038161003816100381610038161003816
2
times Δ1015840
[(1198982
119863119904
minus 1198982
1205941198882
)
2
1198982
ℓ+ (119898
2
119863119904
minus 1198982
1205941198882
)
2
1199022
minus 2 (1198982
119863119904
+ 1198982
1205941198882
) 1199024
+ 1199026
]
Table 1 Parameters appearing in the fit function of the form factors
1198910
1198881
1198882
1198982
fit
119870(1199022
) minus(109 plusmn 038) 290 minus068 1265119887minus(1199022
) 205 plusmn 072GeVminus2 701 2186 1265119887+(1199022
) 242 plusmn 085GeVminus2 563 12951 1265ℎ(1199022
) minus(442 plusmn 155) times 10minus9 GeVminus2 minus191 111 1265
Table 2 Numerical results of decay width for different leptonchannels
Γ (GeV)1205941198882
rarr 119863119904120583]120583
(160 plusmn 067) times 10minus11
1205941198882
rarr 119863119904119890]119890
(162 plusmn 068) times 10minus11
+ 2Re [119870 (1199022
) 119887lowast
+(1199022
)]
times Δ1015840
[minus1199024
+ 1198982
119863119904
(1198982
ℓ+ 1199022
) minus 1198982
1205941198882
(1198982
ℓ+ 1199022
)]
minus 2Re [119887minus(1199022
) 119887lowast
+(1199022
)] Δ1015840
1198982
ℓ1199022
(1198982
119863119904
minus 1198982
1205941198882
)
+
10038161003816100381610038161003816119870 (119902
2
)
10038161003816100381610038161003816
2
times [1198984
119863119904
(1198982
ℓ+ 1199022
) + 1198984
1205941198882
(1198982
ℓ+ 1199022
)
+ 1199024
(1198982
ℓ+ 1199022
) minus 21198982
119863119904
times (1198982
1205941198882
+ 1199022
) (1198982
ℓ+ 1199022
) + 1198982
1205941198882
1199022
(1198982
ℓ+ 5119902
2
)]
+ 3
10038161003816100381610038161003816ℎ (119902
2
)
10038161003816100381610038161003816
2
Δ1015840
1198982
1205941198882
1199022
(1198982
ℓ+ 1199022
)
minus2Re [119870 (1199022
) 119887lowast
minus(1199022
)] Δ1015840
1198982
ℓ1199022
(23)
After performing integration over 1199022 in (23) in the interval
1198982
ℓle 119902
2
le (1198981205941198882
minus 119898119863119904
)2 we obtain the decay widths in
both 119890 and 120583 channels as presented in Table 2 Consideringthe developments in experimental side we hope that it willbe possible to study such decay channels in the experimentin near future Comparison of future data with theoreticalcalculations will help us get useful information on thestructure of 120594
1198882tensor meson as well as the perturbative and
nonperturbative aspects of QCDThe obtained results in thiswork can also be used in the analysis of the 119861
119888meson decay
at LHC as the 119861119888
rarr 1205941198882is expected to have a considerable
contribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Abilikim M N Achasov X C Ai et al ldquoMeasurement of120594119888119869decaying into 120578
1015840K+Kminusrdquo Physical Review D vol 89 ArticleID 074030 2014
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
8 9 10 11 12
8 9 10 11 12K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
Total contributionPerturbative contribution
M2(GeV2
)
Nonperturbative contribution
(a)K(q2=0)
00
minus04
minus08
minus12
00
minus04
minus08
minus12
5 6 7 8
5 6 7 8
Total contributionPerturbative contributionNonperturbative contribution
M9984002
(GeV2)
(b)
Figure 1 (a) 119870(1199022
= 0) as a function of the Borel mass parameter 1198722 at 11987210158402
= 65GeV2 (b) 119870(1199022
= 0) as a function of the Borel massparameter11987210158402 at1198722
= 10GeV2
The functions B1198722B11987210158402Π
non-pert119894
are very lengthy hence wedo not present their explicit expressions here We shouldstress that the contributions of the light quark condensatesare eliminated by applying the double Borel transformationswith respect to the initial and final momenta hence in theB1198722B11987210158402Π
non-pert119894
functions we only consider the two-gluoncondensate contributions
3 Numerical Results and Discussion
In this section we present our numerical results for the formfactors of the semileptonic 120594
1198882rarr 119863
119904ℓ] transition whose
sum rules have been found in the previous section For thisaim we use the following input parameters 119898
119888= (1275 plusmn
0025)GeV 119898119904= 95 plusmn 5MeV [7] 119866
119865= 117 times 10
minus5 GeVminus2⟨1205721199041198662
120587⟩ = (0012 plusmn 0004)GeV4 ⟨119904119904(1GeV)⟩ = minus08(024 plusmn
001)3 GeV3 [13 14] 1198982
0(1GeV) = (08 plusmn 02)GeV2 [15 16]
119891119863119904
= 245 plusmn 157 plusmn 45MeV [17] and 1198911205941198882
= 00111 plusmn 00062
[18]The sum rules for the form factors also contain four
auxiliary parameters two Borelmass parameters1198722 and11987210158402
as well as two continuum thresholds 1199040and 119904
1015840
0 According
to the criteria of the method the physical quantities suchas form factors should be independent of these parametersHence we will look for regions such that the dependenceof form factors on these helping parameters is weak Thecontinuum thresholds 119904
0and 119904
1015840
0are not totally arbitrary but
they depend on the energy of the first excited states withthe same quantum numbers as the interpolating currents ofthe initial and final channels respectively Our numericalcalculations reveal that in the intervals 13GeV2 le 119904
0le
15GeV2 and 45GeV2 le 1199041015840
0le 6GeV2 the form factors
demonstrate weak dependence on continuum thresholdsThe working region for the Borel mass parameters is
determined with the requirements that not only the higherstates and continuum contributions are suppressed but alsothe contributions of the higher order operators are small thatis the sum rules are convergent From these restrictions theworking regions for the Borel parameters are found to be8GeV2 le 119872
2
le 12GeV2 and 5GeV2 le 11987210158402
le 8GeV2Note that the above regions for the continuum thresholds areobtained according to the standard criteria of the QCD sumrules that is the continuum thresholds are independent ofBorel mass parameters However some recent works (see forinstance [19]) show that the standard criteria do not lead tocorrect results and the continuum thresholds should be takenas functions of Borel masses Following [19] we will add anextra 15 systematic error to the uncertainties of the formfactors
As an example we depict the dependence of form factor119870(119902
2
) on 1198722 and 119872
10158402 in Figure 1 at 1199022
= 0 From thisfigure we observe that the form factor 119870(119902
2
) demonstratesa good stability with respect to the variations of Borel massparameters in their working regions It is also clear that theperturbative part exceeds the nonperturbative one substan-tially and constitutes approximately the whole contribution
Having determined the working regions for the auxiliaryparameters we proceed to find the behaviors of the formfactors in terms of 119902
2 The sum rules for the form factorsare truncated at some points below the perturbative cut soto extend our results to the full physical region we look fora parameterization of the form factors such that its resultscoincide with the results of sum rules at reliable region Our
6 Advances in High Energy Physics
minus25
minus20
minus15
minus10
minus05
minus25
minus20
minus15
minus10
minus05
00 05 10 15 20 25
00 05 10 15 20 25
QCD sum ruleFit function
K(q2)
q2(GeV2
)
Figure 2 119870(1199022
) as a function of 1199022 at 1198722
= 10GeV2 and 11987210158402
=
65GeV2
analysis shows that the form factors are well fitted to thefunction
119891 (1199022
) = 1198910exp[119888
1
1199022
1198982
fit+ 1198882(
1199022
1198982
fit)
2
] (22)
where the values of the parameters 1198910 1198881 1198882 and 119898
2
fitobtained using 119872
2
= 10GeV2 and 11987210158402
= 65GeV2 for1205941198882
rarr 119863119904ℓ] transition are presented in Table 1 The
errors appearing in the results belong to the uncertaintiesin the input parameters those coming from determinationof the working regions of the auxiliary parameters and thepreviously discussed systematic uncertainties As an examplewe depict the dependence of the form factor 119870(119902
2
) on 1199022 at
1198722
= 10GeV2 and11987210158402
= 65GeV2 in Figure 2 which showsa good fitting of the sum rules results to those obtained fromthe above fit function
Our final purpose in this section is to obtain the decaywidth of the 120594
1198882rarr 119863
119904ℓ] transition The differential decay
width for this transition is obtained as
119889Γ
1198891199022
=
1198662
1198651198812
119888119904
210321198987
1205941198882
12058731199026(1198982
ℓminus 1199022
)
2
Δ101584032
times
10038161003816100381610038161003816119887minus(1199022
)
10038161003816100381610038161003816
2
Δ1015840
1198982
ℓ1199024
+
10038161003816100381610038161003816119887+(1199022
)
10038161003816100381610038161003816
2
times Δ1015840
[(1198982
119863119904
minus 1198982
1205941198882
)
2
1198982
ℓ+ (119898
2
119863119904
minus 1198982
1205941198882
)
2
1199022
minus 2 (1198982
119863119904
+ 1198982
1205941198882
) 1199024
+ 1199026
]
Table 1 Parameters appearing in the fit function of the form factors
1198910
1198881
1198882
1198982
fit
119870(1199022
) minus(109 plusmn 038) 290 minus068 1265119887minus(1199022
) 205 plusmn 072GeVminus2 701 2186 1265119887+(1199022
) 242 plusmn 085GeVminus2 563 12951 1265ℎ(1199022
) minus(442 plusmn 155) times 10minus9 GeVminus2 minus191 111 1265
Table 2 Numerical results of decay width for different leptonchannels
Γ (GeV)1205941198882
rarr 119863119904120583]120583
(160 plusmn 067) times 10minus11
1205941198882
rarr 119863119904119890]119890
(162 plusmn 068) times 10minus11
+ 2Re [119870 (1199022
) 119887lowast
+(1199022
)]
times Δ1015840
[minus1199024
+ 1198982
119863119904
(1198982
ℓ+ 1199022
) minus 1198982
1205941198882
(1198982
ℓ+ 1199022
)]
minus 2Re [119887minus(1199022
) 119887lowast
+(1199022
)] Δ1015840
1198982
ℓ1199022
(1198982
119863119904
minus 1198982
1205941198882
)
+
10038161003816100381610038161003816119870 (119902
2
)
10038161003816100381610038161003816
2
times [1198984
119863119904
(1198982
ℓ+ 1199022
) + 1198984
1205941198882
(1198982
ℓ+ 1199022
)
+ 1199024
(1198982
ℓ+ 1199022
) minus 21198982
119863119904
times (1198982
1205941198882
+ 1199022
) (1198982
ℓ+ 1199022
) + 1198982
1205941198882
1199022
(1198982
ℓ+ 5119902
2
)]
+ 3
10038161003816100381610038161003816ℎ (119902
2
)
10038161003816100381610038161003816
2
Δ1015840
1198982
1205941198882
1199022
(1198982
ℓ+ 1199022
)
minus2Re [119870 (1199022
) 119887lowast
minus(1199022
)] Δ1015840
1198982
ℓ1199022
(23)
After performing integration over 1199022 in (23) in the interval
1198982
ℓle 119902
2
le (1198981205941198882
minus 119898119863119904
)2 we obtain the decay widths in
both 119890 and 120583 channels as presented in Table 2 Consideringthe developments in experimental side we hope that it willbe possible to study such decay channels in the experimentin near future Comparison of future data with theoreticalcalculations will help us get useful information on thestructure of 120594
1198882tensor meson as well as the perturbative and
nonperturbative aspects of QCDThe obtained results in thiswork can also be used in the analysis of the 119861
119888meson decay
at LHC as the 119861119888
rarr 1205941198882is expected to have a considerable
contribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Abilikim M N Achasov X C Ai et al ldquoMeasurement of120594119888119869decaying into 120578
1015840K+Kminusrdquo Physical Review D vol 89 ArticleID 074030 2014
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
minus25
minus20
minus15
minus10
minus05
minus25
minus20
minus15
minus10
minus05
00 05 10 15 20 25
00 05 10 15 20 25
QCD sum ruleFit function
K(q2)
q2(GeV2
)
Figure 2 119870(1199022
) as a function of 1199022 at 1198722
= 10GeV2 and 11987210158402
=
65GeV2
analysis shows that the form factors are well fitted to thefunction
119891 (1199022
) = 1198910exp[119888
1
1199022
1198982
fit+ 1198882(
1199022
1198982
fit)
2
] (22)
where the values of the parameters 1198910 1198881 1198882 and 119898
2
fitobtained using 119872
2
= 10GeV2 and 11987210158402
= 65GeV2 for1205941198882
rarr 119863119904ℓ] transition are presented in Table 1 The
errors appearing in the results belong to the uncertaintiesin the input parameters those coming from determinationof the working regions of the auxiliary parameters and thepreviously discussed systematic uncertainties As an examplewe depict the dependence of the form factor 119870(119902
2
) on 1199022 at
1198722
= 10GeV2 and11987210158402
= 65GeV2 in Figure 2 which showsa good fitting of the sum rules results to those obtained fromthe above fit function
Our final purpose in this section is to obtain the decaywidth of the 120594
1198882rarr 119863
119904ℓ] transition The differential decay
width for this transition is obtained as
119889Γ
1198891199022
=
1198662
1198651198812
119888119904
210321198987
1205941198882
12058731199026(1198982
ℓminus 1199022
)
2
Δ101584032
times
10038161003816100381610038161003816119887minus(1199022
)
10038161003816100381610038161003816
2
Δ1015840
1198982
ℓ1199024
+
10038161003816100381610038161003816119887+(1199022
)
10038161003816100381610038161003816
2
times Δ1015840
[(1198982
119863119904
minus 1198982
1205941198882
)
2
1198982
ℓ+ (119898
2
119863119904
minus 1198982
1205941198882
)
2
1199022
minus 2 (1198982
119863119904
+ 1198982
1205941198882
) 1199024
+ 1199026
]
Table 1 Parameters appearing in the fit function of the form factors
1198910
1198881
1198882
1198982
fit
119870(1199022
) minus(109 plusmn 038) 290 minus068 1265119887minus(1199022
) 205 plusmn 072GeVminus2 701 2186 1265119887+(1199022
) 242 plusmn 085GeVminus2 563 12951 1265ℎ(1199022
) minus(442 plusmn 155) times 10minus9 GeVminus2 minus191 111 1265
Table 2 Numerical results of decay width for different leptonchannels
Γ (GeV)1205941198882
rarr 119863119904120583]120583
(160 plusmn 067) times 10minus11
1205941198882
rarr 119863119904119890]119890
(162 plusmn 068) times 10minus11
+ 2Re [119870 (1199022
) 119887lowast
+(1199022
)]
times Δ1015840
[minus1199024
+ 1198982
119863119904
(1198982
ℓ+ 1199022
) minus 1198982
1205941198882
(1198982
ℓ+ 1199022
)]
minus 2Re [119887minus(1199022
) 119887lowast
+(1199022
)] Δ1015840
1198982
ℓ1199022
(1198982
119863119904
minus 1198982
1205941198882
)
+
10038161003816100381610038161003816119870 (119902
2
)
10038161003816100381610038161003816
2
times [1198984
119863119904
(1198982
ℓ+ 1199022
) + 1198984
1205941198882
(1198982
ℓ+ 1199022
)
+ 1199024
(1198982
ℓ+ 1199022
) minus 21198982
119863119904
times (1198982
1205941198882
+ 1199022
) (1198982
ℓ+ 1199022
) + 1198982
1205941198882
1199022
(1198982
ℓ+ 5119902
2
)]
+ 3
10038161003816100381610038161003816ℎ (119902
2
)
10038161003816100381610038161003816
2
Δ1015840
1198982
1205941198882
1199022
(1198982
ℓ+ 1199022
)
minus2Re [119870 (1199022
) 119887lowast
minus(1199022
)] Δ1015840
1198982
ℓ1199022
(23)
After performing integration over 1199022 in (23) in the interval
1198982
ℓle 119902
2
le (1198981205941198882
minus 119898119863119904
)2 we obtain the decay widths in
both 119890 and 120583 channels as presented in Table 2 Consideringthe developments in experimental side we hope that it willbe possible to study such decay channels in the experimentin near future Comparison of future data with theoreticalcalculations will help us get useful information on thestructure of 120594
1198882tensor meson as well as the perturbative and
nonperturbative aspects of QCDThe obtained results in thiswork can also be used in the analysis of the 119861
119888meson decay
at LHC as the 119861119888
rarr 1205941198882is expected to have a considerable
contribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Abilikim M N Achasov X C Ai et al ldquoMeasurement of120594119888119869decaying into 120578
1015840K+Kminusrdquo Physical Review D vol 89 ArticleID 074030 2014
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 7
[2] S-K Choi S Olsen K Abe et al ldquoObservation of a near-threshold 120596J120595 mass enhancement in exclusive BrarrK120596J120595decaysrdquo Physical Review Letters vol 94 no 18 Article ID182002 2005
[3] B Aubert R Barate D Boutigny et al ldquoObservation of a broadstructure in the 120587+120587minus119869120595mass spectrum around 426 GeVc2rdquoPhysical Review Letters vol 95 Article ID 142001 2005
[4] S Chen J Fast J W Hinson et al ldquoStudy of 1205941198881and 120594
1198882meson
production in B meson decaysrdquo Physical Review D vol 63Article ID 031102 2001
[5] B I Eisenstein J Ernst G E Gladding et al ldquoExperimentalinvestigation of the two-photon widths of the 120594
1198880and the 120594
1198882
mesonsrdquoPhysical Review Letters vol 87 Article ID 061801 2001[6] K M Ecklund and CLEO Collaboration ldquoTwo-photon widths
of the 120594119888119869states of charmoniumrdquo Physical Review D vol 78
Article ID 091501 2008[7] J Beringer J F Arguin R M Barnett et al ldquoReview of particle
physicsrdquo Physical Review D vol 86 Article ID 010001 2012[8] K Azizi Y Sarac and H Sundu ldquoInvestigation of the TeX
transition via QCD sum rulesrdquo The European Physical JournalC vol 73 p 2638 2013
[9] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979
[10] M C Birse and B Krippa ldquoDetermination of the pion-nucleoncoupling constant from QCD sum rulesrdquo Physics Letters B vol373 no 1ndash3 pp 9ndash15 1996
[11] K Maltman ldquoHigher resonance contamination of 120587NN cou-plings obtained via the three-point function method in QCDsum rulesrdquo Physical Review C vol 57 article 69 1998
[12] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985
[13] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006
[14] B L Ioffe ldquoDetermination of baryon and baryonic masses fromQCD sum rules Strange baryonsrdquo Soviet PhysicsmdashJETP vol 57pp 716ndash721 1982
[15] H G Dosch M Jamin and S Narison ldquoBaryon massesand flavour symmetry breaking of chiral condensatesrdquo PhysicsLetters B vol 220 no 1-2 pp 251ndash257 1989
[16] V M Belyaev and B L Ioffe ldquoDetermination of baryon andbaryonic masses from qcd sum rules strange baryonsrdquo SovietPhysicsmdashJETP vol 57 pp 716ndash721 1983
[17] W Lucha D Melikhov and S Simula ldquoDecay constants ofheavy pseudoscalar mesons from QCD sum rulesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 38 no 10 ArticleID 105002 2011
[18] T M Aliev K Azizi and M Savci ldquoHeavy 120594Q2 tensor mesonsin QCDrdquo Physics Letters B vol 690 no 2 pp 164ndash167 2010
[19] W Lucha D Melikhov and S Simula ldquoEffective continuumthreshold in dispersive sum rulesrdquo Physical Review D vol 79no 9 Article ID 096011 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
