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Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 704529 5 pageshttpdxdoiorg1011552013704529

Research ArticleMass Spectrum of 1 1119875

1Meson State and Mixing Angle of

Strange Axial-Vector Mesons

Xue-Chao Feng1 Ke-Wei Wei2 Jie Wu1 Yun-Qiang Zhang1

Ming-Yang Wu1 and Feng-Chun Jiang1

1 Department of Technology and Physics Zhengzhou University of Light Industry Zhengzhou 450002 China2 College of Physics and Electrical Engineering Anyang Normal University Anyang 455002 China

Correspondence should be addressed to Xue-Chao Feng fxchaozzulieducn

Received 29 December 2012 Revised 6 March 2013 Accepted 26 March 2013

Academic Editor Alexey Petrov

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

In the presence of the 1198871(1235) ℎ

1(1170)119863

1(2420) and119863

1199041(2536)plusmn being the members of the 1 1119875

1meson multiplet we investigate

the mass spectrum of the 1 11198751meson state and the mixing angle of strange axial-vector mesons in the framework of meson-meson

mass mixing and Regge phenomenology Also in the glueball dominance picture the kaon is phenomenologically determined tohave a mass of about 1356MeV The results are compared with the values from different phenomenological models and may beuseful for the assignment of the 1 1119875

1meson multiplet in the future

1 Introduction

Quantum Chromodynamics (QCD) is widely accepted as asuccessful theory of the strong interactions in particlephysics However the understanding of the strong inter-actions is far from complete and it is difficult to interpretthe particle properties for the experimental data from thefirst principles To be able to determine the nature of a newexperimental resonance it is necessary to create a templateto compare the observed states with theoretical predictionsTherefore different phenomenological models such asquenched lattice gauge theory [1] the Dyson-Schwingerformalism [2] and constituent quark model [3] are built todescribe the properties of experimental results [4]

In the recent issue of Review of Particle Physics (PDG)the 1198871(1235) ℎ

1(1170) 119863

1(2420) and 119863

1199041(2536)

plusmn have beenobserved in the experiment and assigned as the membersof the 11119875

1meson multiplet [5] However the assignment

for the strange member 119904119904 state is problematic On the onehand based on the decay results in the 3119875

0model Barnes

et al suggest that the ℎ1(1380) should be a candidate of the

111198751meson [6] On the other hand the mass of ℎ

1(1380)

is inconsistent with the prediction in the potential modelwhich is about 100MeV higher than the measured masses

of the ℎ1(1380) [7] Accordingly the mass and decays of the

ℎ1(1380) need to be further examined experimentallyApart from the 119904119904 member the mass of the 119870

1119861also

remains interesting Due to the states 1198701(11198751) and 119870

1(31198751)

having no definite C-parity and the SU(3) broken the1198701(11198751)

and1198701(31198751) canmix via the spin-orbit interaction to form the

physical states 1198701(1270) and 119870

1(1400) In order to compare

the theoretical prediction with experimental data the mixingangle of 119870

1119860and 119870

1119861has been discussed with different

theoretical approaches [8ndash12] In addition to the light mesonstates the charmonium is one of the most exciting areas ofhigh energy physics In the 11119875

1meson state the ℎ

119888(1119875) has

been observed in the decay 120595(2119904) rarr 1205870ℎ119888with ℎ

119888rarr 120574120578

119888by

CLEO [13] and in the reaction 119901119901 rarr ℎ119888rarr 120574120578119888rarr 120574120574120574 [14]

The experimental signals about ℎ119888(1119875) are still very weak so

farIn allusion to the problems mentioned earlier the pur-

pose of this work is to investigate the mass spectrum of111198751meson state and mixing angle of strange axial-vector

mesonsThe results should be useful for exploring the natureand the assignment of 11119875

1meson state The organization

of this paper is as follows In Section 2 the brief reviewof the mass mixing matrix of isoscalar states is given (for

2 Advances in High Energy Physics

the detailed review see eg [15ndash17]) In Section 3 the massspectrum of 11119875

1meson state is presented in the framework

of Regge phenomenology and glueball-meson mass relationThe mixing angle of 119870

1119860and 119870

1119861is predicted in Section 4

and a summary and conclusion are given in Section 5

2 The Review of Mass Mixing Matrix ofIsoscalar States

In the quarkmodel the two isoscalar states with the same 119869119875119862quantum number will mix to form the physical states Onecan obtain the mass-squared matrix in the 119878 = 119904119904 and 119873 =(119906119906 + 119889119889)radic2 basis [18] as follows

1198722= (1198722

119873+ 2119860119899119899

radic2119860119899119904

radic2119860119899119904

1198722

119904119904+ 119860119904119904

) (1)

where 119872119873and 119872

119904119904are the masses of bare states 119873 and 119878

respectively 119860119899119899 119860119899119904 and 119860

119904119904are the mixing parameters

which describe the 119902119902 harr11990210158401199021015840 transition amplitudes In orderto reduce the number of parameters the following expres-sions are widely used in [7 19]

119860119899119904= 119860119899119899119883 119860

119904119904= 1198601198991198991198832 (2)

where 119883 is a phenomenological parameter describing theSU(3) broken ratio of the nonstrange and strange quarkpropagators via the constituent quark mass ratio In thepresent work we take 119883 = 063 which is determined withinthe nonrelativistic constituent quark model [17 20]

In the meson nonet the physical isoscalar states 120593 and1205931015840 are the eigenstates of mass-squared matrix 1198722 and the

masses square1198722120593and1198722

1205931015840 are the eigenvalues respectively

The physical states 120593 and 1205931015840 can be related to the 119878 = 119904119904 and119873 = (119906119906 + 119889119889)radic2 by

(

1003816100381610038161003816120593⟩

100381610038161003816100381610038161205931015840⟩) = 119880(

|119873⟩

|119878⟩) (3)

and the unitary matrix 119880 can be described as

1198801198722119880dagger= (1198722

1205930

0 1198722

1205931015840

) (4)

The mix of the physical isoscalar states can also be de-scribed as

(

1003816100381610038161003816120593⟩

100381610038161003816100381610038161205931015840⟩) = (

cos 120579 minus sin 120579sin 120579 cos 120579 )(

1205938

1205931

) (5)

with

1205931=119906119906 + 119889119889 + 119904119904

radic3

1205938=119906119906 + 119889119889 minus 2119904119904

radic6

(6)

where 120579 is the nonet mixing angle

From the relations (3) (4) and (5) one will obtain

119880 = (cos 120579 minus sin 120579sin 120579 cos 120579 )(

radic1

3minusradic2

3

radic2

3radic1

3

) (7)

Based on the relations (4) and (7) the following relationsare obtained

1198722

119873+ 2119860 = (radic

1

3cos 120579 minus radic2

3sin 120579)

2

1198722

120593

+ (radic2

3cos 120579 + radic1

3sin 120579)

2

1198722

1205931015840

1198722

119904119904+ 119860119883

2= (radic

1


3sin 120579)

2

1198722

1205931015840

+ (radic2


3sin 120579)

2

1198722

120593

(8)

Considering the fact that 119873 is the orthogonal partner ofthe isovector state of a meson nonet one can expect that 119873degenerates with isovector state 119872

119873= 119872119868=1

Applying therelations (8) for the ground pseudoscalar meson we obtainthe pure 119904119904member mass of the ground pseudoscalar mesonnonet119872

119904119904(111198780) = 685MeV [17] Here and in the following

all themasses used as input for our calculation are taken fromthe PDG [5]

3 Mass Spectrum of 111198751

Meson State inRegge Phenomenology andGlueball Dominance Picture

Regge theory is cornered with the particle spectrum theforces between particles and the high energy behavior ofscattering amplitudes In [21] Li et al investigate the massesof different mesonmultiplets in the framework of Regge phe-nomenology and suggest that the quasilinear Regge trajec-tories could provide a reasonable description for the mesonmass spectrum Khruschov predicts the masses spectrumof excited meson states using the phenomenology formulasderived from the Regge trajectories [22] Anisovich et al alsosuggest that themass of meson states can fit to the quasilinearRegge trajectories with good accuracy [23] In the last decadeRegge trajectories are reanalysed to make predictions for themasses of states not yet to be discovered in experiment andto determine the quantum numbers of the newly discoveredstates [21ndash28]

Based on the assumption that the hadrons with identical119869119875119862 quantumnumbers obey quasilinear form of Regge trajec-tories one has the following relations [21]

119897 = 1205721198941198941015840 (0) + 120572

1015840

11989411989410158401198722

1198941198941015840

119897 = 1205721198941198951015840 (0) + 120572

1015840

11989411989510158401198722

1198941198951015840

119897 = 1205721198951198951015840 (0) + 120572

1015840

11989511989510158401198722

1198951198951015840

(9)

Advances in High Energy Physics 3

Table 1 Comparison of predictions for themasses of the 111198751meson

states with other references (All in MeV)

Present work [5] [15] [7] [31] [32] [33]119872119904119904

1499 1386 149518 1470119872119899119904

1375 137003 1340 1368 1356119872119888119888

3509 352593 3520 3526

where 1198941198941015840 denotes the quark and antiquark flavors and 119897 and1198721198941198941015840are the orbital momentum and mass of the 1198941198941015840 meson

The parameters 12057210158401198941198941015840and 120572

1198941198941015840(0) are respectively the slope

and intercept of the Regge trajectory The intercepts can bedescribed as the following relation by [21 24]

120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (10)

The relation (10) is satisfied in two-dimensional QCD [29]and the quark bremsstrahlung model [30]

With the aid of the parity partners trajectories having thesame slopes we can apply relations (9)-(10) to the 11119878

0and

111198751mesonmultiplets to predict the followingmesonmasses

1205721015840

1198991198991198722(111198780)119899119899+ 1205721015840

1199041199041198722(111198780)119904119904= 21205721015840

1198991199041198722(111198780)119899119904

1205721015840

1198991198991198722(111198780)119899119899+ 1205721015840

1198881198881198722(111198780)119888119888= 21205721015840

1198881198991198722(111198780)119888119899

1205721015840

1199041199041198722(111198780)119904119904+ 1205721015840

1198881198881198722(111198780)119888119888= 21205721015840

1198881199041198722(111198780)119888119904

1205721015840

1198991198991198722(111198751)119899119899+ 1205721015840

1199041199041198722(111198751)119904119904= 21205721015840

1198991199041198722(111198751)119899119904

1205721015840

1198991198991198722(111198751)119899119899+ 1205721015840

1198881198881198722(111198751)119888119888= 21205721015840

1198881198991198722(111198751)119888119899

1205721015840

1199041199041198722(111198751)119904119904+ 1205721015840

1198881198881198722(111198751)119888119888= 21205721015840

1198881199041198722(111198751)119888119904

(11)

In order to eliminate the Regge slopes from the previousformulas we adopt the following relation [24]

1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(12)

For the 111198780meson multiplet all members are established

well For the 111198751meson multiplet only the 119887

1(1235) and

ℎ1(1170) are established well and the strange meson 119870

1119861

and the charmed mesons 1198631(2420) and 119863

1199041(2536)

plusmn are themixtures of spin-singlet 11119875

1and spin-triplet 13119875

1states In

the present work based on the small mass splitting betweencharmedmesons we conclude that the effect ofmixing on thecharmedmesonmasses will be smallTherefore inserting themasses of the corresponding mesons states and119872(11119875

1)119888119904=

1198721198631199041(2536)

as well as119872(111198751)119888119899= 1198721198631(2420)

into relations (11)-(12) the masses of the 119904119904 member 119870

1119861 and ℎ

119888(1119875) for the

111198751meson multiplet are obtained and the results are shown

in Table 1The mass of the 119870

1119861is important for the calculation of

mixing angle of the 1198701119860

and 1198701119861

states In order to checkconsistence of the results in Table 1 we investigate the massof kaon in the glueball dominance picture in the next section[18] And the result is in good agreement with the predictionin the Regge phenomenology

Table 2 Masses (in GeV) of the ground tensor and pseudovectorglueball from different theoretical models

Mass [34] [35] [36] [37] [38] [39] [40] Average119872ten 259 240 239 215 226 24 2417 237119872pse 294 298 267 271 283

In the mass matrix the quark mixing amplitudes in (1)can be expressed as

1198601199021199021015840 = sum

119896

⟨119902119902

100381610038161003816100381610038161003816100381611986711990210158401199021015840

119901119888

1003816100381610038161003816100381610038161003816119896⟩⟨119896

100381610038161003816100381610038161003816119867119902119902

119901119888

10038161003816100381610038161003816100381611990210158401199021015840⟩

1198722

11990210158401199021015840minus1198722

119896

(13)

where 119867119902119902119901119888 is the quark pair creation operator for the flavor119902 and |119896⟩ is a complete set of the intermediate states Basedon the assumption that there is no direct quarkonium-quarkonium mixing and the 119902119902 harr 11990210158401199021015840 transitions are dom-inated by the ground glueball with the corresponding quan-tum numbers we obtain the following relations

119860119899119899=

1198912

119899119899119866

1198722

119899119899minus1198722

119866

119860119904119904=

1198912

119904119904119866

1198722

119904119904minus1198722

119866

(14)

where 119891119902119902119866equiv ⟨119902119902|119867

119902119902|119866⟩|119901120583119875120583=1198722

119902119902

and 119902 refers to the 119899 and119904 quarks The parameters1198722

119899119899= 1198722

119868=1and1198722

119904119904= 2119872

2

119868=12minus

1198722

119868=1 119872119868=1

and 119872119868=12

are the masses of the isovector andisodoublet states of ameson nonet Considering the functions119891119902119902119866

Brisudova et al indicate the product of 119891119899119899119866

and 119891119904119904119866

in QCD to be a constant approximately independent of thequantum numbers of a meson nonet namely [18]

119891119899119899119866119891119904119904119866asymp Const (15)

Following the relations (1) (4) and (14)-(15) we have

(1198722

1198871(1235)

minus1198722

pse) (21198722

1198701119861

minus1198722

1198871(1235)

minus1198722

pse)

(1198722

1198862(1320)

minus1198722

ten) (21198722

119870lowast

2

minus1198722

1198862(1320)

minus1198722

ten)=

119860131198752

119860111198751

(16)

with

119860111198751

=

1198722

ℎ1(1170)

+1198722

ℎ1(1380)

minus 21198722

1198702

1119861

2 + 1198832

119860131198752

=

1198722

1198912(1270)

+1198722

1198911015840

2(1525)

minus 21198722

119870lowast

2

2 + 1198832

(17)

where 119872ten and 119872pse are the masses of ground tensor andpseudovector glueball respectively In order to extract themass of 119870

1119861by using relation (16) we should determine

the approximate masses of ground tensor and pseudovectorglueballs Some predictions on the masses of the glueballgiven by different theoretical models are shown in Table 2 Inthe present work taking themasses of correspondingmesonsin the relation (16) and the average values of glueballmasses inTable 2 as input we obtain themass of the119870

1119861 Our predicted

result 1356MeV is in agreement with our previous work


Table 3 Predicted mixing angle 120601 of the1198701(1270) and 119870

1(1400)

Refs Present work [15] [12] [41] [42] [43] [44]|120601| 616

∘(5955 plusmn 281)

∘5929

∘547∘ or 353∘ 62

∘557∘

60∘

4 The Mixing Angle of 1198701119860

and 1198701119861

According to the introduction in Section 1 the 1198701119860

and 1198701119861

in the 131198751and 11119875

1meson states can mix to produce the


1(1400) In the |11119875

1⟩ and

|131198751⟩ basis the physical states 119870

1(1270) and 119870

1(1400) can

be parameterized as [11]

1198701(1270) = sin120601 100381610038161003816100381610038161

31198751⟩ + cos120601 100381610038161003816100381610038161

11198751⟩

1198701(1400) = cos120601 100381610038161003816100381610038161

31198751⟩ minus sin120601 100381610038161003816100381610038161

11198751⟩

(18)

In this work 120601 is defined as the 1198701119860

and 1198701119861

mixing angleTo obtain the mixing angle 120601 we use the formula [31]

tan2 (2120601) = (1198722

1198701119860

minus1198722

1198701119861

1198722

1198701(1400)

minus1198722

1198701(1270)

)

2

minus 1 (19)

The relation (19) can also be rewritten as

cos (2120601) =1198722

1198701(1400)

minus1198722

1198701(1270)

1198722

1198701119860

minus1198722

1198701119861

(20)

Moreover using the following relation which is independentof the mixing angle the mass of 119870

1119860is determined to be

1303MeV Consider

1198722

1198701119860

+1198722

1198701119861

= 1198722

1198701(1400)

+1198722

1198701(1270)

(21)

With the help of the masses of 1198701119860 1198701119861 1198701(1270) and

1198701(1400) the mixing angle is determined to be plusmn616∘

5 Results and Conclusions

Based on the fact that the 1198871(1235) ℎ

1(1170) 119863

1(2420) and

1198631199041(2536)

plusmn are assigned as the 111198751meson multiplet we

obtain the masses of the 1198701119861 ℎ1(1380) and ℎ

119888(1119875) in the

111198751meson state The mass of the ℎ

1(1380) is determined

to be 1492MeV which is consistent with the value 1470MeVpredicted by Godfrey and Isgur in the potential model [7]The value is about 100MeV higher than themeasuredmass ofthe ℎ1(1380) in the experiment Besides experimental infor-

mation on the ℎ1(1380) is rather restricted which has been

reported only by two collaborations LASSCollaboration [45](with mass 1380 plusmn 20MeV and width 80 plusmn 30MeV) andCrystal Barrel Collaboration [46] (with mass 1440 plusmn 60MeVand width 170 plusmn 80MeV) Therefore we suggest that theassignment for this state needs to be further tested in the newexperiments in the future

The mass of 1198701119861

is useful for determining the mixingangle of the 119870

1(1270) and 119870

1(1400) In the present work

the mass for 1198701119861

is determined to be 1375MeV which isin agreement with the value from the meson-glueball mass

relation and other theoretical approaches The mixing angleof 1198701(1270) and 119870

1(1400) is compared with other different

predictions in Table 3 Our predictions may open a windowfor the assignment of the 11119875

1meson multiplet in the new

experiment in the future

Acknowledgments

This project is supported by the Zhengzhou University ofLight Industry Foundation of China (Grant nos 2009XJJ011and 2012XJJ008) in part by the National Natural ScienceFoundation of China (Grant nos 10975018 and 11147197) andthe Key Project of Scientific and Technological Research ofthe Education Department of Henan Province (Grant nos12B140001 and 13B140332)

References

[1] A M Polyakov ldquoInteraction of goldstone particles in two di-mensions 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] 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

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

[6] T Barnes N Black and P R Page ldquoStrong decays of strangequarkoniardquo Physical Review D vol 68 no 5 Article ID 05401443 pages 2003

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

[8] N Isgur and M B Wise ldquoWeak decays of heavy mesons in thestatic quark approximationrdquo Physics Letters B vol 232 no 1 pp113ndash117 1989

[9] D M Asner A Eppich J Gronberg et al ldquoResonance structureof 120591minus rarr 119870

minus120587+120587minusV120591rdquo Physical Review D vol 62 no 7 Article

ID 072006 11 pages 2000[10] H Y Cheng ldquoHadronic charmed meson decays involving axial

vector mesonsrdquo Physical Review D vol 67 no 9 Article ID094007 10 pages 2003

[11] M Suzuki ldquoStrange axial-vector mesonsrdquo Physical Review Dvol 47 no 3 pp 1252ndash1255 1993

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

[13] J L Rosner N E Adam J P Alexander et al ldquoObservation ofthe ℎ119888( 11198751) state of charmoniumrdquo Physical Review Letters vol

95 no 10 Article ID 102003 10 pages 2005


[14] M Andreotti S Bagnasco W Baldini et al ldquoResults of a searchfor the ℎ

119888( 11198751) state of charmonium in the 120578

119888120574 and J1205951205870 decay

modesrdquo Physical Review D vol 72 no 3 Article ID 032001 11pages 2005

[15] D M Li B Ma and Y Hong ldquoRegarding the axial-vectormesonsrdquo The European Physical Journal A vol 26 pp 141ndash1452005

[16] X C Feng and J L Feng ldquoThe mass of the isodoublet of the 21S0meson nonet revisitedrdquo Chinese Physics vol 16 no 11 pp

3297ndash3299 2007[17] X C Feng P Chen and J M Shang ldquoTowards the assignment

for the ground scalar mesonrdquo Advances in High Energy Physicsvol 2013 Article ID 219456 7 pages 2013

[18] M M Brisudova L Burakovsky and T Goldman ldquoNew glue-ball-meson mass relationsrdquo Physical Review D vol 58 no 11Article ID 114015 7 pages 1998

[19] I Cohen and H J Lipkin ldquoA phenomenological model forpseudoscalar mesonmixingrdquoNuclear Physics B vol 151 pp 16ndash28 1979

[20] D M Li K W Wei and H Yu ldquoA possible assignment for theground scalar meson nonetrdquo The European Physical Journal Avol 25 no 2 pp 263ndash266 2005

[21] DM Li BMa Y X Li Q K Yao andH Yu ldquoMeson spectruminRegge phenomenologyrdquoTheEuropean Physical Journal C vol37 no 3 pp 323ndash333 2004

[22] V V Khruschov ldquoMass spectra of excited meson states consist-ing of u- d-quarks and antiquarksrdquo httparxivorgabshep-ph0504077

[23] A V Anisovich V V Anisovich and A V Sarantsev ldquoSystem-atics of q119902 states in the (n 1198722) and (J 1198722) planesrdquo PhysicalReview D vol 62 no 5 Article ID 051502 5 pages 2000

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

[25] M M Brisudova L Burakovsky and J T Goldman ldquoEffectivefunctional form of Regge trajectoriesrdquo Physical Review D vol61 no 5 Article ID 054013 15 pages 2000

[26] Z C Ye X Wang X Liu and Q Zhao ldquoThe mass spectrumand strong decays of isoscalar tensor mesonsrdquo Physical ReviewD vol 86 no 5 Article ID 054025 14 pages 2012

[27] P Masjuan E R Arriola and W Broniowski ldquoSystematics ofradial and angular-momentum Regge trajectories of light non-strange q119902-statesrdquo Physical Review D vol 85 no 9 Article ID094006 15 pages 2012

[28] E R Arriola and W Broniowski ldquoScalar-isoscalar states in thelarge-119873

119888Regge approachrdquo Physical Review D vol 81 no 5

Article ID 054009 18 pages 2010[29] R C Brower J EllisM G Schmidt and J HWeis ldquoWhat is the

relativistic generalization of a linearly rising potentialrdquoNuclearPhysics B vol 128 no 1 pp 75ndash92 1977

[30] V V Dixit and L A P Balazs ldquoRegge intercepts of multiquarksystems in Feynmanrsquos bremsstrahlung analogyrdquo Physical ReviewD vol 20 no 3 pp 816ndash819 1979

[31] P V Chliapnikov ldquoS- and P-wave meson spectroscopy in thenonrelativistic quark modelrdquo Physics Letters B vol 496 no 3-4pp 129ndash136 2000

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

[33] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia and 119861

119888mesons in the relativistic quark modelrdquo

Physical Review D vol 67 no 1 Article ID 014027 19 pages2003

[34] A B Kaidolv and A Yu Simonov ldquoGlueball masses andPomeron trajectory in nonperturbativeQCDapproachrdquoPhysicsLetters B vol 477 no 1ndash3 pp 163ndash170 2000

[35] C J Morningstar and M Peardon ldquoGlueball spectrum from ananisotropic lattice studyrdquo Physical Review D vol 60 no 3 pp1ndash13 1999

[36] Y Chen A Alexandru and S J Dong ldquoGlueball spectrum andmatrix elements on anisotropic latticesrdquo Physical Review D vol73 no 1 Article ID 014516 21 pages 2006

[37] H B Meyer ldquoGlueball regge trajectoriesrdquo httparxivorgabshep-lat0508002

[38] M J Temper ldquoPhysics from the lattice glueballs in QCDtopology SU(N) for all Nrdquo NATO Science B vol 368 pp 43ndash74 2002

[39] C Liu ldquoA lattice study of the glueball spectrumrdquoChinese PhysicsLetters vol 18 no 2 pp 187ndash189 2001

[40] DQ Liu JMWu andYChen ldquoNew approach to constructingthe operator on the lattice for the calculation of the glueballmassesrdquo High Energy Physics and Nuclear Physics vol 26 no3 pp 222ndash229 2002


[42] J Rosner ldquoP wave mesons with one heavy quarkrdquo CommentsNuclear and Particle Physics vol 16 article 109 1986

[43] J Vijande F Fernandez and A Valcarce ldquoConstituent quarkmodel study of the meson spectrardquo Journal of Physics G vol 31no 5 article 481 2005

[44] A Tayduganov E Kou and A Le Yaouanc ldquoThe strong decaysof 1198701resonancesrdquo Physical Review D vol 85 no 7 Article ID

074011 28 pages 2012[45] D Aston N Awaji T Bienz et al ldquoEvidence for two strangeo-

nium resonances with 119869119875119862 = 1++ and 1+minus in K minus p interactionsat 11 GeVcrdquo Physics Letters B vol 201 no 4 pp 573ndash578 1998

[46] A Abel J Adomeit C Amsler et al ldquoAntiproton-proton anni-hilation at rest into119870

11987111987011987812058701205870rdquo Physics Letters B vol 415 no 3

pp 280ndash288 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


the detailed review see eg [15ndash17]) In Section 3 the massspectrum of 11119875

1meson state is presented in the framework

of Regge phenomenology and glueball-meson mass relationThe mixing angle of 119870

1119860and 119870

1119861is predicted in Section 4

and a summary and conclusion are given in Section 5

2 The Review of Mass Mixing Matrix ofIsoscalar States

In the quarkmodel the two isoscalar states with the same 119869119875119862quantum number will mix to form the physical states Onecan obtain the mass-squared matrix in the 119878 = 119904119904 and 119873 =(119906119906 + 119889119889)radic2 basis [18] as follows

1198722= (1198722

119873+ 2119860119899119899

radic2119860119899119904

radic2119860119899119904

1198722

119904119904+ 119860119904119904

) (1)

where 119872119873and 119872

119904119904are the masses of bare states 119873 and 119878

respectively 119860119899119899 119860119899119904 and 119860

119904119904are the mixing parameters

which describe the 119902119902 harr11990210158401199021015840 transition amplitudes In orderto reduce the number of parameters the following expres-sions are widely used in [7 19]

119860119899119904= 119860119899119899119883 119860

119904119904= 1198601198991198991198832 (2)

where 119883 is a phenomenological parameter describing theSU(3) broken ratio of the nonstrange and strange quarkpropagators via the constituent quark mass ratio In thepresent work we take 119883 = 063 which is determined withinthe nonrelativistic constituent quark model [17 20]

In the meson nonet the physical isoscalar states 120593 and1205931015840 are the eigenstates of mass-squared matrix 1198722 and the

masses square1198722120593and1198722

1205931015840 are the eigenvalues respectively

The physical states 120593 and 1205931015840 can be related to the 119878 = 119904119904 and119873 = (119906119906 + 119889119889)radic2 by

(

1003816100381610038161003816120593⟩

100381610038161003816100381610038161205931015840⟩) = 119880(

|119873⟩

|119878⟩) (3)

and the unitary matrix 119880 can be described as

1198801198722119880dagger= (1198722

1205930

0 1198722

1205931015840

) (4)

The mix of the physical isoscalar states can also be de-scribed as

(

1003816100381610038161003816120593⟩

100381610038161003816100381610038161205931015840⟩) = (

cos 120579 minus sin 120579sin 120579 cos 120579 )(

1205938

1205931

) (5)

with

1205931=119906119906 + 119889119889 + 119904119904

radic3

1205938=119906119906 + 119889119889 minus 2119904119904

radic6

(6)

where 120579 is the nonet mixing angle

From the relations (3) (4) and (5) one will obtain

119880 = (cos 120579 minus sin 120579sin 120579 cos 120579 )(

radic1

3minusradic2

3

radic2

3radic1

3

) (7)

Based on the relations (4) and (7) the following relationsare obtained

1198722

119873+ 2119860 = (radic

1


3sin 120579)

2

1198722

120593

+ (radic2


3sin 120579)

2

1198722

1205931015840

1198722

119904119904+ 119860119883

2= (radic

1


3sin 120579)

2

1198722

1205931015840

+ (radic2


3sin 120579)

2

1198722

120593

(8)

Considering the fact that 119873 is the orthogonal partner ofthe isovector state of a meson nonet one can expect that 119873degenerates with isovector state 119872

119873= 119872119868=1

Applying therelations (8) for the ground pseudoscalar meson we obtainthe pure 119904119904member mass of the ground pseudoscalar mesonnonet119872

119904119904(111198780) = 685MeV [17] Here and in the following

all themasses used as input for our calculation are taken fromthe PDG [5]

3 Mass Spectrum of 111198751

Meson State inRegge Phenomenology andGlueball Dominance Picture

Regge theory is cornered with the particle spectrum theforces between particles and the high energy behavior ofscattering amplitudes In [21] Li et al investigate the massesof different mesonmultiplets in the framework of Regge phe-nomenology and suggest that the quasilinear Regge trajec-tories could provide a reasonable description for the mesonmass spectrum Khruschov predicts the masses spectrumof excited meson states using the phenomenology formulasderived from the Regge trajectories [22] Anisovich et al alsosuggest that themass of meson states can fit to the quasilinearRegge trajectories with good accuracy [23] In the last decadeRegge trajectories are reanalysed to make predictions for themasses of states not yet to be discovered in experiment andto determine the quantum numbers of the newly discoveredstates [21ndash28]

Based on the assumption that the hadrons with identical119869119875119862 quantumnumbers obey quasilinear form of Regge trajec-tories one has the following relations [21]

119897 = 1205721198941198941015840 (0) + 120572

1015840

11989411989410158401198722

1198941198941015840

119897 = 1205721198941198951015840 (0) + 120572

1015840

11989411989510158401198722

1198941198951015840

119897 = 1205721198951198951015840 (0) + 120572

1015840

11989511989510158401198722

1198951198951015840

(9)




Present work [5] [15] [7] [31] [32] [33]119872119904119904

1499 1386 149518 1470119872119899119904

1375 137003 1340 1368 1356119872119888119888

3509 352593 3520 3526





120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (10)



0and


1205721015840

1198991198991198722(111198780)119899119899+ 1205721015840

1199041199041198722(111198780)119904119904= 21205721015840

1198991199041198722(111198780)119899119904

1205721015840

1198991198991198722(111198780)119899119899+ 1205721015840

1198881198881198722(111198780)119888119888= 21205721015840

1198881198991198722(111198780)119888119899

1205721015840

1199041199041198722(111198780)119904119904+ 1205721015840

1198881198881198722(111198780)119888119888= 21205721015840

1198881199041198722(111198780)119888119904

1205721015840

1198991198991198722(111198751)119899119899+ 1205721015840

1199041199041198722(111198751)119904119904= 21205721015840

1198991199041198722(111198751)119899119904

1205721015840

1198991198991198722(111198751)119899119899+ 1205721015840

1198881198881198722(111198751)119888119888= 21205721015840

1198881198991198722(111198751)119888119899

1205721015840

1199041199041198722(111198751)119904119904+ 1205721015840

1198881198881198722(111198751)119888119888= 21205721015840

1198881199041198722(111198751)119888119904

(11)


1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(12)



1(1235) and


1119861


1199041(2536)



1states In


1)119888119904=

1198721198631199041(2536)

as well as119872(111198751)119888119899= 1198721198631(2420)


1119861 and ℎ

119888(1119875) for the





and 1198701119861





1198601199021199021015840 = sum

119896

⟨119902119902

100381610038161003816100381610038161003816100381611986711990210158401199021015840

119901119888

1003816100381610038161003816100381610038161003816119896⟩⟨119896

100381610038161003816100381610038161003816119867119902119902

119901119888

10038161003816100381610038161003816100381611990210158401199021015840⟩

1198722

11990210158401199021015840minus1198722

119896

(13)


119860119899119899=

1198912

119899119899119866

1198722

119899119899minus1198722

119866

119860119904119904=

1198912

119904119904119866

1198722

119904119904minus1198722

119866

(14)

where 119891119902119902119866equiv ⟨119902119902|119867

119902119902|119866⟩|119901120583119875120583=1198722

119902119902


119899119899= 1198722

119868=1and1198722

119904119904= 2119872

2

119868=12minus

1198722

119868=1 119872119868=1

and 119872119868=12



and 119891119904119904119866


119891119899119899119866119891119904119904119866asymp Const (15)


(1198722

1198871(1235)

minus1198722

pse) (21198722

1198701119861

minus1198722

1198871(1235)

minus1198722

pse)

(1198722

1198862(1320)

minus1198722

ten) (21198722

119870lowast

2

minus1198722

1198862(1320)

minus1198722

ten)=

119860131198752

119860111198751

(16)

with

119860111198751

=

1198722

ℎ1(1170)

+1198722

ℎ1(1380)

minus 21198722

1198702

1119861

2 + 1198832

119860131198752

=

1198722

1198912(1270)

+1198722

1198911015840

2(1525)

minus 21198722

119870lowast

2

2 + 1198832

(17)








1(1400)


∘(5955 plusmn 281)

∘5929

∘547∘ or 353∘ 62

∘557∘

60∘


and 1198701119861


and 1198701119861

in the 131198751and 11119875



1(1400) In the |11119875

1⟩ and


1(1270) and 119870

1(1400) can


1198701(1270) = sin120601 100381610038161003816100381610038161

31198751⟩ + cos120601 100381610038161003816100381610038161

11198751⟩

1198701(1400) = cos120601 100381610038161003816100381610038161

31198751⟩ minus sin120601 100381610038161003816100381610038161

11198751⟩

(18)


and 1198701119861


tan2 (2120601) = (1198722

1198701119860

minus1198722

1198701119861

1198722

1198701(1400)

minus1198722

1198701(1270)

)

2

minus 1 (19)


cos (2120601) =1198722

1198701(1400)

minus1198722

1198701(1270)

1198722

1198701119860

minus1198722

1198701119861

(20)



1303MeV Consider

1198722

1198701119860

+1198722

1198701119861

= 1198722

1198701(1400)

+1198722

1198701(1270)

(21)





1(1170) 119863

1(2420) and

1198631199041(2536)



119888(1119875) in the






The mass of 1198701119861


1(1270) and 119870


the mass for 1198701119861







Acknowledgments


References




















119888120574 and J1205951205870 decay









































pp 280ndash288 1997






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Present work [5] [15] [7] [31] [32] [33]119872119904119904

1499 1386 149518 1470119872119899119904

1375 137003 1340 1368 1356119872119888119888

3509 352593 3520 3526





120572119894119894(0) + 120572

119895119895(0) = 2120572

119894119895(0) (10)



0and


1205721015840

1198991198991198722(111198780)119899119899+ 1205721015840

1199041199041198722(111198780)119904119904= 21205721015840

1198991199041198722(111198780)119899119904

1205721015840

1198991198991198722(111198780)119899119899+ 1205721015840

1198881198881198722(111198780)119888119888= 21205721015840

1198881198991198722(111198780)119888119899

1205721015840

1199041199041198722(111198780)119904119904+ 1205721015840

1198881198881198722(111198780)119888119888= 21205721015840

1198881199041198722(111198780)119888119904

1205721015840

1198991198991198722(111198751)119899119899+ 1205721015840

1199041199041198722(111198751)119904119904= 21205721015840

1198991199041198722(111198751)119899119904

1205721015840

1198991198991198722(111198751)119899119899+ 1205721015840

1198881198881198722(111198751)119888119888= 21205721015840

1198881198991198722(111198751)119888119899

1205721015840

1199041199041198722(111198751)119904119904+ 1205721015840

1198881198881198722(111198751)119888119888= 21205721015840

1198881199041198722(111198751)119888119904

(11)


1

1205721015840

1198941198941015840

+1

1205721015840

1198951198951015840

=2

1205721015840

1198941198951015840

(12)



1(1235) and


1119861


1199041(2536)



1states In


1)119888119904=

1198721198631199041(2536)

as well as119872(111198751)119888119899= 1198721198631(2420)


1119861 and ℎ

119888(1119875) for the





and 1198701119861





1198601199021199021015840 = sum

119896

⟨119902119902

100381610038161003816100381610038161003816100381611986711990210158401199021015840

119901119888

1003816100381610038161003816100381610038161003816119896⟩⟨119896

100381610038161003816100381610038161003816119867119902119902

119901119888

10038161003816100381610038161003816100381611990210158401199021015840⟩

1198722

11990210158401199021015840minus1198722

119896

(13)


119860119899119899=

1198912

119899119899119866

1198722

119899119899minus1198722

119866

119860119904119904=

1198912

119904119904119866

1198722

119904119904minus1198722

119866

(14)

where 119891119902119902119866equiv ⟨119902119902|119867

119902119902|119866⟩|119901120583119875120583=1198722

119902119902


119899119899= 1198722

119868=1and1198722

119904119904= 2119872

2

119868=12minus

1198722

119868=1 119872119868=1

and 119872119868=12



and 119891119904119904119866


119891119899119899119866119891119904119904119866asymp Const (15)


(1198722

1198871(1235)

minus1198722

pse) (21198722

1198701119861

minus1198722

1198871(1235)

minus1198722

pse)

(1198722

1198862(1320)

minus1198722

ten) (21198722

119870lowast

2

minus1198722

1198862(1320)

minus1198722

ten)=

119860131198752

119860111198751

(16)

with

119860111198751

=

1198722

ℎ1(1170)

+1198722

ℎ1(1380)

minus 21198722

1198702

1119861

2 + 1198832

119860131198752

=

1198722

1198912(1270)

+1198722

1198911015840

2(1525)

minus 21198722

119870lowast

2

2 + 1198832

(17)








1(1400)


∘(5955 plusmn 281)

∘5929

∘547∘ or 353∘ 62

∘557∘

60∘


and 1198701119861


and 1198701119861

in the 131198751and 11119875



1(1400) In the |11119875

1⟩ and


1(1270) and 119870

1(1400) can


1198701(1270) = sin120601 100381610038161003816100381610038161

31198751⟩ + cos120601 100381610038161003816100381610038161

11198751⟩

1198701(1400) = cos120601 100381610038161003816100381610038161

31198751⟩ minus sin120601 100381610038161003816100381610038161

11198751⟩

(18)


and 1198701119861


tan2 (2120601) = (1198722

1198701119860

minus1198722

1198701119861

1198722

1198701(1400)

minus1198722

1198701(1270)

)

2

minus 1 (19)


cos (2120601) =1198722

1198701(1400)

minus1198722

1198701(1270)

1198722

1198701119860

minus1198722

1198701119861

(20)



1303MeV Consider

1198722

1198701119860

+1198722

1198701119861

= 1198722

1198701(1400)

+1198722

1198701(1270)

(21)





1(1170) 119863

1(2420) and

1198631199041(2536)



119888(1119875) in the






The mass of 1198701119861


1(1270) and 119870


the mass for 1198701119861







Acknowledgments


References




















119888120574 and J1205951205870 decay









































pp 280ndash288 1997






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1(1400)


∘(5955 plusmn 281)

∘5929

∘547∘ or 353∘ 62

∘557∘

60∘


and 1198701119861


and 1198701119861

in the 131198751and 11119875



1(1400) In the |11119875

1⟩ and


1(1270) and 119870

1(1400) can


1198701(1270) = sin120601 100381610038161003816100381610038161

31198751⟩ + cos120601 100381610038161003816100381610038161

11198751⟩

1198701(1400) = cos120601 100381610038161003816100381610038161

31198751⟩ minus sin120601 100381610038161003816100381610038161

11198751⟩

(18)


and 1198701119861


tan2 (2120601) = (1198722

1198701119860

minus1198722

1198701119861

1198722

1198701(1400)

minus1198722

1198701(1270)

)

2

minus 1 (19)


cos (2120601) =1198722

1198701(1400)

minus1198722

1198701(1270)

1198722

1198701119860

minus1198722

1198701119861

(20)



1303MeV Consider

1198722

1198701119860

+1198722

1198701119861

= 1198722

1198701(1400)

+1198722

1198701(1270)

(21)





1(1170) 119863

1(2420) and

1198631199041(2536)



119888(1119875) in the






The mass of 1198701119861


1(1270) and 119870


the mass for 1198701119861







Acknowledgments


References




















119888120574 and J1205951205870 decay









































pp 280ndash288 1997






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119888120574 and J1205951205870 decay









































pp 280ndash288 1997






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Mass Spectrum of 1 Meson State and Mixing Angle of Strange ...downloads.hindawi.com/journals/ahep/2013/704529.pdf · Advances in High Energy Physics the detailed review, see, e.g.,