Research Article Thermal Properties of Light Tensor Mesons ...Advances in High Energy Physics In the present work we investigate the properties of light 2 (1320) , 2 (1270) ,and 2

Research ArticleThermal Properties of Light Tensor Mesons via QCD Sum Rules

K Azizi1 A Tuumlrkan2 E Veli Veliev2 and H Sundu2

1Department of Physics Faculty of Arts and Sciences Dogus University Acibadem Kadikoy 34722 Istanbul Turkey2Department of Physics Kocaeli University 41380 Izmit Turkey

Correspondence should be addressed to K Azizi kazizidogusedutr

Received 27 August 2014 Revised 20 February 2015 Accepted 25 February 2015

Academic Editor Kadayam S Viswanathan

Copyright copy 2015 K Azizi 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

The thermal properties of1198912(1270) 119886

2(1320) and119870lowast

2(1430) light tensormesons are investigated in the framework of QCD sum rules

at finite temperature In particular the masses and decay constants of the light tensor mesons are calculated taking into account thenew operators appearing at finite temperature The numerical results show that at the point at which the temperature-dependentcontinuum threshold vanishes the decay constants decrease with amount of (70ndash85) compared to their vacuum values whilethe masses diminish about (60ndash72) depending on the kinds of the mesons under consideration The results obtained at zerotemperature are in good consistency with the experimental data as well as the existing theoretical predictions

1 Introduction

The study of strong interaction at low energies is one ofthe most important problems of the high energy physicsThis can play a crucial role in exploring the structure ofmesons baryons and vacuum properties of strong interac-tion The tensor particles can provide a different perspectivefor understanding the low energy QCD dynamics In recentdecades great efforts have been made both experimentallyand theoretically to investigate the tensor particles in orderto understand their nature and internal structure

The investigation of hadronic properties at finite baryondensity and temperature in QCD also plays an essentialrole in interpretation of the results of heavy-ion collisionexperiments and obtaining the QCD phase diagram TheCompressed BaryonicMatter (CBM) experiment of the FAIRproject at GSI is important for understanding the way ofChiral symmetry realization in the low energy region andconsequently the confinement ofQCDAccording to thermalQCD the hadronic matter undergoes quark gluon-plasmaphase at a critical temperatureThese kinds of phasemay existin the neutron stars and early universe Hence calculation ofthe parameters of hadrons via thermal QCD may provide uswith useful information on these subjects

The restoration of Chiral symmetry at high temperaturerequires the medium modifications of hadronic parameters[1] There are many nonperturbative approaches to hadronphysics The QCD sum rule method [2 3] is one of themost attractive and applicable tools in this respect In thisapproach hadrons are represented by their interpolatingquark currents and the correlation function of these currentsis calculated using the operator product expansion (OPE)The thermal version of this approach is based on some basicassumptions so that the Wilson expansion and the quark-hadron duality approximation remain valid but the vacuumcondensates are replaced by their thermal expectation values[4] At finite temperature the Lorentz invariance is brokenand due to the residual 119874(3) symmetry some new operatorsappear in the Wilson expansion [5ndash7] These operators areexpressed in terms of the four-vector velocity of the mediumand the energy momentum tensor There are numerousworks in the literature on the medium modifications ofparameters of (pseudo)scalar and (axial)vector mesons usingdifferent theoretical approaches for example Chiral model[8] coupled channel approach [9 10] and QCD sum rules[5 6 11ndash17] Recently we applied this method to calculatesome hadronic parameters related to the charmed 119863lowast

2(2460)

and charmed-strange119863lowast1199042(2573) tensor [18] mesons

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 794243 7 pageshttpdxdoiorg1011552015794243

2 Advances in High Energy Physics

In the present work we investigate the properties oflight 119886

2(1320) 119891

2(1270) and 119870lowast

2(1430) tensor mesons in the

framework of QCD sum rules at finite temperature We alsocompare the results obtained at zero temperature with thepredictions of some previous studies on the parameters of thesame mesons in vacuum [19ndash21]

The present paper is organized as follows In the nextsection considering the new operators raised at finite tem-perature we evaluate the corresponding thermal correlationfunction to obtain the QCD sum rules for the parameters ofthe mesons under consideration The last section is devotedto the numerical analysis of the sum rules obtained as wellas investigation of the sensitivity of the masses and decayconstants of the light tensor mesons on temperature

2 Thermal QCD Sum Rules forMasses and Decay Constants ofLight Tensor Mesons

In this section we present the basics of the thermal QCD sumrules and apply this method to some light tensor mesons like1198912(1270) 119886

2(1320) and119870lowast

2(1430) to compute their mass and

decay constantThe starting point is to consider the followingthermal correlation function

Π120583]120572120573 (119902 119879) = 119894 int 1198894119909119890119894119902sdot(119909minus119910)119879119903 120588T [119869

120583] (119909) 119869120572120573 (119910)]

(1)

where 120588 = 119890minus120573119867119879119903(119890minus120573119867) is the thermal density matrix ofQCD 120573 = 1119879 with 119879 being temperature 119867 is the QCDHamiltonianT indicates the time ordered product and 119869

120583] isthe interpolating current of tensor mesonsThe interpolatingfields for these mesons can be written as

119869119870lowast

2

120583] (119909) =119894

2[119904 (119909) 120574

120583

harr

D] (119909) 119889 (119909) + 119904 (119909) 120574]harr

D120583 (119909) 119889 (119909)]

1198691198912120583] (119909) =

119894

2radic2[119906 (119909) 120574

120583

harr

D] (119909) 119906 (119909) + 119906 (119909) 120574]harr

D120583 (119909) 119906 (119909)

+ 119889 (119909) 120574120583

harr

D] (119909) 119889 (119909)

+ 119889 (119909) 120574]harr

D120583 (119909) 119889 (119909)]

1198691198862120583] (119909) =

119894

2radic2[119906 (119909) 120574

120583

harr

D] (119909) 119906 (119909) + 119906 (119909) 120574]harr

D120583 (119909) 119906 (119909)

minus 119889 (119909) 120574120583

harr

D] (119909) 119889 (119909)

minus 119889 (119909) 120574]harr

D120583 (119909) 119889 (119909)]

(2)

whereharr

D120583(119909) denotes the derivative with respect to four-119909simultaneously acting on left and right It is given as

harr

D120583 (119909) =1

2[D120583(119909) minus D

120583(119909)] (3)

where

D120583(119909) =

120583(119909) minus 119894

119892

2120582119886119860119886120583(119909)

D120583(119909) =

120583(119909) + 119894

119892

2120582119886119860119886120583(119909)

(4)

with 120582119886 (119886 = 1 8) and 119860119886120583(119909) being the Gell-Mann matrices

and external gluon fields respectively The currents containderivatives with respect to the space-time hence we considerthe two currents at points 119909 and 119910 in (1) but for simplicitywe will set 119910 = 0 after applying derivative with respect to 119910

It is well known that in thermal QCD sum rule approachthe thermal correlation function can be calculated in twodifferent ways Firstly it is calculated in terms of hadronicparameters such as masses and decay constants Secondlyit is calculated in terms of the QCD parameters such asquark masses quark condensates and quark-gluon couplingconstants The coefficients of sufficient structures from bothrepresentations of the same correlation function are thenequated to find the sum rules for the physical quantitiesunder consideration We apply Borel transformation andcontinuum subtraction to both sides of the sum rules in orderto further suppress the contributions of the higher states andcontinuum

Let us focus on the calculation of the hadronic side of thecorrelation function For this aim we insert a complete set ofintermediate physical states having the same quantum num-bers as the interpolating current into (1) After performingintegral over four-119909 and setting 119910 = 0 we get

Π120583]120572120573 (119902 119879)

=⟨0

10038161003816100381610038161003816119869120583] (0)10038161003816100381610038161003816 119870lowast

2(1198912) (1198862)⟩ ⟨119870lowast

2(1198912) (1198862)10038161003816100381610038161003816119869120572120573 (0)

10038161003816100381610038161003816 0⟩

1198982119870lowast

2(1198912)(1198862)minus 1199022

+ sdot sdot sdot

(5)

where dots indicate the contributions of the higher andcontinuum states The matrix element ⟨0|119869

120583](0)|119870lowast

2(1198912)(1198862)⟩

creating the tensor mesons from vacuum can be written interms of the decay constant 119891

119870lowast

2(1198912)(1198862) as

⟨010038161003816100381610038161003816119869120583] (0)

10038161003816100381610038161003816 119870lowast

2(1198912) (1198862)⟩ = 119891

119870lowast

2(1198912)(1198862)1198983119870lowast

2(1198912)(1198862)120576(120582)120583] (6)

where 120576(120582)120583] is the polarization tensor Using (6) in (5) the final

expression of the physical side is obtained as

Π120583]120572120573 (119902 119879)

=1198912119870lowast

2(1198912)(1198862)1198986119870lowast

2(1198912)(1198862)

1198982119870lowast

2(1198912)(1198862)minus 1199022

1

2(119892120583120572119892]120573 + 119892

120583120573119892]120572)

+ other structures + sdot sdot sdot

(7)

Advances in High Energy Physics 3

where the only structure that we will use in our calculationshas been shown explicitly To obtain the above expression wehave used the summation over polarization tensors as

sum120582

120576(120582)120583] 120576lowast(120582)

120572120573=1

2119879120583120572119879]120573 +

1

2119879120583120573119879]120572 minus

1

3119879120583]119879120572120573 (8)

where

119879120583] = minus119892

120583] +119902120583119902]

1198982119870lowast

2(1198912)(1198862)

(9)

Now we concentrate on the OPE side of the thermalcorrelation function In OPE representation the coefficientof the selected structure can be separated into perturbativeand nonperturbative parts

Π(119902 119879) = Πpert (119902 119879) + Πnon-pert (119902 119879) (10)

The perturbative or short-distance contributions are calcu-lated using the perturbation theory This part in spectralrepresentation is written as

Πpert (119902 119879) = int119889119904120588 (119904)

119904 minus 1199022 (11)

where 120588(119904) is the spectral density and it is given by theimaginary part of the correlation function that is

120588 (119904) =1

120587119868119898 [Πpert

(119904 119879)] (12)

The nonperturbative or long-distance contributions are rep-resented in terms of thermal expectation values of thequark and gluon condensates as well as thermal average ofthe energy density Our main task in the following is tocalculate the spectral density as well as the nonperturbativecontributions For this aim we use the explicit forms of theinterpolating currents for the tensor mesons in (1) Aftercontracting out all quark fields using Wickrsquos theorem we get

Π119870lowast

2

120583]120572120573 (119902 119879)

= minus119894

4int1198894119909119890119894119902sdot(119909minus119910)

sdot 119879119903 [119878119904(119910 minus 119909) 120574

120583

harr

D] (119909)harr

D120573 (119910) 119878119889 (119909 minus 119910) 120574120572]

+ [120573 larrrarr 120572] + [] larrrarr 120583]

+ [120573 larrrarr 120572 ] larrrarr 120583]

Π1198912(1198862)

120583]120572120573 (119902 119879)

= minus119894

8int1198894119909119890119894119902sdot(119909minus119910)

sdot 119879119903 [119878119906(119910 minus 119909) 120574

120583

harr

D] (119909)harr

D120573 (119910) 119878119906 (119909 minus 119910) 120574120572

+ 119878119889(119910 minus 119909) 120574

120583

harr

D] (119909)harr

D120573 (119910) 119878119889 (119909 minus 119910) 120574120572]

+ [120573 larrrarr 120572] + [] larrrarr 120583]

+ [120573 larrrarr 120572 ] larrrarr 120583]

(13)

To proceed we need to know the thermal light quark prop-agator 119878

119902=119906119889119904(119909 minus 119910) in coordinate space which is given as

[18 23]

119878119894119895119902(119909 minus 119910) = 119894 119909 minus 119910

21205872 (119909 minus 119910)4120575119894119895minus

119898119902

41205872 (119909 minus 119910)2120575119894119895minus⟨119902119902⟩

12120575119894119895

minus(119909 minus 119910)

2

19211989820⟨119902119902⟩ [1 minus 119894

119898119902

6(119909 minus 119910)] 120575119894119895

+119894

3[(119909 minus 119910) (

119898119902

16⟨119902119902⟩ minus

1

12⟨119906Θ119891119906⟩)

+1

3(119906 sdot (119909 minus 119910) 119906 ⟨119906Θ

119891119906⟩)] 120575119894119895

minus119894119892119904

321205872 (119909 minus 119910)2119866120583]

sdot ((119909 minus 119910) 120590120583] + 120590120583] (119909 minus 119910)) 120575119894119895

(14)

where ⟨119902119902⟩ is the temperature-dependent quark condensateΘ119891120583] is the fermionic part of the energy momentum tensor

and 119906120583is the four-velocity vector of the heat bath In the rest

frame of the heat bath 119906120583= (1 0 0 0) and 1199062 = 1

The next step is to use the expressions of the propagatorsand apply the derivatives with respect to 119909 and 119910 in (13)After lengthy but straightforward calculations the spectraldensities at different channels are obtained as

120588119870lowast

2

(119904) = 119873119888(119898119889119898119904119904

321205872+

1199042

1601205872)

1205881198912(1198862)(119904) = 119873

119888((1198982119906+ 1198982119889) 119904

961205872+

1199042

1601205872)

(15)

where119873119888= 3 is the number of colors From a similar way for

the nonperturbative contributions we get

Πnon-pert119870lowast

2

(119902 119879) =(6119898119904minus 5119898119889)11989820

1441199022⟨119889119889⟩

+(6119898119889minus 5119898119904)11989820

1441199022⟨119904119904⟩

minus2⟨119906Θ119891119906⟩ (119902 sdot 119906)

2

91199022

Πnon-pert1198912(1198862)

(119902 119879) =11989811988911989820

1441199022⟨119889119889⟩ +

11989811990611989820

1441199022⟨119906119906⟩

minus2 ⟨119906Θ119891119906⟩ (119902 sdot 119906)

2

91199022

(16)

4 Advances in High Energy Physics

m(G

eV)

M2 (GeV2)

f2(1270) s0 = 235GeV2

a2(1320) s0 = 255GeV2

Klowast2 (1430) s0 = 315GeV2

14

12

16

12

16

16 18 20 22 24 26 28 30

14 16 18 20 22 24 26 28 30

(a)

f

0030

0035

0040

0045

0050

0055

0060

0030

0035

0040

0045

0050

0055

0060

M2 (GeV2)

14 16 18 20 22 24 26 28 30

14 16 18 20 22 24 26 28 30

f2(1270) s0 = 235GeV2

a2(1320) s0 = 255GeV2

Klowast2 (1430) s0 = 315GeV2

(b)

Figure 1 Variations of the masses and decay constants of the119870lowast2(1430) 119891

2(1270) and 119886

2(1320)mesons with respect to1198722 at fixed values of

the continuum threshold and at zero temperature

After matching the hadronic and OPE representationsapplying Borel transformation with respect to 1199022 and per-forming continuum subtraction we obtain the followingtemperature-dependent sum rule

1198912119870lowast

2(1198912)(1198862)(119879)119898

6

119870lowast

2(1198912)(1198862)(119879) exp[

minus1198982119870lowast

2(1198912)(1198862)(119879)

1198722]

= int1199040(119879)

(119898119902+119898119889)2

119889119904 120588119870lowast

2(1198912)(1198862)(119904) exp [ minus119904

1198722]

+ BΠnon-pert119870lowast

2(1198912)(1198862)(119902 119879)

(17)

where B denotes the Borel transformation with respect to11990221198722 is the Borel mass parameter 119904

0(119879) is the temperature-

dependent continuum threshold and 119898119902can be 119898

119906 119898119889 or

119898119904depending on the kind of tensor mesonThe temperature-

dependent mass of the considered tensor states is found as

1198982119870lowast

2(1198912)(1198862)(119879)

= (int1199040(119879)

(119898119902+119898119889)2

119889119904 120588119870lowast

2(1198912)(1198862)(119904) 119904 exp [ minus119904

1198722]

minus119889

119889 (11198722)[BΠ

non-pert119870lowast

2(1198912)(1198862)])

sdot (int1199040(119879)

(119898119902+119898119889)2

119889119904 120588119870lowast

2(1198912)(1198862)(119904) exp [ minus119904

1198722]

+ BΠnon-pert119870lowast

2(1198912)(1198862))

minus1

(18)

3 Numerical Analysis

In this section we discuss the sensitivity of the massesand decay constants of the 119891

2 1198862 and 119870lowast

2tensor mesons

to temperature and compare the results obtained at zerotemperature with the predictions of vacuum sum rules [1921] as well as the existing experimental data [22] For thisaim we use some input parameters as 119898

119906= (23+07

minus05)MeV

119898119889

= (48+07minus03

)MeV and 119898119904

= (95 plusmn 5)MeV [22] and⟨0|119906119906|0⟩ = ⟨0|119889119889|0⟩ = minus(024 plusmn 001)3 GeV3 [24] and⟨0|119904119904|0⟩ = 08⟨0|119906119906|0⟩ [25]

In further analysis we need to know the expressionof the light quark condensate at finite temperature calcu-lated at different works (see eg [26 27]) In the presentstudy we use the parametrization obtained in [27] whichis also consistent with the lattice results [28 29] Forthe temperature-dependent continuum threshold we alsouse the parametrization obtained in [27] in terms of thetemperature-dependent light quark condensate and contin-uum threshold in vacuum (119904

0) The continuum threshold 119904

0

is not completely arbitrary and is correlated with the energyof the first excited state with the same quantum numbersas the chosen interpolating currents Our analysis revealsthat in the intervals (22ndash25) GeV2 (24ndash27) GeV2 and(30ndash33) GeV2 respectively for 119891

2 1198862 and 119870lowast

2channels the

results weakly depend on the continuum threshold Hencewe consider these intervals as working regions of 119904

0for the

channels under considerationAccording to the general philosophy of the method used

the physical quantities under consideration should also bepractically independent of the Borel mass parameter 1198722The working regions for this parameter are determined byrequiring that not only are the higher state and continuum

Advances in High Energy Physics 5

T (GeV)

06

000 005 010 015 020

000 005 010 015 020

09

12

15

06

09

12

15

s0 = 3GeV2

s0 = 315GeV2

s0 = 33GeV2

mKlowast 2(1430)(G

eV)

(a)

T (GeV)000

000

002

004

000

002

004

005 010 015 020

000 005 010 015 020

s0 = 3GeV2

s0 = 315GeV2

s0 = 33GeV2

fKlowast 2(1430)

(b)

Figure 2 Temperature dependence of the mass and decay constant of the 119870lowast2(1430)meson

T (GeV)000 005 010 015 020

000 005 010 015 020

s0 =

s0 = 235GeV2

s0 = 25GeV2

22GeV2

mf2(1270)(G

eV)

04

06

08

10

12

14

04

06

08

10

12

14

(a)

ff2(1270)

T (GeV)000

000

001

002

003

004

005

0039

0040

0041

0042

005 010 015 020

000 005 010 015 020

s0 =

s0 = 235GeV2

s0 = 25GeV2

22GeV2

(b)

Figure 3 Temperature dependence of the mass and decay constant of the 1198912(1270)meson

contributions suppressed but also the contributions of thehighest order operator are small Taking into account theseconditions we find that in the interval 14GeV2 le 1198722 le3GeV2 the results weakly depend on1198722 Figure 1 indicatesthe dependence of the masses and decay constants on theBorel mass parameter at zero temperature From this figurewe see that the results demonstrate good stability with respectto the variations of1198722 in its working region

Now we proceed to discuss how the physical quantitiesunder consideration behave in terms of temperature in theworking regions of the auxiliary parameters 1198722 and 119904

0

For this aim we present the dependence of the masses

and decay constants on temperature at 1198722 = 22GeV2in Figures 2 3 and 4 Note that we plot these figuresup to the temperature at which the temperature-dependentcontinuum threshold vanishes that is 119879 ≃ 183MeVFrom these figures we see that the masses and decayconstants diminish by increasing the temperature Near tothe temperature 119879 ≃ 183MeV the decay constants of the1198912(1270) 119886

2(1320) and 119870lowast

2(1430) decrease with amount

of 81 70 and 85 compared to their vacuum valuesrespectively while the masses decrease about 70 72 and60 for 119891

2(1270) 119886

2(1320) and 119870lowast

2(1430) states respec-

tively

6 Advances in High Energy Physics

T (GeV)000 005 010 015 020

000 005 010 015 020

s0 =

s0 = 255GeV2

s0 = 27GeV2

24GeV2

ma2(1320)(G

eV)

03

06

09

12

15

03

06

09

12

15

(a)

T (GeV)000

001

002

003

004

005

005 010 015 0200

2

4

6

8

10000 005 010 015 020

s0 =

s0 = 255GeV2

s0 = 27GeV2

24GeV2

fa2(1320)

(b)

Figure 4 Temperature dependence of the mass and decay constant of the 1198862(1320)meson

Table 1 Values of the masses and decay constants of the119870lowast2 1198912 and 119886

2mesons at zero temperature

Present work Experiment [22] Vacuum sum rules [19 20]relativistic quark model [20]

119898119870lowast

2(1430)

(GeV) 148 plusmn 012 14256 plusmn 00015 144 plusmn 010 [21] 1424 [20]119891119870lowast

2(1430)

0043 plusmn 0002 mdash 0050 plusmn 0002 [21]1198981198912(1270)

(GeV) 130 plusmn 008 12751 plusmn 00012 125 [19]1198911198912(1270)

0042 plusmn 0002 mdash 0040 [19]1198981198862(1320)

(GeV) 135 plusmn 011 13183 plusmn 00006 125 [19]1198911198862(1320)

0042 plusmn 0002 mdash mdash

Our final task is to compare the results of this workobtained at zero temperature with those of the vacuum sumrules as well as other existing theoretical predictions andexperimental data This comparison is made in Table 1 Fromthis table we see that the results on the masses and decayconstants obtained at zero temperature are roughly consistentwith existing experimental data as well as the vacuum sumrules and relativistic quark model predictions within theuncertainties Our predictions on the decay constants of thelight tensor mesons can be checked in future experimentsThe results obtained in the present work can be used intheoretical determination of the electromagnetic propertiesof the light tensor mesons as well as their weak decayparameters and their strong couplings with other hadronsOur results on the thermal behavior of the masses and decayconstants can also be useful in analysis of the results of futureheavy-ion collision experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work has been supported in part by the Scientificand Technological Research Council of Turkey (TUBITAK)under the Research Projects 110T284 and 114F018

References

[1] G E Brown andMRho ldquoChiral restoration in hot andor densematterrdquo Physics Reports vol 269 no 6 pp 333ndash380 1996

[2] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics theoretical foundationsrdquo Nuclear PhysicsSection B vol 147 no 5 pp 385ndash447 1979

[3] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics ApplicationsrdquoNuclear Physics B vol 147 no5 pp 448ndash518 1979

[4] A I Bochkarev and M E Shaposhnikov ldquoThe spectrum ofhot hadronic matter and finite-temperature QCD sum rulesrdquoNuclear Physics B vol 268 no 1 pp 220ndash252 1986

[5] T Hatsuda Y Koike and S H Lee ldquoFinite-temperature QCDsum rules reexamined 984858 120596 and A

1mesonsrdquo Nuclear Physics B

vol 394 no 1 pp 221ndash264 1993

Advances in High Energy Physics 7

[6] S Mallik ldquoOperator product expansion at finite temperaturerdquoPhysics Letters B vol 416 no 3-4 pp 373ndash378 1998

[7] E V Shuryak ldquoCorrelation functions in the QCD vacuumrdquoReviews of Modern Physics vol 65 no 1 pp 1ndash46 1993

[8] SMallik and S Sarkar ldquoVector and axial-vectormesons at finitetemperaturerdquo The European Physical Journal CmdashParticles andFields vol 25 no 3 pp 445ndash452 2002

[9] T Waas N Kaiser and W Weise ldquoLow energy K N interactionin nuclear matterrdquo Physics Letters B vol 365 pp 12ndash16 1996

[10] L Tolos D Cabrera and A Ramos ldquoStrange mesons in nuclearmatter at finite temperaturerdquo Physical Review C vol 78 ArticleID 045205 2008

[11] E V Veliev ldquoOperator product expansion for the thermalcorrelator of scalar currentsrdquo Journal of Physics G Nuclear andParticle Physics vol 35 Article ID 035004 2008

[12] E V Veliev K Azizi H Sundu and N Aksit ldquoInvestigation ofheavy-heavy pseudoscalar mesons in thermal QCD sum rulesrdquoJournal of Physics G vol 39 no 1 Article ID 015002 2012

[13] E V Veliev and G Kaya ldquoLeptonic decay constants of 119863119904and

119861119904mesons at finite temperaturerdquoThe European Physical Journal

C vol 63 no 1 pp 87ndash91 2009[14] F Klingl S Kim S H Lee P Morath andWWeise ldquoMasses of

119869120595 and 120578119888 in the nuclear medium QCD sum rule approachrdquoPhysical Review Letters vol 82 no 17 pp 3396ndash3399 1999

[15] CADominguezM Loewe and J C Rojas ldquoHeavy-light quarkpseudoscalar and vector mesons at finite temperaturerdquo Journalof High Energy Physics vol 2007 no 8 article 040 2007

[16] C A Dominguez and M Loewe ldquoDeconfinement and chiral-symmetry restoration at finite temperaturerdquo Physics Letters Bvol 233 pp 201ndash204 1989

[17] E V Veliev K Azizi H Sundu G Kaya and A TurkanldquoThermal qcd sum rules study of vector charmonium andbottomonium statesrdquo The European Physical Journal A vol 47no 9 p 110 2011

[18] K Azizi H Sundu A Turkan and E V Veliev ldquoThermalproperties of 119863lowast

1199042(2460) and 119863lowast

1199042(2573) tensor mesons using

QCD sum rulesrdquo Journal of Physics G Nuclear and ParticlePhysics vol 41 no 3 Article ID 035003 2014

[19] T M Aliev and M A Shifman ldquoOld tensor mesons in QCDsum rulesrdquo Physics Letters B vol 112 pp 401ndash405 1982

[20] D Ebert R N Faustov and V O Galkin ldquoMass spectra andRegge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

[21] T M Aliev K Azizi and V Bashiry ldquoOn the mass anddecay constant of Klowast

2(1430) tensor mesonrdquo Journal of Physics G

Nuclear and Particle Physics vol 37 no 2 Article ID 0250012010

[22] J Beringer J F Arguin R M Barnett et al ldquoReview of particlephysicsrdquo Physical Review D vol 86 Article ID 010001 2012

[23] Z-G Wang Z-C Liu and X-H Zhang ldquoAnalysis of theY(4140) and related molecular states with QCD sum rulesrdquoTheEuropean Physical Journal C vol 64 no 3 pp 373ndash386 2009

[24] B L Ioffe ldquoQCD (Quantum chromodynamics) at low energiesrdquoProgress in Particle and Nuclear Physics vol 56 pp 232ndash2772006

[25] S Narison ldquoOpen charm and beauty chiral multiplets in QCDrdquoPhysics Letters B vol 605 no 3-4 pp 319ndash325 2005

[26] A Barducci R Casalbuoni S deCurtis RGatto andG PettinildquoCurrent quark mass and chiral-symmetry breaking in QCD

at finite temperature in a mean-field approximationrdquo PhysicalReview D vol 46 no 5 pp 2203ndash2211 1992

[27] A Ayala A Bashir C A Dominguez E Gutierrez M Loeweand A Raya ldquoQCD phase diagram from finite energy sumrulesrdquo Physical Review D vol 84 Article ID 056004 2011

[28] A Bazavov T Bhattacharya M Cheng et al ldquoEquation of stateand QCD transition at finite temperaturerdquo Physical Review Dvol 80 Article ID 014504 2009

[29] M Cheng S Ejiri P Hegde et al ldquoEquation of state for physicalquark massesrdquo Physical Review D vol 81 no 5 Article ID054504 8 pages 2010

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