Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 219456 7 pageshttpdxdoiorg1011552013219456

Research ArticleTowards the Assignment for the Ground Scalar Meson

Xue-Chao Feng Peng Chen and Ji-Min Shang

Department of Technology and Physics Zhengzhou University of Light Industry Zhengzhou 450002 China

Correspondence should be addressed to Xue-Chao Feng fxchaozzulieducn

Received 5 November 2012 Revised 24 December 2012 Accepted 24 December 2012

Academic Editor C Q Geng

Copyright copy 2013 Xue-Chao Feng et alThis 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

Combining three complementary approaches nonrelativistic constituent quark model Regge phenomenology and meson-mesonmixing we obtain themass relations which relate different mesonmultipletsWith the help of knownmesonmasses these relationscan be used for the prediction of unknown meson states masses The utility of this new analysis technique is applied for the scalarmeson and results in a possible allocation scheme for the ground scalar meson

1 Introduction

It is well known that the strong interactions are describedby the non-Abelian gauge field theory quantum chromody-namics (QCD) However we are still far from deriving thespectrumofmesons fromfirst principles so farTherefore thephenomenological analysis of meson states is important forthe hadronic physics [1ndash3]

The study of scalar mesons plays an important role inparticle physics The scalar mesons with vacuum quantumnumbers could provide a full understanding of the symmetrybreaking mechanisms in QCD and the confinement [4]Moreover scalar mesons are very important intermediatestate in Goldstone boson interactions away from thresholdwhere chiral perturbation theory is not applicable [5] There-fore the investigation of scalar mesons becomes an attentionfocus in theory and experiments [5ndash13] Nevertheless theinterpretation of the nature of scalar mesons is a long-standing problem of particle physics to this day Accordingto the new edition of Particle Data Group (PDG) forthe light quark sector the number of resonances found inexperiment below 2GeV exceeds the number of states thatconventional quark models can accommodate [14] There areten resonances are observed in the experiments includingthe isoscalar resonances119891

0(600)119891

0(980)119891

0(1370)119891

0(1500)

and1198910(1710) the isovector resonances 119886

0(980) and 119886

0(1450)

and the isodoublet resonances 119870lowast

0(1430) 119870

lowast

0(800) and

119870lowast

0(1950) Up to now none of the ten reported scalar states

can be definitely assigned as ground scalar nonet members

Considering that some resonances cannot be explained by theconventional 119902119902 quark model different theoretical modelshave also been established such as 119870119870 molecule [15] MITbag model [16] QCD sum rules [17] and chiral Lagrangians[18] For the heavy-light quark meson two resonances119863lowast

119878119869(2317) and 119863

lowast

0(2400) are observed in the experiments

The resonances 119863lowast

119878119869(2317) with a mass 23178 plusmn 06MeV

and width 3493 plusmn 06MeV are assigned as charm-strangemeson of scalar state However the experimental value isconsiderably smaller than the results in many theoreticalpredictions [19 20] The disagreement results in the differentinterpretations about the119863lowast

119878119869(2317) such as themolecule state

[21] the four-quark state [22] or the 119863120587 atom [23] On theother hand considering the fact that the mass 119863

lowast

119878119869(2317)

is smaller than its partner 119863lowast

0(2400) we also argue that

the assignment of the 119863lowast

119878119869(2317) needs further tests in the

experimentsIn the present work based on the relations derive from

three complementary approaches nonrelativistic constituentquark model Regge phenomenology and meson-mesonmixing we investigate the scalar meson mass spectrum andpropose a possible assignment for the ground scalar mesonnonet

2 Nonrelativistic Constituent QuarkModel for Scalar Meson

In the framework of the non-relativistic constituent quarkmodel one typically assumes that the 119902119902 wave functions are

2 Advances in High Energy Physics

a solution of a non-relativistic Schrodinger equation with theBreit-Fermi Hamitonian119867 [24ndash26] which contain

119867120595119899 (119903) = (119898

119902+ 119898119902+

119898119902+ 119898119902

2119898119902119898119902

1199012minus (

1198983

119902+ 1198983

119902

81198983

1199021198983

119902

)

1199014

+119881119873119877 (119903) + 119867

119871119878+119878119878+119879)120595119899 (119903)

(1)

with

119881119873119877

(119903) = 119881119907(119903) + 119881

119904(119903)

119867119871119878+119878119878+119879

= 119867119871119878

+ 119867119878119878

+ 119867119879

(2)

where 119898119902and 119898

119902are the constituent quark masses 119881

119873119877

is the confining potential consisting of vector and scalarcontributions 119867

119871119878 119867119878119878 and 119867

119879 are the spin-spin spin-

orbit and tensor terms respectively Following we apply theBreit-Fermi Hamitonian to the S-wave and P-wave mesonmultiplets

For the S-wave mesons consider the fact the matrixelement of the tensor and spin-orbit interactions vanishesone arrive at the following relation

119872 = 119898119902+ 119898119902+ 1198860+ 1198900

⟨119878119902119878119902⟩

119898119902119898119902

(3)

119878119902and 119878

119902are the constituent quark spins and 119890

0is the

spin-spin interaction coefficient The constant 1198860is small

compared to the constituent quark masses 119898119902and 119898

119902and

may be absorbed into the constituent quark masses [24 2930] Therefore the relation (3) can be expressed as

119872 = 119898119902+ 119898119902+ 1198900

⟨119878119902119878119902⟩

119898119902119898119902

(4)

For the 119875-wave mesons the phenomenological form ofthe matrix element of the Breit-Fermi Hamitonian has thefollowing relation

119872 = 119898119902+ 119898119902+ 1198861+ 1198871(

1

119898119902

+1

119898119902

)

+ 1198881(

1

1198982

119902

+1

1198982

119902

) +1198891

119898119902119898119902

+ 1198921[

[

(119898119902+ 119898119902)2

+ 2119898119902119898119902

41198982

1199021198982

119902

⟨119871 sdot 119878⟩

minus

1198982

119902minus 1198982

119902

41198982

1199021198982

119902

⟨119871 (119878119902minus 119878119902)⟩]

]

+ 1198901

⟨119878119902119878119902⟩

119898119902119898119902

+ 1198911(

1

1198983

119902

+1

1198983

119902

) + ℎ1

119886 (119895 1)

119898119902119898119902

(5)

with

119886 (119895 1) =4

5⟨11987821198712minus

3

2(119871119878) minus 3(119871119878)

2⟩

⟨119871119878⟩ =1

2[119895 (119895 + 1) minus 3119878 minus 2]

(6)

where 1198861 1198871 1198881 1198891 11989011198911 1198921 and ℎ

1are the constants and the

angular momentum parts of the matrix elements are shownin Table 1 With the help of the parameters in Table 1 fromthe relations (4) and (5) we obtain the following formulas

119872120587+ 3119872120588

2119872119870+ 6119872119870lowast minus 119872

120587minus 3119872120588

=119898119899

119898119904

(7)

119872119904119904(131198752) minus 119872

119904119904(131198750)

119872119899119899

(131198752) minus 119872

119899119899(131198750)= (

119898119899

119898119904

)

2

(8)

where119872119899119899is themass of isovector state119872

119904119904refers to themass

of the bare strangemember of the isoscalar states and119898119904and

119898119899(119899 stands for nonstrange 119906- and 119889-quark) are the corre-

sponding constituent quark masses In view of relation (8)we can obtain the scalar mesons masses from the masses ofother meson states which quark model assignment is firmlyestablishedDue to the isovectormeson act as a beacon for themass scales of meson nonet we should firstly determine themass of isovector state In the following section we are goingto calculate the mass of isovector states in the framework ofRegge phenomenology and mass mixing matrix

3 Mass Mixing Matrix of Isoscalar States

In the quark model mesons are 1199021199021015840 bound states of quark

119902 and antiquark 1199021015840 Three light flavors of quarks -119906 -119889

and -119904 combined with three antiquarks yield nine mesonnonet which contain two bare isoscalar states In generalthe two bare isoscalar states can mix which result in twophysical isoscalar states Therefore the mix of isoscalar statesis important for the meson due to providing a clue for theassignment of a resonance in a meson nonet

In the nonstrange 119873 = (119906119906 + 119889119889)radic2 and strange 119878 = 119904119904

basis the mass-squared matrix describing the mixing of thetwo physical isoscalar states can be written as [31]

1198722= (

1198722

119873+ 2119860 radic2119860119883

radic2119860119883 1198722

119904119904+ 119860119883

2) (9)

where 119872119873and 119872

119904119904are the masses of bare states 119873 and 119878

respectively 119860 are the mixing parameters which describethe 119902119902 harr 119902

10158401199021015840 transition amplitudes The 119883 that is a

phenomenological parameter describes the 119878119880(3) brokenratio of the nonstrange (119906- and 119889-quark) and strange quarkpropagators via the constituent quark mass ratio The valueof119883 is determined to be 063 by inserting the correspondingmasses into relation (7) [29] Here and below all the massesused as input for our calculation are taken from PDG [14]

In the meson nonet the physical isoscalar states 120593 and1205931015840 are the eigenstates of mass-squared matrix 119872

2 as well

Advances in High Energy Physics 3

Table 1 Angular momentum part of the matrix elements of relations (4) and (5)

1198992119904+1

119897119869

111198780

131198781

131198752

131198750

⟨119871119878⟩ 1 minus2119886(119895 1) minus25 minus4⟨119878119902119878119902⟩ minus34 14 14 14

⟨119871(119878119902minus 119878119902)⟩ 0 0

Table 2 Predicted mass of bare 119904119904 for meson nonet 120579quad is the nonet mixing angle from the quadratic mass formulae [14]

Nonet 111198780

131198781

131198752

131198633

211198780

119872119904119904(MeV) Present work 685 1018 1530 1858 1323

[27] 697 1009 1546[28] 702 1010 1538[19] 960 1020 1530 1900 1630

120579quad (∘) minus115 387 296 32 minus224

as the square masses 1198722

120593and 119872

2

1205931015840 are the eigenvalues

respectivelyThe physical states 120593 and 1205931015840 can be related to the

119904119904 and119873 = (119906119906 + 119889119889)radic2 by

(phy1

phy2

) = 119880(119873

119878) (10)

and the unitary matrix 119880 can be described as

1198801198722119880dagger= (

1198722

phy1

0

0 1198722

phy2

) (11)

The mix of the physical isoscalar states can be alsodescribed as

(phy1

phy2

) = (cos 120579 minus sin 120579

sin 120579 cos 120579)(

1205938

1205931

) (12)

with

1205931=

(119906119906 + 119889119889 + 119904119904)

radic3

1205938=

(119906119906 + 119889119889 minus 2119904119904)

radic6

(13)

where 120579 is the nonet mixing angleFrom the relations (9) (10) (11) and (12) one will obtain

119880 = (cos 120579 minus sin 120579

sin 120579 cos 120579 )(

radic1

3minusradic

2

3

radic2

3radic

1

3

) (14)

Based on the relations (9) and (11) one can have thefollowing relations

1198722

119873+ 2119860 = (radic

1

3cos 120579 minus radic

2

3sin 120579)

2

1198722

phy1

+ (radic2

3cos 120579 + radic

1

3sin 120579)

2

1198722

phy2

1198722

119904119904+ 119860119883

2= (radic

1

3cos 120579 minus radic

2

3sin 120579)

2

1198722

phy2

+ (radic2

3cos 120579 + radic

1

3sin 120579)

2

1198722

phy1

(15)

Applying the relations (15) for themeson states 111198780 131198781

131198752 131198633 and 2

11198780 we obtain the bare 119904119904 member masses

and the results are present in Table 2 Here and below all themasses and themixing angle used as input for our calculationare taken from the PDG [14]

4 Regge Phenomenology and Mass ofIsovector Scalar Meson

Regge phenomenology is widely used for the calculation ofthe meson and baryonium spectrum [28 36ndash38] In [28]Li et al indicate that the quasi-linear Regge trajectoriescould provide a reasonable description for the meson massspectrum Anisovich et al also conclude that the mesonstates can fit to the quasi-linear Regge trajectories at the(119873 1198722) and (119869 1198722) planes with good accuracy [36] Inthe last decade Regge trajectories are reanalysis to makepredictions for the masses of states not yet to be discoveredin experiment or to determine the quantum numbers of thenewly discovered states

4 Advances in High Energy Physics

Table 3 Predicted mass119872119899119899 119872119899119904 119872119904119904 and119872

119888119904of ground scalar meson (All inMeV and the values used as input for our analysis are shown

in boldface)

Nonet Mass 119872119899119899

119872119899119904

119872119904119904

119872119888119899

119872119888119904

119872119888119888

131198750

Present work 1024 1279 1414 2318 2423 3415[32] 960 plusmn 30 1220

+50

minus1501220 plusmn 40

[24] 983 plusmn 10 1204 plusmn 10 1397 plusmn 16[19] 1090 1240 1360 2400 2480 3440

231198750

Present work 1611 1826 1924 2835 3099 3958[33] 2880 2970

[34 35] 2919 3054[19] 1780 1890 1990 3920

331198750

Present work 2034 2243 2390 3270 3653 4434

Based on the assumption that the hadrons with identical119869PC quantum numbers obey quasi-linear form of Reggetrajectories one have the following relations [28]

119897 = 1205721198941198941015840(0) + 120572

1015840

11989411989410158401198722

1198941198941015840

119897 = 1205721198941198951015840(0) + 120572

1015840

11989411989510158401198722

1198941198951015840

119897 = 1205721198951198951015840(0) + 120572

1015840

11989511989510158401198722

1198951198951015840

(16)

where 1198941198941015840 denotes the quark and antiquark flavors and 119897 and

1198721198941198941015840are the orbital momentum and mass of the 119894119894

1015840 mesonThe parameters 1205721015840

1198941198941015840and 120572

1198941198941015840(0) are respectively the slope and

intercept of the Regge trajectory The intercepts and slopescan be expressed by [28 39]

120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (17)

1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(18)

The relation (17) is satisfied in two-dimensional QCD[40] the dual-analytic model [41] and the quark bremss-trahlung model [42] The relation (18) is deprived fromthe topological and the 119902119902-string picture of hadrons [43]Recently based on the available data for mesonic resonancesof light medium and heavy flavors Filipponi et al [44]and Burakovsky and Goldman [45] construct different formslopes (19) of linear Regge trajectories for all quark flavorsOne has the following

1205721015840

119894119895=

09GeVminus2

1 + 022((119898119894+ 119898119895) GeV)

32

1205721015840

119894119895=

1205721015840

1205874 + (1205874)radic1205721015840(119898119894+ 119898119895) 2

(19)

According to the relations (19) we find the slope of Reggetrajectory depends on the constituent quark masses throughthe combination (119898

119894+ 119898119895) The relations (19) are obtained

by fitting experimental data and these relations are possible

to update with the increasing amount of experimental dataIn the present work we adopt the assumption presented by[28 31 46] that the slopes of the parity partners (eg the 0+1minus 2+ and 1

minus are parity partners) trajectories coincide andfurther that the slopes do not depend on charge conjugationin accordance with the C-invariance of QCD Under thisassumption the corresponding slopes of the 13119875

2are the same

as those of the 131198750trajectories and we have the following

relations

1 +

1198722(131198750)1198991198991198722(131198752)119899119904

minus 1198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119904

=

41198722(131198750)1198991198991198722(131198752)119899119904

minus 41198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991198991198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119899

1 +

1198722(131198750)1198881198881198722(131198752)119888119899

minus 1198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881199041198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119899

=

41198722(131198750)1198881198881198722(131198752)119888119899

minus 41198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881198881198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119888

1 +

1198722(131198750)1198881198881198722(131198752)119888119904minus 1198722(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119904

=

41198722(131198750)1198881198881198722(131198752)119888119904minus 4119872

2(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881198881198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119888

(20)

In relations (20) we eliminate the slopes and obtainmasses expressions which relate six meson states Insertingthe masses of the corresponding mesons into relations (20)we obtain the masses of 119872

119899119899 119872119899119904 119872119904119904 and 119872

119888119904of ground

scalar meson In our previous work a series of relationsbetweenmesonmass and the parameters of radial trajectoriesare established in the framework of Regge phenomenology[47 48] The relations are still applicable for the scalarmeson Inserting the ground scalar meson masses and theparameters in [47 48] the radial excited meson masses arepredicted The results are compared with other predictions

Advances in High Energy Physics 5

1 2 3

N

4

3

2

1

nbar(n)

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(a)

nbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(b)

sbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(c)

Figure 1 The quasi-linear trajectory for11987331198750meson nonet

and shown in Table 3 Moreover the radial excited masses ofthe light quark sector are shown in Figure 1

5 Results and Discussion

According to the introduction in Section 1 the isovectormember of nonet is important for the assignment of isoscalarmeson due to providing a beacon for the mass scale ofmeson nonet However both states the 119886

0(980) and 119886

0(1450)

are observed in the experiments and could be considered ascandidates In Table 3 our results favor the interpretation ofthe state 119886

0(980) rather than the 119886

0(1450) as the isovector

ground scalar meson The assignment is supported by thedifferent theoretical models prediction In the naive quarkmodel for 119878 = 1 states spin-orbit coupling results in massorder 119898(2

++) ge 119898(1

++) ge 119898(0

++) which supports the fact

that 1198860(980) (not the 119886

0(1450)) is the isovector member of

scalar nonet [49 50] In the scheme of K-matrix analysisAnvisoch also indicate that the mass of the 1

31198750isovector

state is about 960 plusmn 30MeV which is in excellent agreementwith our prediction [32]

Apart from the isovector member the isoscalar sectoris also ambiguous both experimentally and theoreticallyIn Table 3 we propose the bare nonstrange and strangemembers masses of isoscalar meson

In the last decade the heavy-light mesons that attractmore attention with many new resonances have beenobserved in the experiments such as 119863lowast

1199040(2317) 119863lowast

1199041(2460)

119863lowast

119904119869(2715) and 119863

lowast

119904119869(2860) For the 119863

lowast

1199040(2317) it was first

observed in 119863+

119878120587 by the Babar Collaboration at the Stanford

Linear Accelerator Center with a mass of about 2317MeVand width of about 350MeV [51] Subsequently this statehas also been reported by the Bell Collaboration [52] andthe CLEO [53] In our present work the mass of the charm-strange meson is about 100MeV higher than the measuredmass of 119863lowast

1199040(2317) which is consistent with the predictions

by Godfrey and Isgur [19] Consequently we suggest that theassignment for the119863lowast

1199040(2317) as a 13119875

0charm-strange meson

needs further testing in the experimentIn summary the understanding of scalar meson is quite

puzzling for the theorists and experimentalists Until nowthere is no one state that can be assigned as the scalar

6 Advances in High Energy Physics

meson member without dispute In the present work com-bining three complementary approaches non-relativisticconstituent quarkmodel Regge phenomenology andmeson-meson mixing we investigate the assignment of groundscalar meson Our results are useful for the assignment of thescalar meson in the future

Moreover we also find a bridge which relate the heavyand light flavor mesons in the process of analyzing scalarmeson The bridge is significant for the assignment of thediscovered states and searching the new resonances whichhave not been observed in the experiments

Acknowledgment

This project supported by the Zhengzhou University of LightIndustry Foundation China (Grant nos 2009XJJ011 and2012XJJ008)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in twodimensions Applications to ferromagnets and massive Yang-Mills fieldsrdquo Physics Letters B vol 59 no 1 pp 79ndash81 1975

[2] P Maris and C D Roberts ldquoDyson-schwinger equations a toolfor hadron physicsrdquo International Journal of Modern Physics Evol 12 no 3 article 297 2003

[3] M Gell-Mann ldquoA schematic model of baryons and mesonsrdquoPhysics Letters vol 8 no 3 pp 214ndash215 1964

[4] F E Close andN A Tornqvist ldquoScalarmesons above and below1 GeVrdquo Journal of Physics G vol 28 no 10 article R249 2002

[5] A H Fariborz ldquoIsosinglet scalar mesons below 2 GeV and thescalar glueball massrdquo International Journal of Modern Physics Avol 19 no 13 article 2095 2004

[6] Y H Liu D M Li J X Shao X G Wang and Z Y ZhangldquoRegarding the scalar mesonsrdquo Physical Review D vol 77 no 3Article ID 034025 5 pages 2008

[7] DM Li KWWei and Y Hong ldquoA possible assignment for theground scalar meson nonetrdquo European Physical Journal A vol25 pp 263ndash266 2005

[8] M Schumacher ldquoStructure of scalar mesons and the Higgssector of strong interactionrdquo Journal of Physics G vol 38 no8 Article ID 083001 2011

[9] Z T Xia and W Zuo ldquoA mixing scheme for the structure off0(600) and f

0(1370)rdquo Nuclear Physics A vol 848 pp 317ndash329

2010[10] X Liu and Z J Xiao ldquoLight scalar mesons and charmless

hadronic B119888rarr SPSVdecays in the perturbativeQCDapproachrdquo

Physical Review Letters vol 82 Article ID 054029 2010[11] Z G Wang ldquoAnalysis of the nonet scalar mesons as tetraquark

states with new QCD sum rulesrdquo International Journal ofTheoretical Physics vol 51 no 2 pp 507ndash517 2012

[12] A H Fariborz R Jora and J Schechter ldquoLight scalar puzzle inQCDrdquo AIP Conference Proceedings vol 1361 pp 127ndash131 2011

[13] W de Paula and T Frederico ldquoScalar mesons within a dynami-cal holographic QCD modelrdquo Physics Letters B vol 693 no 3pp 287ndash291 2010

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

[15] J Weinstein and N Isgur ldquo119870119870 moleculesrdquo Physical Review Dvol 41 no 7 pp 2236ndash2257 1990

[16] R L Jaffe ldquoMultiquark hadrons I Phenomenology of 1198762119876minus2mesonsrdquo Physical Review D vol 15 no 1 pp 267ndash280 1977

[17] F Shi T G Steele V Elias K B Sprague Y Xue and A HFariborz ldquoHolder inequalities and isospin splitting of the quarkscalar mesonsrdquoNuclear Physics A vol 671 no 1ndash4 pp 416ndash4462000

[18] D Black A H Fariborz S Moussa S Nasri and J SchechterldquoUnitarized pseudoscalar meson scattering amplitudes fromthree flavor linear sigma modelsrdquo Physical Review D vol 64no 1 Article ID 014031 45 pages 2001

[19] S Godfrey and N Isgur ldquoMesons in a relativized quark modelwith chromodynamicsrdquo Physical Review D vol 32 no 1 pp189ndash231 1985

[20] M Di Pierro and E J Eichten ldquoExcited heavy-light systems andhadronic transitionsrdquo Physical Review D vol 64 no 11 ArticleID 114004 20 pages 2001

[21] T Barnes F E Close and H J Lipkin ldquoImplications of a Dkmolecule at 232-GeVrdquo Physical Review D vol 68 Article ID054006 5 pages 2003

[22] H Y Cheng and W S Hou ldquoB decays as spectroscope forcharmed four-quark statesrdquo Physics Letters B vol 566 no 3-4pp 193ndash200 2003

[23] A P Szczepaniak ldquoDescription of the 119863lowast119904resonance as the D120587

atomrdquo Physics Letters B vol 567 no 1-2 pp 23ndash26 2003[24] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in the

nonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

[25] W Lucha F F Schoberl and D Gromes ldquoBound states ofquarksrdquo Physics Reports vol 200 no 4 pp 127ndash240 1991

[26] D Flamm and F Schoberl Introduction to Quark Model of Ele-mentary Particles vol 1 Gordon and Breach Science PublishersLondon UK 1982

[27] KWWei X P Dong and G Lu ldquoMasses of 119904119904 states and nonetmixing anglesrdquo International Journal of Modern Physics A vol26 no 12 article 2065 2011

[28] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectrumin regge phenomenologyrdquoThe European Physical Journal C vol37 no 3 pp 323ndash333 2004

[29] D M Li and Z Li ldquoStrange axial-vector mesons mixing anglerdquoThe European Physical Journal vol 28 pp 369ndash373 2006

[30] L Burakovsky and T Goldman ldquoRegarding the enigmas of P-wave meson spectroscopyrdquo Physical Review D vol 57 no 5 pp2879ndash2888 1998

[31] M M Brisudova L Burakovsky and T Goldman ldquoNewglueball-meson mass relationsrdquo Physical Review D vol 58 no11 Article ID 114015 7 pages 1998

[32] A V Anisovich and A V Sarantsev ldquoK-matrix analysis ofthe K120587 S-wave in the mass region 900-2100 MeV and nonetclassification of scalar 119902119902-states rdquo Physics Letters B vol 413 no1-2 pp 137ndash146 1997

[33] A M Badalian and B L G Bakker ldquoHigher excitations of theD and D

119904mesonsrdquo Physical Review D vol 84 no 3 Article ID

034006 10 pages 2011[34] D Ebert R N Faustov and V O Galkin ldquoMass spectra and

Regge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

Advances in High Energy Physics 7

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Soft MatterJournal of

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ThermodynamicsJournal of

2 Advances in High Energy Physics

a solution of a non-relativistic Schrodinger equation with theBreit-Fermi Hamitonian119867 [24ndash26] which contain

119867120595119899 (119903) = (119898

119902+ 119898119902+

119898119902+ 119898119902

2119898119902119898119902

1199012minus (

1198983

119902+ 1198983

119902

81198983

1199021198983

119902

)

1199014

+119881119873119877 (119903) + 119867

119871119878+119878119878+119879)120595119899 (119903)

(1)

with

119881119873119877

(119903) = 119881119907(119903) + 119881

119904(119903)

119867119871119878+119878119878+119879

= 119867119871119878

+ 119867119878119878

+ 119867119879

(2)

where 119898119902and 119898

119902are the constituent quark masses 119881

119873119877

is the confining potential consisting of vector and scalarcontributions 119867

119871119878 119867119878119878 and 119867

119879 are the spin-spin spin-

orbit and tensor terms respectively Following we apply theBreit-Fermi Hamitonian to the S-wave and P-wave mesonmultiplets

For the S-wave mesons consider the fact the matrixelement of the tensor and spin-orbit interactions vanishesone arrive at the following relation

119872 = 119898119902+ 119898119902+ 1198860+ 1198900

⟨119878119902119878119902⟩

119898119902119898119902

(3)

119878119902and 119878

119902are the constituent quark spins and 119890

0is the

spin-spin interaction coefficient The constant 1198860is small

compared to the constituent quark masses 119898119902and 119898

119902and

may be absorbed into the constituent quark masses [24 2930] Therefore the relation (3) can be expressed as

119872 = 119898119902+ 119898119902+ 1198900

⟨119878119902119878119902⟩

119898119902119898119902

(4)

For the 119875-wave mesons the phenomenological form ofthe matrix element of the Breit-Fermi Hamitonian has thefollowing relation

119872 = 119898119902+ 119898119902+ 1198861+ 1198871(

1

119898119902

+1

119898119902

)

+ 1198881(

1

1198982

119902

+1

1198982

119902

) +1198891

119898119902119898119902

+ 1198921[

[

(119898119902+ 119898119902)2

+ 2119898119902119898119902

41198982

1199021198982

119902

⟨119871 sdot 119878⟩

minus

1198982

119902minus 1198982

119902

41198982

1199021198982

119902

⟨119871 (119878119902minus 119878119902)⟩]

]

+ 1198901

⟨119878119902119878119902⟩

119898119902119898119902

+ 1198911(

1

1198983

119902

+1

1198983

119902

) + ℎ1

119886 (119895 1)

119898119902119898119902

(5)

with

119886 (119895 1) =4

5⟨11987821198712minus

3

2(119871119878) minus 3(119871119878)

2⟩

⟨119871119878⟩ =1

2[119895 (119895 + 1) minus 3119878 minus 2]

(6)

where 1198861 1198871 1198881 1198891 11989011198911 1198921 and ℎ

1are the constants and the

angular momentum parts of the matrix elements are shownin Table 1 With the help of the parameters in Table 1 fromthe relations (4) and (5) we obtain the following formulas

119872120587+ 3119872120588

2119872119870+ 6119872119870lowast minus 119872

120587minus 3119872120588

=119898119899

119898119904

(7)

119872119904119904(131198752) minus 119872

119904119904(131198750)

119872119899119899

(131198752) minus 119872

119899119899(131198750)= (

119898119899

119898119904

)

2

(8)

where119872119899119899is themass of isovector state119872

119904119904refers to themass

of the bare strangemember of the isoscalar states and119898119904and

119898119899(119899 stands for nonstrange 119906- and 119889-quark) are the corre-

sponding constituent quark masses In view of relation (8)we can obtain the scalar mesons masses from the masses ofother meson states which quark model assignment is firmlyestablishedDue to the isovectormeson act as a beacon for themass scales of meson nonet we should firstly determine themass of isovector state In the following section we are goingto calculate the mass of isovector states in the framework ofRegge phenomenology and mass mixing matrix

3 Mass Mixing Matrix of Isoscalar States

In the quark model mesons are 1199021199021015840 bound states of quark

119902 and antiquark 1199021015840 Three light flavors of quarks -119906 -119889

and -119904 combined with three antiquarks yield nine mesonnonet which contain two bare isoscalar states In generalthe two bare isoscalar states can mix which result in twophysical isoscalar states Therefore the mix of isoscalar statesis important for the meson due to providing a clue for theassignment of a resonance in a meson nonet

In the nonstrange 119873 = (119906119906 + 119889119889)radic2 and strange 119878 = 119904119904

basis the mass-squared matrix describing the mixing of thetwo physical isoscalar states can be written as [31]

1198722= (

1198722

119873+ 2119860 radic2119860119883

radic2119860119883 1198722

119904119904+ 119860119883

2) (9)

where 119872119873and 119872

119904119904are the masses of bare states 119873 and 119878

respectively 119860 are the mixing parameters which describethe 119902119902 harr 119902

10158401199021015840 transition amplitudes The 119883 that is a

phenomenological parameter describes the 119878119880(3) brokenratio of the nonstrange (119906- and 119889-quark) and strange quarkpropagators via the constituent quark mass ratio The valueof119883 is determined to be 063 by inserting the correspondingmasses into relation (7) [29] Here and below all the massesused as input for our calculation are taken from PDG [14]

In the meson nonet the physical isoscalar states 120593 and1205931015840 are the eigenstates of mass-squared matrix 119872

2 as well

Advances in High Energy Physics 3

Table 1 Angular momentum part of the matrix elements of relations (4) and (5)

1198992119904+1

119897119869

111198780

131198781

131198752

131198750

⟨119871119878⟩ 1 minus2119886(119895 1) minus25 minus4⟨119878119902119878119902⟩ minus34 14 14 14

⟨119871(119878119902minus 119878119902)⟩ 0 0

Table 2 Predicted mass of bare 119904119904 for meson nonet 120579quad is the nonet mixing angle from the quadratic mass formulae [14]

Nonet 111198780

131198781

131198752

131198633

211198780

119872119904119904(MeV) Present work 685 1018 1530 1858 1323

[27] 697 1009 1546[28] 702 1010 1538[19] 960 1020 1530 1900 1630

120579quad (∘) minus115 387 296 32 minus224

as the square masses 1198722

120593and 119872

2

1205931015840 are the eigenvalues

respectivelyThe physical states 120593 and 1205931015840 can be related to the

119904119904 and119873 = (119906119906 + 119889119889)radic2 by

(phy1

phy2

) = 119880(119873

119878) (10)

and the unitary matrix 119880 can be described as

1198801198722119880dagger= (

1198722

phy1

0

0 1198722

phy2

) (11)

The mix of the physical isoscalar states can be alsodescribed as

(phy1

phy2

) = (cos 120579 minus sin 120579

sin 120579 cos 120579)(

1205938

1205931

) (12)

with

1205931=

(119906119906 + 119889119889 + 119904119904)

radic3

1205938=

(119906119906 + 119889119889 minus 2119904119904)

radic6

(13)

where 120579 is the nonet mixing angleFrom the relations (9) (10) (11) and (12) one will obtain

119880 = (cos 120579 minus sin 120579

sin 120579 cos 120579 )(

radic1

3minusradic

2

3

radic2

3radic

1

3

) (14)

Based on the relations (9) and (11) one can have thefollowing relations

1198722

119873+ 2119860 = (radic

1

3cos 120579 minus radic

2

3sin 120579)

2

1198722

phy1

+ (radic2

3cos 120579 + radic

1

3sin 120579)

2

1198722

phy2

1198722

119904119904+ 119860119883

2= (radic

1

3cos 120579 minus radic

2

3sin 120579)

2

1198722

phy2

+ (radic2

3cos 120579 + radic

1

3sin 120579)

2

1198722

phy1

(15)

Applying the relations (15) for themeson states 111198780 131198781

131198752 131198633 and 2

11198780 we obtain the bare 119904119904 member masses

and the results are present in Table 2 Here and below all themasses and themixing angle used as input for our calculationare taken from the PDG [14]

4 Regge Phenomenology and Mass ofIsovector Scalar Meson

Regge phenomenology is widely used for the calculation ofthe meson and baryonium spectrum [28 36ndash38] In [28]Li et al indicate that the quasi-linear Regge trajectoriescould provide a reasonable description for the meson massspectrum Anisovich et al also conclude that the mesonstates can fit to the quasi-linear Regge trajectories at the(119873 1198722) and (119869 1198722) planes with good accuracy [36] Inthe last decade Regge trajectories are reanalysis to makepredictions for the masses of states not yet to be discoveredin experiment or to determine the quantum numbers of thenewly discovered states

4 Advances in High Energy Physics

Table 3 Predicted mass119872119899119899 119872119899119904 119872119904119904 and119872

119888119904of ground scalar meson (All inMeV and the values used as input for our analysis are shown

in boldface)

Nonet Mass 119872119899119899

119872119899119904

119872119904119904

119872119888119899

119872119888119904

119872119888119888

131198750

Present work 1024 1279 1414 2318 2423 3415[32] 960 plusmn 30 1220

+50

minus1501220 plusmn 40

[24] 983 plusmn 10 1204 plusmn 10 1397 plusmn 16[19] 1090 1240 1360 2400 2480 3440

231198750

Present work 1611 1826 1924 2835 3099 3958[33] 2880 2970

[34 35] 2919 3054[19] 1780 1890 1990 3920

331198750

Present work 2034 2243 2390 3270 3653 4434

Based on the assumption that the hadrons with identical119869PC quantum numbers obey quasi-linear form of Reggetrajectories one have the following relations [28]

119897 = 1205721198941198941015840(0) + 120572

1015840

11989411989410158401198722

1198941198941015840

119897 = 1205721198941198951015840(0) + 120572

1015840

11989411989510158401198722

1198941198951015840

119897 = 1205721198951198951015840(0) + 120572

1015840

11989511989510158401198722

1198951198951015840

(16)

where 1198941198941015840 denotes the quark and antiquark flavors and 119897 and

1198721198941198941015840are the orbital momentum and mass of the 119894119894

1015840 mesonThe parameters 1205721015840

1198941198941015840and 120572

1198941198941015840(0) are respectively the slope and

intercept of the Regge trajectory The intercepts and slopescan be expressed by [28 39]

120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (17)

1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(18)

The relation (17) is satisfied in two-dimensional QCD[40] the dual-analytic model [41] and the quark bremss-trahlung model [42] The relation (18) is deprived fromthe topological and the 119902119902-string picture of hadrons [43]Recently based on the available data for mesonic resonancesof light medium and heavy flavors Filipponi et al [44]and Burakovsky and Goldman [45] construct different formslopes (19) of linear Regge trajectories for all quark flavorsOne has the following

1205721015840

119894119895=

09GeVminus2

1 + 022((119898119894+ 119898119895) GeV)

32

1205721015840

119894119895=

1205721015840

1205874 + (1205874)radic1205721015840(119898119894+ 119898119895) 2

(19)

According to the relations (19) we find the slope of Reggetrajectory depends on the constituent quark masses throughthe combination (119898

119894+ 119898119895) The relations (19) are obtained

by fitting experimental data and these relations are possible

to update with the increasing amount of experimental dataIn the present work we adopt the assumption presented by[28 31 46] that the slopes of the parity partners (eg the 0+1minus 2+ and 1

minus are parity partners) trajectories coincide andfurther that the slopes do not depend on charge conjugationin accordance with the C-invariance of QCD Under thisassumption the corresponding slopes of the 13119875

2are the same

as those of the 131198750trajectories and we have the following

relations

1 +

1198722(131198750)1198991198991198722(131198752)119899119904

minus 1198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119904

=

41198722(131198750)1198991198991198722(131198752)119899119904

minus 41198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991198991198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119899

1 +

1198722(131198750)1198881198881198722(131198752)119888119899

minus 1198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881199041198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119899

=

41198722(131198750)1198881198881198722(131198752)119888119899

minus 41198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881198881198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119888

1 +

1198722(131198750)1198881198881198722(131198752)119888119904minus 1198722(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119904

=

41198722(131198750)1198881198881198722(131198752)119888119904minus 4119872

2(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881198881198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119888

(20)

In relations (20) we eliminate the slopes and obtainmasses expressions which relate six meson states Insertingthe masses of the corresponding mesons into relations (20)we obtain the masses of 119872

119899119899 119872119899119904 119872119904119904 and 119872

119888119904of ground

scalar meson In our previous work a series of relationsbetweenmesonmass and the parameters of radial trajectoriesare established in the framework of Regge phenomenology[47 48] The relations are still applicable for the scalarmeson Inserting the ground scalar meson masses and theparameters in [47 48] the radial excited meson masses arepredicted The results are compared with other predictions

Advances in High Energy Physics 5

1 2 3

N

4

3

2

1

nbar(n)

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(a)

nbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(b)

sbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(c)

Figure 1 The quasi-linear trajectory for11987331198750meson nonet

and shown in Table 3 Moreover the radial excited masses ofthe light quark sector are shown in Figure 1

5 Results and Discussion

According to the introduction in Section 1 the isovectormember of nonet is important for the assignment of isoscalarmeson due to providing a beacon for the mass scale ofmeson nonet However both states the 119886

0(980) and 119886

0(1450)

are observed in the experiments and could be considered ascandidates In Table 3 our results favor the interpretation ofthe state 119886

0(980) rather than the 119886

0(1450) as the isovector

ground scalar meson The assignment is supported by thedifferent theoretical models prediction In the naive quarkmodel for 119878 = 1 states spin-orbit coupling results in massorder 119898(2

++) ge 119898(1

++) ge 119898(0

++) which supports the fact

that 1198860(980) (not the 119886

0(1450)) is the isovector member of

scalar nonet [49 50] In the scheme of K-matrix analysisAnvisoch also indicate that the mass of the 1

31198750isovector

state is about 960 plusmn 30MeV which is in excellent agreementwith our prediction [32]

Apart from the isovector member the isoscalar sectoris also ambiguous both experimentally and theoreticallyIn Table 3 we propose the bare nonstrange and strangemembers masses of isoscalar meson

In the last decade the heavy-light mesons that attractmore attention with many new resonances have beenobserved in the experiments such as 119863lowast

1199040(2317) 119863lowast

1199041(2460)

119863lowast

119904119869(2715) and 119863

lowast

119904119869(2860) For the 119863

lowast

1199040(2317) it was first

observed in 119863+

119878120587 by the Babar Collaboration at the Stanford

Linear Accelerator Center with a mass of about 2317MeVand width of about 350MeV [51] Subsequently this statehas also been reported by the Bell Collaboration [52] andthe CLEO [53] In our present work the mass of the charm-strange meson is about 100MeV higher than the measuredmass of 119863lowast

1199040(2317) which is consistent with the predictions

by Godfrey and Isgur [19] Consequently we suggest that theassignment for the119863lowast

1199040(2317) as a 13119875

0charm-strange meson

needs further testing in the experimentIn summary the understanding of scalar meson is quite

puzzling for the theorists and experimentalists Until nowthere is no one state that can be assigned as the scalar

6 Advances in High Energy Physics

meson member without dispute In the present work com-bining three complementary approaches non-relativisticconstituent quarkmodel Regge phenomenology andmeson-meson mixing we investigate the assignment of groundscalar meson Our results are useful for the assignment of thescalar meson in the future

Moreover we also find a bridge which relate the heavyand light flavor mesons in the process of analyzing scalarmeson The bridge is significant for the assignment of thediscovered states and searching the new resonances whichhave not been observed in the experiments

Acknowledgment

This project supported by the Zhengzhou University of LightIndustry Foundation China (Grant nos 2009XJJ011 and2012XJJ008)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in twodimensions Applications to ferromagnets and massive Yang-Mills fieldsrdquo Physics Letters B vol 59 no 1 pp 79ndash81 1975

[2] P Maris and C D Roberts ldquoDyson-schwinger equations a toolfor hadron physicsrdquo International Journal of Modern Physics Evol 12 no 3 article 297 2003

[3] M Gell-Mann ldquoA schematic model of baryons and mesonsrdquoPhysics Letters vol 8 no 3 pp 214ndash215 1964

[4] F E Close andN A Tornqvist ldquoScalarmesons above and below1 GeVrdquo Journal of Physics G vol 28 no 10 article R249 2002

[5] A H Fariborz ldquoIsosinglet scalar mesons below 2 GeV and thescalar glueball massrdquo International Journal of Modern Physics Avol 19 no 13 article 2095 2004

[6] Y H Liu D M Li J X Shao X G Wang and Z Y ZhangldquoRegarding the scalar mesonsrdquo Physical Review D vol 77 no 3Article ID 034025 5 pages 2008

[7] DM Li KWWei and Y Hong ldquoA possible assignment for theground scalar meson nonetrdquo European Physical Journal A vol25 pp 263ndash266 2005

[8] M Schumacher ldquoStructure of scalar mesons and the Higgssector of strong interactionrdquo Journal of Physics G vol 38 no8 Article ID 083001 2011

[9] Z T Xia and W Zuo ldquoA mixing scheme for the structure off0(600) and f

0(1370)rdquo Nuclear Physics A vol 848 pp 317ndash329

2010[10] X Liu and Z J Xiao ldquoLight scalar mesons and charmless

hadronic B119888rarr SPSVdecays in the perturbativeQCDapproachrdquo

Physical Review Letters vol 82 Article ID 054029 2010[11] Z G Wang ldquoAnalysis of the nonet scalar mesons as tetraquark

states with new QCD sum rulesrdquo International Journal ofTheoretical Physics vol 51 no 2 pp 507ndash517 2012

[12] A H Fariborz R Jora and J Schechter ldquoLight scalar puzzle inQCDrdquo AIP Conference Proceedings vol 1361 pp 127ndash131 2011

[13] W de Paula and T Frederico ldquoScalar mesons within a dynami-cal holographic QCD modelrdquo Physics Letters B vol 693 no 3pp 287ndash291 2010

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

[15] J Weinstein and N Isgur ldquo119870119870 moleculesrdquo Physical Review Dvol 41 no 7 pp 2236ndash2257 1990

[16] R L Jaffe ldquoMultiquark hadrons I Phenomenology of 1198762119876minus2mesonsrdquo Physical Review D vol 15 no 1 pp 267ndash280 1977

[17] F Shi T G Steele V Elias K B Sprague Y Xue and A HFariborz ldquoHolder inequalities and isospin splitting of the quarkscalar mesonsrdquoNuclear Physics A vol 671 no 1ndash4 pp 416ndash4462000

[18] D Black A H Fariborz S Moussa S Nasri and J SchechterldquoUnitarized pseudoscalar meson scattering amplitudes fromthree flavor linear sigma modelsrdquo Physical Review D vol 64no 1 Article ID 014031 45 pages 2001

[19] S Godfrey and N Isgur ldquoMesons in a relativized quark modelwith chromodynamicsrdquo Physical Review D vol 32 no 1 pp189ndash231 1985

[20] M Di Pierro and E J Eichten ldquoExcited heavy-light systems andhadronic transitionsrdquo Physical Review D vol 64 no 11 ArticleID 114004 20 pages 2001

[21] T Barnes F E Close and H J Lipkin ldquoImplications of a Dkmolecule at 232-GeVrdquo Physical Review D vol 68 Article ID054006 5 pages 2003

[22] H Y Cheng and W S Hou ldquoB decays as spectroscope forcharmed four-quark statesrdquo Physics Letters B vol 566 no 3-4pp 193ndash200 2003

[23] A P Szczepaniak ldquoDescription of the 119863lowast119904resonance as the D120587

atomrdquo Physics Letters B vol 567 no 1-2 pp 23ndash26 2003[24] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in the

nonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

[25] W Lucha F F Schoberl and D Gromes ldquoBound states ofquarksrdquo Physics Reports vol 200 no 4 pp 127ndash240 1991

[26] D Flamm and F Schoberl Introduction to Quark Model of Ele-mentary Particles vol 1 Gordon and Breach Science PublishersLondon UK 1982

[27] KWWei X P Dong and G Lu ldquoMasses of 119904119904 states and nonetmixing anglesrdquo International Journal of Modern Physics A vol26 no 12 article 2065 2011

[28] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectrumin regge phenomenologyrdquoThe European Physical Journal C vol37 no 3 pp 323ndash333 2004

[29] D M Li and Z Li ldquoStrange axial-vector mesons mixing anglerdquoThe European Physical Journal vol 28 pp 369ndash373 2006

[30] L Burakovsky and T Goldman ldquoRegarding the enigmas of P-wave meson spectroscopyrdquo Physical Review D vol 57 no 5 pp2879ndash2888 1998

[31] M M Brisudova L Burakovsky and T Goldman ldquoNewglueball-meson mass relationsrdquo Physical Review D vol 58 no11 Article ID 114015 7 pages 1998

[32] A V Anisovich and A V Sarantsev ldquoK-matrix analysis ofthe K120587 S-wave in the mass region 900-2100 MeV and nonetclassification of scalar 119902119902-states rdquo Physics Letters B vol 413 no1-2 pp 137ndash146 1997

[33] A M Badalian and B L G Bakker ldquoHigher excitations of theD and D

119904mesonsrdquo Physical Review D vol 84 no 3 Article ID

034006 10 pages 2011[34] D Ebert R N Faustov and V O Galkin ldquoMass spectra and

Regge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

Advances in High Energy Physics 7

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

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 3

Table 1 Angular momentum part of the matrix elements of relations (4) and (5)

1198992119904+1

119897119869

111198780

131198781

131198752

131198750

⟨119871119878⟩ 1 minus2119886(119895 1) minus25 minus4⟨119878119902119878119902⟩ minus34 14 14 14

⟨119871(119878119902minus 119878119902)⟩ 0 0

Table 2 Predicted mass of bare 119904119904 for meson nonet 120579quad is the nonet mixing angle from the quadratic mass formulae [14]

Nonet 111198780

131198781

131198752

131198633

211198780

119872119904119904(MeV) Present work 685 1018 1530 1858 1323

[27] 697 1009 1546[28] 702 1010 1538[19] 960 1020 1530 1900 1630

120579quad (∘) minus115 387 296 32 minus224

as the square masses 1198722

120593and 119872

2

1205931015840 are the eigenvalues

respectivelyThe physical states 120593 and 1205931015840 can be related to the

119904119904 and119873 = (119906119906 + 119889119889)radic2 by

(phy1

phy2

) = 119880(119873

119878) (10)

and the unitary matrix 119880 can be described as

1198801198722119880dagger= (

1198722

phy1

0

0 1198722

phy2

) (11)

The mix of the physical isoscalar states can be alsodescribed as

(phy1

phy2

) = (cos 120579 minus sin 120579

sin 120579 cos 120579)(

1205938

1205931

) (12)

with

1205931=

(119906119906 + 119889119889 + 119904119904)

radic3

1205938=

(119906119906 + 119889119889 minus 2119904119904)

radic6

(13)

where 120579 is the nonet mixing angleFrom the relations (9) (10) (11) and (12) one will obtain

119880 = (cos 120579 minus sin 120579

sin 120579 cos 120579 )(

radic1

3minusradic

2

3

radic2

3radic

1

3

) (14)

Based on the relations (9) and (11) one can have thefollowing relations

1198722

119873+ 2119860 = (radic

1

3cos 120579 minus radic

2

3sin 120579)

2

1198722

phy1

+ (radic2

3cos 120579 + radic

1

3sin 120579)

2

1198722

phy2

1198722

119904119904+ 119860119883

2= (radic

1

3cos 120579 minus radic

2

3sin 120579)

2

1198722

phy2

+ (radic2

3cos 120579 + radic

1

3sin 120579)

2

1198722

phy1

(15)

Applying the relations (15) for themeson states 111198780 131198781

131198752 131198633 and 2

11198780 we obtain the bare 119904119904 member masses

and the results are present in Table 2 Here and below all themasses and themixing angle used as input for our calculationare taken from the PDG [14]

4 Regge Phenomenology and Mass ofIsovector Scalar Meson

Regge phenomenology is widely used for the calculation ofthe meson and baryonium spectrum [28 36ndash38] In [28]Li et al indicate that the quasi-linear Regge trajectoriescould provide a reasonable description for the meson massspectrum Anisovich et al also conclude that the mesonstates can fit to the quasi-linear Regge trajectories at the(119873 1198722) and (119869 1198722) planes with good accuracy [36] Inthe last decade Regge trajectories are reanalysis to makepredictions for the masses of states not yet to be discoveredin experiment or to determine the quantum numbers of thenewly discovered states

4 Advances in High Energy Physics

Table 3 Predicted mass119872119899119899 119872119899119904 119872119904119904 and119872

119888119904of ground scalar meson (All inMeV and the values used as input for our analysis are shown

in boldface)

Nonet Mass 119872119899119899

119872119899119904

119872119904119904

119872119888119899

119872119888119904

119872119888119888

131198750

Present work 1024 1279 1414 2318 2423 3415[32] 960 plusmn 30 1220

+50

minus1501220 plusmn 40

[24] 983 plusmn 10 1204 plusmn 10 1397 plusmn 16[19] 1090 1240 1360 2400 2480 3440

231198750

Present work 1611 1826 1924 2835 3099 3958[33] 2880 2970

[34 35] 2919 3054[19] 1780 1890 1990 3920

331198750

Present work 2034 2243 2390 3270 3653 4434

Based on the assumption that the hadrons with identical119869PC quantum numbers obey quasi-linear form of Reggetrajectories one have the following relations [28]

119897 = 1205721198941198941015840(0) + 120572

1015840

11989411989410158401198722

1198941198941015840

119897 = 1205721198941198951015840(0) + 120572

1015840

11989411989510158401198722

1198941198951015840

119897 = 1205721198951198951015840(0) + 120572

1015840

11989511989510158401198722

1198951198951015840

(16)

where 1198941198941015840 denotes the quark and antiquark flavors and 119897 and

1198721198941198941015840are the orbital momentum and mass of the 119894119894

1015840 mesonThe parameters 1205721015840

1198941198941015840and 120572

1198941198941015840(0) are respectively the slope and

intercept of the Regge trajectory The intercepts and slopescan be expressed by [28 39]

120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (17)

1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(18)

The relation (17) is satisfied in two-dimensional QCD[40] the dual-analytic model [41] and the quark bremss-trahlung model [42] The relation (18) is deprived fromthe topological and the 119902119902-string picture of hadrons [43]Recently based on the available data for mesonic resonancesof light medium and heavy flavors Filipponi et al [44]and Burakovsky and Goldman [45] construct different formslopes (19) of linear Regge trajectories for all quark flavorsOne has the following

1205721015840

119894119895=

09GeVminus2

1 + 022((119898119894+ 119898119895) GeV)

32

1205721015840

119894119895=

1205721015840

1205874 + (1205874)radic1205721015840(119898119894+ 119898119895) 2

(19)

According to the relations (19) we find the slope of Reggetrajectory depends on the constituent quark masses throughthe combination (119898

119894+ 119898119895) The relations (19) are obtained

by fitting experimental data and these relations are possible

to update with the increasing amount of experimental dataIn the present work we adopt the assumption presented by[28 31 46] that the slopes of the parity partners (eg the 0+1minus 2+ and 1

minus are parity partners) trajectories coincide andfurther that the slopes do not depend on charge conjugationin accordance with the C-invariance of QCD Under thisassumption the corresponding slopes of the 13119875

2are the same

as those of the 131198750trajectories and we have the following

relations

1 +

1198722(131198750)1198991198991198722(131198752)119899119904

minus 1198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119904

=

41198722(131198750)1198991198991198722(131198752)119899119904

minus 41198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991198991198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119899

1 +

1198722(131198750)1198881198881198722(131198752)119888119899

minus 1198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881199041198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119899

=

41198722(131198750)1198881198881198722(131198752)119888119899

minus 41198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881198881198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119888

1 +

1198722(131198750)1198881198881198722(131198752)119888119904minus 1198722(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119904

=

41198722(131198750)1198881198881198722(131198752)119888119904minus 4119872

2(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881198881198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119888

(20)

In relations (20) we eliminate the slopes and obtainmasses expressions which relate six meson states Insertingthe masses of the corresponding mesons into relations (20)we obtain the masses of 119872

119899119899 119872119899119904 119872119904119904 and 119872

119888119904of ground

scalar meson In our previous work a series of relationsbetweenmesonmass and the parameters of radial trajectoriesare established in the framework of Regge phenomenology[47 48] The relations are still applicable for the scalarmeson Inserting the ground scalar meson masses and theparameters in [47 48] the radial excited meson masses arepredicted The results are compared with other predictions

Advances in High Energy Physics 5

1 2 3

N

4

3

2

1

nbar(n)

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(a)

nbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(b)

sbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(c)

Figure 1 The quasi-linear trajectory for11987331198750meson nonet

and shown in Table 3 Moreover the radial excited masses ofthe light quark sector are shown in Figure 1

5 Results and Discussion

According to the introduction in Section 1 the isovectormember of nonet is important for the assignment of isoscalarmeson due to providing a beacon for the mass scale ofmeson nonet However both states the 119886

0(980) and 119886

0(1450)

are observed in the experiments and could be considered ascandidates In Table 3 our results favor the interpretation ofthe state 119886

0(980) rather than the 119886

0(1450) as the isovector

ground scalar meson The assignment is supported by thedifferent theoretical models prediction In the naive quarkmodel for 119878 = 1 states spin-orbit coupling results in massorder 119898(2

++) ge 119898(1

++) ge 119898(0

++) which supports the fact

that 1198860(980) (not the 119886

0(1450)) is the isovector member of

scalar nonet [49 50] In the scheme of K-matrix analysisAnvisoch also indicate that the mass of the 1

31198750isovector

state is about 960 plusmn 30MeV which is in excellent agreementwith our prediction [32]

Apart from the isovector member the isoscalar sectoris also ambiguous both experimentally and theoreticallyIn Table 3 we propose the bare nonstrange and strangemembers masses of isoscalar meson

In the last decade the heavy-light mesons that attractmore attention with many new resonances have beenobserved in the experiments such as 119863lowast

1199040(2317) 119863lowast

1199041(2460)

119863lowast

119904119869(2715) and 119863

lowast

119904119869(2860) For the 119863

lowast

1199040(2317) it was first

observed in 119863+

119878120587 by the Babar Collaboration at the Stanford

Linear Accelerator Center with a mass of about 2317MeVand width of about 350MeV [51] Subsequently this statehas also been reported by the Bell Collaboration [52] andthe CLEO [53] In our present work the mass of the charm-strange meson is about 100MeV higher than the measuredmass of 119863lowast

1199040(2317) which is consistent with the predictions

by Godfrey and Isgur [19] Consequently we suggest that theassignment for the119863lowast

1199040(2317) as a 13119875

0charm-strange meson

needs further testing in the experimentIn summary the understanding of scalar meson is quite

puzzling for the theorists and experimentalists Until nowthere is no one state that can be assigned as the scalar

6 Advances in High Energy Physics

meson member without dispute In the present work com-bining three complementary approaches non-relativisticconstituent quarkmodel Regge phenomenology andmeson-meson mixing we investigate the assignment of groundscalar meson Our results are useful for the assignment of thescalar meson in the future

Moreover we also find a bridge which relate the heavyand light flavor mesons in the process of analyzing scalarmeson The bridge is significant for the assignment of thediscovered states and searching the new resonances whichhave not been observed in the experiments

Acknowledgment

This project supported by the Zhengzhou University of LightIndustry Foundation China (Grant nos 2009XJJ011 and2012XJJ008)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in twodimensions Applications to ferromagnets and massive Yang-Mills fieldsrdquo Physics Letters B vol 59 no 1 pp 79ndash81 1975

[2] P Maris and C D Roberts ldquoDyson-schwinger equations a toolfor hadron physicsrdquo International Journal of Modern Physics Evol 12 no 3 article 297 2003

[3] M Gell-Mann ldquoA schematic model of baryons and mesonsrdquoPhysics Letters vol 8 no 3 pp 214ndash215 1964

[4] F E Close andN A Tornqvist ldquoScalarmesons above and below1 GeVrdquo Journal of Physics G vol 28 no 10 article R249 2002

[5] A H Fariborz ldquoIsosinglet scalar mesons below 2 GeV and thescalar glueball massrdquo International Journal of Modern Physics Avol 19 no 13 article 2095 2004

[6] Y H Liu D M Li J X Shao X G Wang and Z Y ZhangldquoRegarding the scalar mesonsrdquo Physical Review D vol 77 no 3Article ID 034025 5 pages 2008

[7] DM Li KWWei and Y Hong ldquoA possible assignment for theground scalar meson nonetrdquo European Physical Journal A vol25 pp 263ndash266 2005

[8] M Schumacher ldquoStructure of scalar mesons and the Higgssector of strong interactionrdquo Journal of Physics G vol 38 no8 Article ID 083001 2011

[9] Z T Xia and W Zuo ldquoA mixing scheme for the structure off0(600) and f

0(1370)rdquo Nuclear Physics A vol 848 pp 317ndash329

2010[10] X Liu and Z J Xiao ldquoLight scalar mesons and charmless

hadronic B119888rarr SPSVdecays in the perturbativeQCDapproachrdquo

Physical Review Letters vol 82 Article ID 054029 2010[11] Z G Wang ldquoAnalysis of the nonet scalar mesons as tetraquark

states with new QCD sum rulesrdquo International Journal ofTheoretical Physics vol 51 no 2 pp 507ndash517 2012

[12] A H Fariborz R Jora and J Schechter ldquoLight scalar puzzle inQCDrdquo AIP Conference Proceedings vol 1361 pp 127ndash131 2011

[13] W de Paula and T Frederico ldquoScalar mesons within a dynami-cal holographic QCD modelrdquo Physics Letters B vol 693 no 3pp 287ndash291 2010

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

[15] J Weinstein and N Isgur ldquo119870119870 moleculesrdquo Physical Review Dvol 41 no 7 pp 2236ndash2257 1990

[16] R L Jaffe ldquoMultiquark hadrons I Phenomenology of 1198762119876minus2mesonsrdquo Physical Review D vol 15 no 1 pp 267ndash280 1977

[17] F Shi T G Steele V Elias K B Sprague Y Xue and A HFariborz ldquoHolder inequalities and isospin splitting of the quarkscalar mesonsrdquoNuclear Physics A vol 671 no 1ndash4 pp 416ndash4462000

[18] D Black A H Fariborz S Moussa S Nasri and J SchechterldquoUnitarized pseudoscalar meson scattering amplitudes fromthree flavor linear sigma modelsrdquo Physical Review D vol 64no 1 Article ID 014031 45 pages 2001

[19] S Godfrey and N Isgur ldquoMesons in a relativized quark modelwith chromodynamicsrdquo Physical Review D vol 32 no 1 pp189ndash231 1985

[20] M Di Pierro and E J Eichten ldquoExcited heavy-light systems andhadronic transitionsrdquo Physical Review D vol 64 no 11 ArticleID 114004 20 pages 2001

[21] T Barnes F E Close and H J Lipkin ldquoImplications of a Dkmolecule at 232-GeVrdquo Physical Review D vol 68 Article ID054006 5 pages 2003

[22] H Y Cheng and W S Hou ldquoB decays as spectroscope forcharmed four-quark statesrdquo Physics Letters B vol 566 no 3-4pp 193ndash200 2003

[23] A P Szczepaniak ldquoDescription of the 119863lowast119904resonance as the D120587

atomrdquo Physics Letters B vol 567 no 1-2 pp 23ndash26 2003[24] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in the

nonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

[25] W Lucha F F Schoberl and D Gromes ldquoBound states ofquarksrdquo Physics Reports vol 200 no 4 pp 127ndash240 1991

[26] D Flamm and F Schoberl Introduction to Quark Model of Ele-mentary Particles vol 1 Gordon and Breach Science PublishersLondon UK 1982

[27] KWWei X P Dong and G Lu ldquoMasses of 119904119904 states and nonetmixing anglesrdquo International Journal of Modern Physics A vol26 no 12 article 2065 2011

[28] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectrumin regge phenomenologyrdquoThe European Physical Journal C vol37 no 3 pp 323ndash333 2004

[29] D M Li and Z Li ldquoStrange axial-vector mesons mixing anglerdquoThe European Physical Journal vol 28 pp 369ndash373 2006

[30] L Burakovsky and T Goldman ldquoRegarding the enigmas of P-wave meson spectroscopyrdquo Physical Review D vol 57 no 5 pp2879ndash2888 1998

[31] M M Brisudova L Burakovsky and T Goldman ldquoNewglueball-meson mass relationsrdquo Physical Review D vol 58 no11 Article ID 114015 7 pages 1998

[32] A V Anisovich and A V Sarantsev ldquoK-matrix analysis ofthe K120587 S-wave in the mass region 900-2100 MeV and nonetclassification of scalar 119902119902-states rdquo Physics Letters B vol 413 no1-2 pp 137ndash146 1997

[33] A M Badalian and B L G Bakker ldquoHigher excitations of theD and D

119904mesonsrdquo Physical Review D vol 84 no 3 Article ID

034006 10 pages 2011[34] D Ebert R N Faustov and V O Galkin ldquoMass spectra and

Regge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

Advances in High Energy Physics 7

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

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

4 Advances in High Energy Physics

Table 3 Predicted mass119872119899119899 119872119899119904 119872119904119904 and119872

119888119904of ground scalar meson (All inMeV and the values used as input for our analysis are shown

in boldface)

Nonet Mass 119872119899119899

119872119899119904

119872119904119904

119872119888119899

119872119888119904

119872119888119888

131198750

Present work 1024 1279 1414 2318 2423 3415[32] 960 plusmn 30 1220

+50

minus1501220 plusmn 40

[24] 983 plusmn 10 1204 plusmn 10 1397 plusmn 16[19] 1090 1240 1360 2400 2480 3440

231198750

Present work 1611 1826 1924 2835 3099 3958[33] 2880 2970

[34 35] 2919 3054[19] 1780 1890 1990 3920

331198750

Present work 2034 2243 2390 3270 3653 4434

Based on the assumption that the hadrons with identical119869PC quantum numbers obey quasi-linear form of Reggetrajectories one have the following relations [28]

119897 = 1205721198941198941015840(0) + 120572

1015840

11989411989410158401198722

1198941198941015840

119897 = 1205721198941198951015840(0) + 120572

1015840

11989411989510158401198722

1198941198951015840

119897 = 1205721198951198951015840(0) + 120572

1015840

11989511989510158401198722

1198951198951015840

(16)

where 1198941198941015840 denotes the quark and antiquark flavors and 119897 and

1198721198941198941015840are the orbital momentum and mass of the 119894119894

1015840 mesonThe parameters 1205721015840

1198941198941015840and 120572

1198941198941015840(0) are respectively the slope and

intercept of the Regge trajectory The intercepts and slopescan be expressed by [28 39]

120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (17)

1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(18)

The relation (17) is satisfied in two-dimensional QCD[40] the dual-analytic model [41] and the quark bremss-trahlung model [42] The relation (18) is deprived fromthe topological and the 119902119902-string picture of hadrons [43]Recently based on the available data for mesonic resonancesof light medium and heavy flavors Filipponi et al [44]and Burakovsky and Goldman [45] construct different formslopes (19) of linear Regge trajectories for all quark flavorsOne has the following

1205721015840

119894119895=

09GeVminus2

1 + 022((119898119894+ 119898119895) GeV)

32

1205721015840

119894119895=

1205721015840

1205874 + (1205874)radic1205721015840(119898119894+ 119898119895) 2

(19)

According to the relations (19) we find the slope of Reggetrajectory depends on the constituent quark masses throughthe combination (119898

119894+ 119898119895) The relations (19) are obtained

by fitting experimental data and these relations are possible

to update with the increasing amount of experimental dataIn the present work we adopt the assumption presented by[28 31 46] that the slopes of the parity partners (eg the 0+1minus 2+ and 1

minus are parity partners) trajectories coincide andfurther that the slopes do not depend on charge conjugationin accordance with the C-invariance of QCD Under thisassumption the corresponding slopes of the 13119875

2are the same

as those of the 131198750trajectories and we have the following

relations

1 +

1198722(131198750)1198991198991198722(131198752)119899119904

minus 1198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119904

=

41198722(131198750)1198991198991198722(131198752)119899119904

minus 41198722(131198750)1198991199041198722(131198752)119904119904

1198722(131198750)1198991198991198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119899119899

1 +

1198722(131198750)1198881198881198722(131198752)119888119899

minus 1198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881199041198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119899

=

41198722(131198750)1198881198881198722(131198752)119888119899

minus 41198722(131198750)1198881198991198722(131198752)119899119899

1198722(131198750)1198881198881198722(131198752)119899119899

minus 1198722(131198750)1198991198991198722(131198752)119888119888

1 +

1198722(131198750)1198881198881198722(131198752)119888119904minus 1198722(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881199041198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119904

=

41198722(131198750)1198881198881198722(131198752)119888119904minus 4119872

2(131198750)1198881199041198722(131198752)119904119904

1198722(131198750)1198881198881198722(131198752)119904119904minus 1198722(131198750)1199041199041198722(131198752)119888119888

(20)

In relations (20) we eliminate the slopes and obtainmasses expressions which relate six meson states Insertingthe masses of the corresponding mesons into relations (20)we obtain the masses of 119872

119899119899 119872119899119904 119872119904119904 and 119872

119888119904of ground

scalar meson In our previous work a series of relationsbetweenmesonmass and the parameters of radial trajectoriesare established in the framework of Regge phenomenology[47 48] The relations are still applicable for the scalarmeson Inserting the ground scalar meson masses and theparameters in [47 48] the radial excited meson masses arepredicted The results are compared with other predictions

Advances in High Energy Physics 5

1 2 3

N

4

3

2

1

nbar(n)

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(a)

nbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(b)

sbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(c)

Figure 1 The quasi-linear trajectory for11987331198750meson nonet

and shown in Table 3 Moreover the radial excited masses ofthe light quark sector are shown in Figure 1

5 Results and Discussion

According to the introduction in Section 1 the isovectormember of nonet is important for the assignment of isoscalarmeson due to providing a beacon for the mass scale ofmeson nonet However both states the 119886

0(980) and 119886

0(1450)

are observed in the experiments and could be considered ascandidates In Table 3 our results favor the interpretation ofthe state 119886

0(980) rather than the 119886

0(1450) as the isovector

ground scalar meson The assignment is supported by thedifferent theoretical models prediction In the naive quarkmodel for 119878 = 1 states spin-orbit coupling results in massorder 119898(2

++) ge 119898(1

++) ge 119898(0

++) which supports the fact

that 1198860(980) (not the 119886

0(1450)) is the isovector member of

scalar nonet [49 50] In the scheme of K-matrix analysisAnvisoch also indicate that the mass of the 1

31198750isovector

state is about 960 plusmn 30MeV which is in excellent agreementwith our prediction [32]

Apart from the isovector member the isoscalar sectoris also ambiguous both experimentally and theoreticallyIn Table 3 we propose the bare nonstrange and strangemembers masses of isoscalar meson

In the last decade the heavy-light mesons that attractmore attention with many new resonances have beenobserved in the experiments such as 119863lowast

1199040(2317) 119863lowast

1199041(2460)

119863lowast

119904119869(2715) and 119863

lowast

119904119869(2860) For the 119863

lowast

1199040(2317) it was first

observed in 119863+

119878120587 by the Babar Collaboration at the Stanford

Linear Accelerator Center with a mass of about 2317MeVand width of about 350MeV [51] Subsequently this statehas also been reported by the Bell Collaboration [52] andthe CLEO [53] In our present work the mass of the charm-strange meson is about 100MeV higher than the measuredmass of 119863lowast

1199040(2317) which is consistent with the predictions

by Godfrey and Isgur [19] Consequently we suggest that theassignment for the119863lowast

1199040(2317) as a 13119875

0charm-strange meson

needs further testing in the experimentIn summary the understanding of scalar meson is quite

puzzling for the theorists and experimentalists Until nowthere is no one state that can be assigned as the scalar

6 Advances in High Energy Physics

meson member without dispute In the present work com-bining three complementary approaches non-relativisticconstituent quarkmodel Regge phenomenology andmeson-meson mixing we investigate the assignment of groundscalar meson Our results are useful for the assignment of thescalar meson in the future

Moreover we also find a bridge which relate the heavyand light flavor mesons in the process of analyzing scalarmeson The bridge is significant for the assignment of thediscovered states and searching the new resonances whichhave not been observed in the experiments

Acknowledgment

This project supported by the Zhengzhou University of LightIndustry Foundation China (Grant nos 2009XJJ011 and2012XJJ008)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in twodimensions Applications to ferromagnets and massive Yang-Mills fieldsrdquo Physics Letters B vol 59 no 1 pp 79ndash81 1975

[2] P Maris and C D Roberts ldquoDyson-schwinger equations a toolfor hadron physicsrdquo International Journal of Modern Physics Evol 12 no 3 article 297 2003

[3] M Gell-Mann ldquoA schematic model of baryons and mesonsrdquoPhysics Letters vol 8 no 3 pp 214ndash215 1964

[4] F E Close andN A Tornqvist ldquoScalarmesons above and below1 GeVrdquo Journal of Physics G vol 28 no 10 article R249 2002

[5] A H Fariborz ldquoIsosinglet scalar mesons below 2 GeV and thescalar glueball massrdquo International Journal of Modern Physics Avol 19 no 13 article 2095 2004

[6] Y H Liu D M Li J X Shao X G Wang and Z Y ZhangldquoRegarding the scalar mesonsrdquo Physical Review D vol 77 no 3Article ID 034025 5 pages 2008

[7] DM Li KWWei and Y Hong ldquoA possible assignment for theground scalar meson nonetrdquo European Physical Journal A vol25 pp 263ndash266 2005

[8] M Schumacher ldquoStructure of scalar mesons and the Higgssector of strong interactionrdquo Journal of Physics G vol 38 no8 Article ID 083001 2011

[9] Z T Xia and W Zuo ldquoA mixing scheme for the structure off0(600) and f

0(1370)rdquo Nuclear Physics A vol 848 pp 317ndash329

2010[10] X Liu and Z J Xiao ldquoLight scalar mesons and charmless

hadronic B119888rarr SPSVdecays in the perturbativeQCDapproachrdquo

Physical Review Letters vol 82 Article ID 054029 2010[11] Z G Wang ldquoAnalysis of the nonet scalar mesons as tetraquark

states with new QCD sum rulesrdquo International Journal ofTheoretical Physics vol 51 no 2 pp 507ndash517 2012

[12] A H Fariborz R Jora and J Schechter ldquoLight scalar puzzle inQCDrdquo AIP Conference Proceedings vol 1361 pp 127ndash131 2011

[13] W de Paula and T Frederico ldquoScalar mesons within a dynami-cal holographic QCD modelrdquo Physics Letters B vol 693 no 3pp 287ndash291 2010

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

[15] J Weinstein and N Isgur ldquo119870119870 moleculesrdquo Physical Review Dvol 41 no 7 pp 2236ndash2257 1990

[16] R L Jaffe ldquoMultiquark hadrons I Phenomenology of 1198762119876minus2mesonsrdquo Physical Review D vol 15 no 1 pp 267ndash280 1977

[17] F Shi T G Steele V Elias K B Sprague Y Xue and A HFariborz ldquoHolder inequalities and isospin splitting of the quarkscalar mesonsrdquoNuclear Physics A vol 671 no 1ndash4 pp 416ndash4462000

[18] D Black A H Fariborz S Moussa S Nasri and J SchechterldquoUnitarized pseudoscalar meson scattering amplitudes fromthree flavor linear sigma modelsrdquo Physical Review D vol 64no 1 Article ID 014031 45 pages 2001

[19] S Godfrey and N Isgur ldquoMesons in a relativized quark modelwith chromodynamicsrdquo Physical Review D vol 32 no 1 pp189ndash231 1985

[20] M Di Pierro and E J Eichten ldquoExcited heavy-light systems andhadronic transitionsrdquo Physical Review D vol 64 no 11 ArticleID 114004 20 pages 2001

[21] T Barnes F E Close and H J Lipkin ldquoImplications of a Dkmolecule at 232-GeVrdquo Physical Review D vol 68 Article ID054006 5 pages 2003

[22] H Y Cheng and W S Hou ldquoB decays as spectroscope forcharmed four-quark statesrdquo Physics Letters B vol 566 no 3-4pp 193ndash200 2003

[23] A P Szczepaniak ldquoDescription of the 119863lowast119904resonance as the D120587

atomrdquo Physics Letters B vol 567 no 1-2 pp 23ndash26 2003[24] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in the

nonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

[25] W Lucha F F Schoberl and D Gromes ldquoBound states ofquarksrdquo Physics Reports vol 200 no 4 pp 127ndash240 1991

[26] D Flamm and F Schoberl Introduction to Quark Model of Ele-mentary Particles vol 1 Gordon and Breach Science PublishersLondon UK 1982

[27] KWWei X P Dong and G Lu ldquoMasses of 119904119904 states and nonetmixing anglesrdquo International Journal of Modern Physics A vol26 no 12 article 2065 2011

[28] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectrumin regge phenomenologyrdquoThe European Physical Journal C vol37 no 3 pp 323ndash333 2004

[29] D M Li and Z Li ldquoStrange axial-vector mesons mixing anglerdquoThe European Physical Journal vol 28 pp 369ndash373 2006

[30] L Burakovsky and T Goldman ldquoRegarding the enigmas of P-wave meson spectroscopyrdquo Physical Review D vol 57 no 5 pp2879ndash2888 1998

[31] M M Brisudova L Burakovsky and T Goldman ldquoNewglueball-meson mass relationsrdquo Physical Review D vol 58 no11 Article ID 114015 7 pages 1998

[32] A V Anisovich and A V Sarantsev ldquoK-matrix analysis ofthe K120587 S-wave in the mass region 900-2100 MeV and nonetclassification of scalar 119902119902-states rdquo Physics Letters B vol 413 no1-2 pp 137ndash146 1997

[33] A M Badalian and B L G Bakker ldquoHigher excitations of theD and D

119904mesonsrdquo Physical Review D vol 84 no 3 Article ID

034006 10 pages 2011[34] D Ebert R N Faustov and V O Galkin ldquoMass spectra and

Regge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

Advances in High Energy Physics 7

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

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

1 2 3

N

4

3

2

1

nbar(n)

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(a)

nbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(b)

sbar(s)

1 2 3

N

5

4

6

3

2

1

M2(G

eV)

Present workReference [40]

Reference [21]Reference [19]

(c)

Figure 1 The quasi-linear trajectory for11987331198750meson nonet

and shown in Table 3 Moreover the radial excited masses ofthe light quark sector are shown in Figure 1

5 Results and Discussion

According to the introduction in Section 1 the isovectormember of nonet is important for the assignment of isoscalarmeson due to providing a beacon for the mass scale ofmeson nonet However both states the 119886

0(980) and 119886

0(1450)

are observed in the experiments and could be considered ascandidates In Table 3 our results favor the interpretation ofthe state 119886

0(980) rather than the 119886

0(1450) as the isovector

ground scalar meson The assignment is supported by thedifferent theoretical models prediction In the naive quarkmodel for 119878 = 1 states spin-orbit coupling results in massorder 119898(2

++) ge 119898(1

++) ge 119898(0

++) which supports the fact

that 1198860(980) (not the 119886

0(1450)) is the isovector member of

scalar nonet [49 50] In the scheme of K-matrix analysisAnvisoch also indicate that the mass of the 1

31198750isovector

state is about 960 plusmn 30MeV which is in excellent agreementwith our prediction [32]

Apart from the isovector member the isoscalar sectoris also ambiguous both experimentally and theoreticallyIn Table 3 we propose the bare nonstrange and strangemembers masses of isoscalar meson

In the last decade the heavy-light mesons that attractmore attention with many new resonances have beenobserved in the experiments such as 119863lowast

1199040(2317) 119863lowast

1199041(2460)

119863lowast

119904119869(2715) and 119863

lowast

119904119869(2860) For the 119863

lowast

1199040(2317) it was first

observed in 119863+

119878120587 by the Babar Collaboration at the Stanford

Linear Accelerator Center with a mass of about 2317MeVand width of about 350MeV [51] Subsequently this statehas also been reported by the Bell Collaboration [52] andthe CLEO [53] In our present work the mass of the charm-strange meson is about 100MeV higher than the measuredmass of 119863lowast

1199040(2317) which is consistent with the predictions

by Godfrey and Isgur [19] Consequently we suggest that theassignment for the119863lowast

1199040(2317) as a 13119875

0charm-strange meson

needs further testing in the experimentIn summary the understanding of scalar meson is quite

puzzling for the theorists and experimentalists Until nowthere is no one state that can be assigned as the scalar

6 Advances in High Energy Physics

meson member without dispute In the present work com-bining three complementary approaches non-relativisticconstituent quarkmodel Regge phenomenology andmeson-meson mixing we investigate the assignment of groundscalar meson Our results are useful for the assignment of thescalar meson in the future

Moreover we also find a bridge which relate the heavyand light flavor mesons in the process of analyzing scalarmeson The bridge is significant for the assignment of thediscovered states and searching the new resonances whichhave not been observed in the experiments

Acknowledgment

This project supported by the Zhengzhou University of LightIndustry Foundation China (Grant nos 2009XJJ011 and2012XJJ008)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in twodimensions Applications to ferromagnets and massive Yang-Mills fieldsrdquo Physics Letters B vol 59 no 1 pp 79ndash81 1975

[2] P Maris and C D Roberts ldquoDyson-schwinger equations a toolfor hadron physicsrdquo International Journal of Modern Physics Evol 12 no 3 article 297 2003

[3] M Gell-Mann ldquoA schematic model of baryons and mesonsrdquoPhysics Letters vol 8 no 3 pp 214ndash215 1964

[4] F E Close andN A Tornqvist ldquoScalarmesons above and below1 GeVrdquo Journal of Physics G vol 28 no 10 article R249 2002

[5] A H Fariborz ldquoIsosinglet scalar mesons below 2 GeV and thescalar glueball massrdquo International Journal of Modern Physics Avol 19 no 13 article 2095 2004

[6] Y H Liu D M Li J X Shao X G Wang and Z Y ZhangldquoRegarding the scalar mesonsrdquo Physical Review D vol 77 no 3Article ID 034025 5 pages 2008

[7] DM Li KWWei and Y Hong ldquoA possible assignment for theground scalar meson nonetrdquo European Physical Journal A vol25 pp 263ndash266 2005

[8] M Schumacher ldquoStructure of scalar mesons and the Higgssector of strong interactionrdquo Journal of Physics G vol 38 no8 Article ID 083001 2011

[9] Z T Xia and W Zuo ldquoA mixing scheme for the structure off0(600) and f

0(1370)rdquo Nuclear Physics A vol 848 pp 317ndash329

2010[10] X Liu and Z J Xiao ldquoLight scalar mesons and charmless

hadronic B119888rarr SPSVdecays in the perturbativeQCDapproachrdquo

Physical Review Letters vol 82 Article ID 054029 2010[11] Z G Wang ldquoAnalysis of the nonet scalar mesons as tetraquark

states with new QCD sum rulesrdquo International Journal ofTheoretical Physics vol 51 no 2 pp 507ndash517 2012

[12] A H Fariborz R Jora and J Schechter ldquoLight scalar puzzle inQCDrdquo AIP Conference Proceedings vol 1361 pp 127ndash131 2011

[13] W de Paula and T Frederico ldquoScalar mesons within a dynami-cal holographic QCD modelrdquo Physics Letters B vol 693 no 3pp 287ndash291 2010

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

[15] J Weinstein and N Isgur ldquo119870119870 moleculesrdquo Physical Review Dvol 41 no 7 pp 2236ndash2257 1990

[16] R L Jaffe ldquoMultiquark hadrons I Phenomenology of 1198762119876minus2mesonsrdquo Physical Review D vol 15 no 1 pp 267ndash280 1977

[17] F Shi T G Steele V Elias K B Sprague Y Xue and A HFariborz ldquoHolder inequalities and isospin splitting of the quarkscalar mesonsrdquoNuclear Physics A vol 671 no 1ndash4 pp 416ndash4462000

[18] D Black A H Fariborz S Moussa S Nasri and J SchechterldquoUnitarized pseudoscalar meson scattering amplitudes fromthree flavor linear sigma modelsrdquo Physical Review D vol 64no 1 Article ID 014031 45 pages 2001

[19] S Godfrey and N Isgur ldquoMesons in a relativized quark modelwith chromodynamicsrdquo Physical Review D vol 32 no 1 pp189ndash231 1985

[20] M Di Pierro and E J Eichten ldquoExcited heavy-light systems andhadronic transitionsrdquo Physical Review D vol 64 no 11 ArticleID 114004 20 pages 2001

[21] T Barnes F E Close and H J Lipkin ldquoImplications of a Dkmolecule at 232-GeVrdquo Physical Review D vol 68 Article ID054006 5 pages 2003

[22] H Y Cheng and W S Hou ldquoB decays as spectroscope forcharmed four-quark statesrdquo Physics Letters B vol 566 no 3-4pp 193ndash200 2003

[23] A P Szczepaniak ldquoDescription of the 119863lowast119904resonance as the D120587

atomrdquo Physics Letters B vol 567 no 1-2 pp 23ndash26 2003[24] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in the

nonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

[25] W Lucha F F Schoberl and D Gromes ldquoBound states ofquarksrdquo Physics Reports vol 200 no 4 pp 127ndash240 1991

[26] D Flamm and F Schoberl Introduction to Quark Model of Ele-mentary Particles vol 1 Gordon and Breach Science PublishersLondon UK 1982

[27] KWWei X P Dong and G Lu ldquoMasses of 119904119904 states and nonetmixing anglesrdquo International Journal of Modern Physics A vol26 no 12 article 2065 2011

[28] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectrumin regge phenomenologyrdquoThe European Physical Journal C vol37 no 3 pp 323ndash333 2004

[29] D M Li and Z Li ldquoStrange axial-vector mesons mixing anglerdquoThe European Physical Journal vol 28 pp 369ndash373 2006

[30] L Burakovsky and T Goldman ldquoRegarding the enigmas of P-wave meson spectroscopyrdquo Physical Review D vol 57 no 5 pp2879ndash2888 1998

[31] M M Brisudova L Burakovsky and T Goldman ldquoNewglueball-meson mass relationsrdquo Physical Review D vol 58 no11 Article ID 114015 7 pages 1998

[32] A V Anisovich and A V Sarantsev ldquoK-matrix analysis ofthe K120587 S-wave in the mass region 900-2100 MeV and nonetclassification of scalar 119902119902-states rdquo Physics Letters B vol 413 no1-2 pp 137ndash146 1997

[33] A M Badalian and B L G Bakker ldquoHigher excitations of theD and D

119904mesonsrdquo Physical Review D vol 84 no 3 Article ID

034006 10 pages 2011[34] D Ebert R N Faustov and V O Galkin ldquoMass spectra and

Regge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

Advances in High Energy Physics 7

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

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

meson member without dispute In the present work com-bining three complementary approaches non-relativisticconstituent quarkmodel Regge phenomenology andmeson-meson mixing we investigate the assignment of groundscalar meson Our results are useful for the assignment of thescalar meson in the future

Moreover we also find a bridge which relate the heavyand light flavor mesons in the process of analyzing scalarmeson The bridge is significant for the assignment of thediscovered states and searching the new resonances whichhave not been observed in the experiments

Acknowledgment

This project supported by the Zhengzhou University of LightIndustry Foundation China (Grant nos 2009XJJ011 and2012XJJ008)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in twodimensions Applications to ferromagnets and massive Yang-Mills fieldsrdquo Physics Letters B vol 59 no 1 pp 79ndash81 1975

[2] P Maris and C D Roberts ldquoDyson-schwinger equations a toolfor hadron physicsrdquo International Journal of Modern Physics Evol 12 no 3 article 297 2003

[3] M Gell-Mann ldquoA schematic model of baryons and mesonsrdquoPhysics Letters vol 8 no 3 pp 214ndash215 1964

[4] F E Close andN A Tornqvist ldquoScalarmesons above and below1 GeVrdquo Journal of Physics G vol 28 no 10 article R249 2002

[5] A H Fariborz ldquoIsosinglet scalar mesons below 2 GeV and thescalar glueball massrdquo International Journal of Modern Physics Avol 19 no 13 article 2095 2004

[6] Y H Liu D M Li J X Shao X G Wang and Z Y ZhangldquoRegarding the scalar mesonsrdquo Physical Review D vol 77 no 3Article ID 034025 5 pages 2008

[7] DM Li KWWei and Y Hong ldquoA possible assignment for theground scalar meson nonetrdquo European Physical Journal A vol25 pp 263ndash266 2005

[8] M Schumacher ldquoStructure of scalar mesons and the Higgssector of strong interactionrdquo Journal of Physics G vol 38 no8 Article ID 083001 2011

[9] Z T Xia and W Zuo ldquoA mixing scheme for the structure off0(600) and f

0(1370)rdquo Nuclear Physics A vol 848 pp 317ndash329

2010[10] X Liu and Z J Xiao ldquoLight scalar mesons and charmless

hadronic B119888rarr SPSVdecays in the perturbativeQCDapproachrdquo

Physical Review Letters vol 82 Article ID 054029 2010[11] Z G Wang ldquoAnalysis of the nonet scalar mesons as tetraquark

states with new QCD sum rulesrdquo International Journal ofTheoretical Physics vol 51 no 2 pp 507ndash517 2012

[12] A H Fariborz R Jora and J Schechter ldquoLight scalar puzzle inQCDrdquo AIP Conference Proceedings vol 1361 pp 127ndash131 2011

[13] W de Paula and T Frederico ldquoScalar mesons within a dynami-cal holographic QCD modelrdquo Physics Letters B vol 693 no 3pp 287ndash291 2010

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

[15] J Weinstein and N Isgur ldquo119870119870 moleculesrdquo Physical Review Dvol 41 no 7 pp 2236ndash2257 1990

[16] R L Jaffe ldquoMultiquark hadrons I Phenomenology of 1198762119876minus2mesonsrdquo Physical Review D vol 15 no 1 pp 267ndash280 1977

[17] F Shi T G Steele V Elias K B Sprague Y Xue and A HFariborz ldquoHolder inequalities and isospin splitting of the quarkscalar mesonsrdquoNuclear Physics A vol 671 no 1ndash4 pp 416ndash4462000

[18] D Black A H Fariborz S Moussa S Nasri and J SchechterldquoUnitarized pseudoscalar meson scattering amplitudes fromthree flavor linear sigma modelsrdquo Physical Review D vol 64no 1 Article ID 014031 45 pages 2001

[19] S Godfrey and N Isgur ldquoMesons in a relativized quark modelwith chromodynamicsrdquo Physical Review D vol 32 no 1 pp189ndash231 1985

[20] M Di Pierro and E J Eichten ldquoExcited heavy-light systems andhadronic transitionsrdquo Physical Review D vol 64 no 11 ArticleID 114004 20 pages 2001

[21] T Barnes F E Close and H J Lipkin ldquoImplications of a Dkmolecule at 232-GeVrdquo Physical Review D vol 68 Article ID054006 5 pages 2003

[22] H Y Cheng and W S Hou ldquoB decays as spectroscope forcharmed four-quark statesrdquo Physics Letters B vol 566 no 3-4pp 193ndash200 2003

[23] A P Szczepaniak ldquoDescription of the 119863lowast119904resonance as the D120587

atomrdquo Physics Letters B vol 567 no 1-2 pp 23ndash26 2003[24] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in the

nonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

[25] W Lucha F F Schoberl and D Gromes ldquoBound states ofquarksrdquo Physics Reports vol 200 no 4 pp 127ndash240 1991

[26] D Flamm and F Schoberl Introduction to Quark Model of Ele-mentary Particles vol 1 Gordon and Breach Science PublishersLondon UK 1982

[27] KWWei X P Dong and G Lu ldquoMasses of 119904119904 states and nonetmixing anglesrdquo International Journal of Modern Physics A vol26 no 12 article 2065 2011

[28] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectrumin regge phenomenologyrdquoThe European Physical Journal C vol37 no 3 pp 323ndash333 2004

[29] D M Li and Z Li ldquoStrange axial-vector mesons mixing anglerdquoThe European Physical Journal vol 28 pp 369ndash373 2006

[30] L Burakovsky and T Goldman ldquoRegarding the enigmas of P-wave meson spectroscopyrdquo Physical Review D vol 57 no 5 pp2879ndash2888 1998

[31] M M Brisudova L Burakovsky and T Goldman ldquoNewglueball-meson mass relationsrdquo Physical Review D vol 58 no11 Article ID 114015 7 pages 1998

[32] A V Anisovich and A V Sarantsev ldquoK-matrix analysis ofthe K120587 S-wave in the mass region 900-2100 MeV and nonetclassification of scalar 119902119902-states rdquo Physics Letters B vol 413 no1-2 pp 137ndash146 1997

[33] A M Badalian and B L G Bakker ldquoHigher excitations of theD and D

119904mesonsrdquo Physical Review D vol 84 no 3 Article ID

034006 10 pages 2011[34] D Ebert R N Faustov and V O Galkin ldquoMass spectra and

Regge trajectories of light mesons in the relativistic quarkmodelrdquo Physical Review D vol 79 no 11 Article ID 114029 11pages 2009

Advances in High Energy Physics 7

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

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

[35] D Ebert R N Faustov and V O Galkin ldquoHeavy-light mesonspectroscopy and regge trajectories in the relativistic quarkmodelrdquo The European Physical Journal C vol 66 no 1-2 pp197ndash206 2010

[36] A V Anisovich V V Anisovich andA V Sarantsev ldquoSystemat-ics of 119902119902 states in the (nM2) and (JM2) planes rdquo Physical ReviewD vol 62 no 5 Article ID 051502 5 pages 2000

[37] A Zhang ldquoCharmonium spectrum and new observed statesrdquoPhysics Letters B vol 647 no 2-3 pp 140ndash144 2007

[38] X H Guo K W Wei and X H Wu ldquoSome mass relations formesons and baryons in Regge phenomenologyrdquo Physical ReviewD vol 78 no 5 Article ID 056005 21 pages 2008

[39] L Burakosky and T Goldman ldquoOn the regge slopes intramul-tiplet relationrdquo Physics Letters B vol 434 no 3-4 pp 251ndash2561998

[40] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is therelativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[41] N A Kobylinsky E S Martynov and A B PrognimakldquoInterrelations between intercepts of regge trajectories and newmass formulasrdquoUkrainskii Fizicheskii Zhurnal vol 24 pp 969ndash974 1979

[42] V V Dixit and L A Balazs ldquoRegge intercepts of multiquarksystems in feynmanrsquos bremsstrahlung analogyrdquo Physics LettersD vol 20 no 3 pp 816ndash819 1979

[43] A B Kaidalov ldquoHadronic mass-relations from topologicalexpansion and string modelrdquo Zeitschrift fur Physik C Particlesand Fields vol 12 no 1 pp 63ndash66 1982

[44] S Filipponi G Pancheri and Y Srivastava ldquoRegge trajectoriesfor all flavorsrdquo Physical Review Letters vol 80 no 9 pp 1838ndash1840 1998

[45] L Burakovsky and T Goldman ldquoComment on ldquoregge trajecto-ries for all flavorsrdquordquo Physical Review Letters vol 82 no 2 pp457ndash457 1999

[46] D M Li B Ma and Y H Liu ldquoUnderstanding the masses ofthe c119904 states in regge phenomenologyrdquo The European PhysicalJournal C vol 51 pp 359ndash365 2007

[47] Y H Liu X C Feng and J J Zhang ldquoInterpreting the massesof mesons in regge phenomenologyrdquo The European PhysicalJournal vol 43 pp 379ndash388 2010

[48] X C Feng Y Jia T Q Chang and J Y Li ldquoRadial excitationmass spectrum of tensor meson nonetrdquo Acta Physica PolonicaB vol 42 no 1 pp 25ndash32 2011

[49] L Maiani F Piccinini A D Polosa and V Riquer ldquoPositiveparity scalar mesons in the 1-2 GeV mass rangerdquo EuropeanPhysical Journal C vol 50 no 3 pp 609ndash616 2007

[50] E Borchi and R Gatto ldquoDynamical theory of nonleptonichyperon P waves in SU(6) symmetryrdquo Physical Review Lettersvol 14 no 13 pp 507ndash509 1965

[51] B Aubert et al ldquoObservation of a narrowmeson state decayingto119863119878

+1205870 at a mass of 232 GeVc2rdquo Physical Review Letters vol

90 no 24 Article ID 242001 7 pages 2003[52] P Krokovny K Abe T Abe et al ldquoObservation of the D

119904119869(2317)

and D119904119869(2457) in B decaysrdquo Physical Review Letters vol 91 no

26 Article ID 262002 6 pages 2003[53] D Besson S Anderson V V Frolov et al ldquoObservation of a

narrow resonance of mass 246 GeVc2 decaying to119863119878

lowast+1205870 and

confirmation of the DsJ (2317) staterdquo Physical Review D vol 68no 3 Article ID 032002 10 pages 2003

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