The structure, mixing angle, mass and couplings of the light scalar f(0)(500) and f(0)(980) mesons

Physics Letters B 781 (2018) 279–282

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

The

structure,

mixing

angle,

mass

and

couplings

of

the

light

scalar

f

₀

(

500

)

and

f

₀

(

980

)

mesons

S.S. Agaev

a

,

K. Azizi

b

,

c

,

∗

,

H. Sundu

d

a_{InstituteforPhysicalProblems,BakuStateUniversity,Az-1148Baku,Azerbaijan} b_{DepartmentofPhysics,Doˇgu ¸sUniversity,Acibadem-Kadiköy,34722Istanbul,Turkey}

c_{SchoolofPhysics,InstituteforResearchinFundamentalSciences(IPM),P. O. Box19395-5531,Tehran,Iran} d_{DepartmentofPhysics,KocaeliUniversity,41380 Izmit,Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received30November2017

Receivedinrevisedform16January2018 Accepted31March2018

Availableonline5April2018 Editor: J.-P.Blaizot

Themixingangle,massandcouplingsofthelightscalarmesons f0(500)and f0(980)arecalculatedin theframeworkofQCDtwo-pointsumruleapproachbyassumingthattheyaretetraquarkswithdiquark– antidiquarkstructures. Themesonsaretreatedasmixturesoftheheavy|H= ([su][¯su¯]+ [sd][¯sd¯])/√2 and light|L= [ud][¯ud¯] scalar diquark–antidiquark components. We extract from corresponding sum rulesthemixingangles

ϕ

H and

ϕ

Lofthesestatesandevaluatethemassesandcouplingsoftheparticles

f0(500)and f0(980).

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Lightscalarmesonsthat resideintheregionm

<

1GeV ofthe mesonspectroscopyaresourcesoflong-standingproblemsforthe conventional quark model. The standard approach when treating mesons as bound states of a quark and an antiquark qq meets

¯

withevident troublesto include f0

(

500

)

and f0

(

980

)

, aswell as

some other lightparticles into thisscheme:There are discrepan-ciesbetweenpredictionsofthismodelforamasshierarchyoflight scalarsandmeasuredmassesoftheseparticles.Therefore,for in-stance,the f0

(

980

)

mesonwasalreadyconsideredasafour–quark

statewithq2_q

_¯

2_content[1].

During passed decades physicists made great efforts to un-derstand features of the light scalar mesons: They were treated as meson–meson molecules [2–5], or considered as diquark– antidiquarkbound states[6,7].These modelsstimulated not only qualitative analysis of the light scalar mesons, but also allowed one to calculate their parameters using different methods. Thus, inRef. [8] masses ofthe f0

(

500

)

, f0

(

980

)

,a0

(

980

)

and K₀∗

(

800

)

mesonswere evaluatedin thecontextof therelativisticdiquark– antidiquark approach and nice agreements with the data were found.Therearegrowingunderstandingthatthemesonsfromthe lightscalars’nonetareexoticparticlesoratleastcontain substan-tialmultiquark components:lattice simulations andexperimental dataseemsupportthesesuggestions.Furtherinformationon

rele-*

Correspondingauthor.

E-mailaddress:kazizi@dogus.edu.tr(K. Azizi).

vanttheoreticalideasandmodels,aswellasonexperimentaldata canbefoundinoriginalandreviewarticles[9–13].

Intensive studiesof thelight scalars astetraquark stateswere carriedoutusingQCDsumrulesmethod[14–22].Essentialpartof these investigations confirmed assignment of thelight scalars as tetraquarkstatesdespitethefactthattoexplainexperimentaldata in some of them authors had to introduce various modifications toapurediquark–antidiquarkpictureandtotreattheparticlesas a mixtureofdiquark–antidiquarkswithdifferentflavor structures [18], or assuperpositions of diquark–antidiquark andqq compo-

¯

nents [20–22]. There was alsothe article(see, Ref. [19]), results ofwhich didnotsupport an interpretationofthe lightscalars as diquark–antidiquarkboundstates.

Asisseen,theoreticalanalysesperformedevenwithinthesame methodleadtodifferentconclusions abouttheinternalstructures ofthemesonsfromthelightscalarnonet.Oneshouldaddtothis picture also large errorsfrom whichsuffer experimental dataon themassesandwidthsoftheseparticles[23] tounderstand diffi-cultyofexistingproblems.

2. Mixingschemes

An approach to the nonet of light scalars as mixtures of tetraquarks belonging to different representations of the color group wasrecentlyproposed inRef.[24].Inaccordancewiththis approachthenonetofthe lightspin-0mesonscan beconsidered as tetraquarks composed of the color (3c) and flavor (3f) an-titripletscalardiquarks.Then,thesetetraquarksintheflavorspace forma nonet ofthe scalar particles 3f

⊗

3f

=

8f

⊕

1f.In order https://doi.org/10.1016/j.physletb.2018.03.085

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

280 S.S. Agaev et al. / Physics Letters B 781 (2018) 279–282

toembrace thesecond nonetofthescalar mesonsoccupyingthe regionabove 1GeV spin-1diquarksbelongingtothecolor–flavor representation(6c

,

3f) can beused. Thetetraquarks builtofthe spin-1diquarkshavethesameflavorstructureasonesconstructed fromspin-0diquarks,andthereforecanmixwiththem.

InthepresentLetterwe restrictourselvesby consideringonly the first nonet ofthe scalar particles. Therefore, inwhat follows weneglecttheirpossiblemixingwithtetraquarkscomposedofthe spin-1 diquarks. The flavor singlet and octet components of this nonethavethestructures

|

1f

 =

1

√

3



[

su

][

su

] + [

ds

][

ds

] + [

ud

][

ud

]



,

|

8f

 =

1

√

6



[

ds

][

ds

] + [

su

][

su

] −

2

[

ud

][

ud

]



,

thatintheexactSUf

(

3

)

symmetrycanbedirectlyidentifiedwith thephysical mesons.Butthe realscalarparticles are mixturesof thesestates, andin thesinglet-octet basis and one-angle mixing schemehavethedecomposition



|

f



|

f





=

U

(θ )



|

1f



|

8f





,

U

(θ )

=



cos

θ

−

sin

θ

sin

θ

cos

θ



,

(1)

where,forthesakeofsimplicity,wedenote f

=

f0

(

500

)

and f

=

f0

(

980

)

, and

θ

is the corresponding mixing angle. Alternatively,

onecanintroducetheheavy-lightbasis

|

H

 =

√

1 2



[

su

][

su

] + [

ds

][

ds

]



,

|

L

 = [

ud

][

ud

],

(2)

andforthephysicalmesonsgettheexpansion



|

f



|

f





=

U

(

ϕ

)



|

H



|

L





.

(3)

Here we use

ϕ

as the mixingangle in the heavy-lightbasis. An emerged situation isfamiliar to one from analysisof the mixing problemsinthenonetofthepseudoscalarmesons,namelyinthe

η

−

η

system[25–27].Theheavy-lightbasisinthecaseunder con-siderationissimilartothequark–flavorbasisemployedthere.The mixinganglesinthetwo basisare connectedbythesimple rela-tion

tan

θ

=

√

2 tan

ϕ

+

1

tan

ϕ

−

√

2

.

(4)

Ingeneral, one mayintroducealso two-anglesmixing scheme ifitleadstoabetterdescriptionoftheexperimentaldata



|

f



|

f





=

U

(

ϕ

H,

ϕ

L

)



|

H



|

L





,

U

(

ϕ

H,

ϕ

L

)

=



cos

ϕ

H

−

sin

ϕ

L sin

ϕ

H cos

ϕ

L



.

(5)

Thecouplingsinthe f

−

fsystemcanbedefinedintheform



0

|

Ji

|

f

(

p

)

 =

Fi_fmf

,



0

|

Ji

|

f

(

p

)

 =

Fifmf

,

i

=

H

,

L

.

(6) We suggest that the couplings followpattern of state mixingin both one- and two-angles scheme. In the general case of two-anglesmixingschemethisimpliesfulfillment oftheequality



FH_f FL_f FH_f FLf



=

U

(

ϕ

H,

ϕ

L

)



FH 0 0 FL



,

(7)

whereFH andFL maybeformallyinterpretedascouplingsofthe “particles”

|

H



and

|

L



.

Currents JH

₍

_x

₎

_{and J}L

₍

_x

₎

_{inEq.(6) thatcorrespondto}

_|

_H

_

_and

|

L



statesaregivenbytheexpressions

JH

(

x

)

=



dab

_

dce

√

2



uT_a

(

x

)

C

γ

5sb

(

x

)



uc

(

x

)

γ

5C sTe

(

x

)

+



dT_a

(

x

)

C

γ

5sb

(

x

)



dc

(

x

)

γ

5C sTe

(

x

)



,

(8) and JL

(

x

)

=



dab



dce



u_aT

(

x

)

C

γ

5db

(

x

)



uc

(

x

)

γ

5Cd T e

(

x

)

,

(9)

wherea

,

b

,

c

,

d and e arecolorindicesandC isthecharge conju-gationoperator.Thentheinterpolatingcurrentsforphysicalstates

Jf

₍

_x

₎

_{and J}f

₍

_x

₎

_taketheforms



Jf

(

x

)

Jf

(

x

)



=

U

(

ϕ

H,

ϕ

L

)



JH

₍

_x

₎

JL

(

x

)



.

(10)

Inthesimplecaseofone-anglemixingschemeEq. (10) transforms tothefamiliarsuperpositions

Jf

(

x

)

=

JH

(

x

)

cos

ϕ

−

JL

(

x

)

sin

ϕ

,

Jf

(

x

)

=

JH

(

x

)

sin

ϕ

+

JL

(

x

)

cos

ϕ

.

(11)

Thesecurrentsortheirmorecomplicatedformsinthetwo-angles mixingschememaybeusedinQCDsumrulecalculationsto eval-uatethemassesandcouplingsofthemesons f and f.

3. Sumrules

Atthefirststageofourcalculationswederivethesumrulefor themixingangle

ϕ

ofthe f

−

fsystem.Tothisend,weusethe heavy-lightbasisandone-anglemixingschemeandstartfromthe correlationfunction[28]

(

p

)

=

i

d4xeip·x



0

|

T

{

Jf

(

x

)

Jf†

(

0

)

}|

0

.

(12)

The sumruleobtainedusing

(

p

)

allowustofixthemixing an-gle

ϕ

.In fact,because the currents Jf

₍

_x

₎

_{and J}f

₍

_x

₎

_createonly

|

f



and

|

f



mesons, respectively,aphenomenological expression forthecorrelator



Phys

(

p

)

equalstozero.Thenthesecond ingredi-entofthesumrule,namelyexpressionofthecorrelationfunction calculated in terms of quark–gluon degrees of freedom



OPE

(

p

)

should beequaltozero.Because



OPE

(

p

)

dependsonthemixing angle

ϕ

,itisnotdifficulttofind

tan 2

ϕ

=

2



OPEH L

(

p

)



OPE_LL

(

p

)

− 

OPE_{H H}

(

p

)

,

(13) where



i j

(

p

)

=

i

d4xeip·x



0

|

T

{

Ji

(

x

)

Jj†

(

0

)

}|

0

.

(14)

Inderiving ofEq. (13) webenefitedfromthefact that



OPE_{H L}

(

p

)

=



OPE_{L H}

(

p

)

,whichcanbeprovedbyexplicitcalculations.After apply-ing the Boreltransformation andperforming requiredcontinuum subtractionsonecanemployittoevaluate

ϕ

.

Having found the mixing angle we proceed andevaluate the spectroscopicparametersofthemesons f and f.Thecorrelation functions



f

(

p

)

=

i

d4xeip·x



0

|

T

{

Jf

(

x

)

Jf †

(

0

)

}|

0

,



f

(

p

)

=

i

d4xeip·x



0

|

T

{

Jf

(

x

)

Jf†

(

0

)

}|

0

,

(15)

S.S. Agaev et al. / Physics Letters B 781 (2018) 279–282 281

Fig. 1. The tan 2ϕ(a), and the masses mf (b) and mf(c) in the two-angles mixing scheme as functions of the Borel parameter M2_{at fixed s} 0.

are appropriate for thesepurposes and can be utilized to derive therelevant sumrules. Theexpression of



f

(

p

)

interms ofthe physicalparametersofthe f mesonisgivenbythefollowing sim-pleformula



Phys_f

(

p

)

=



0

|

J

f

_|

_f

₍

_p

₎

_

_f

₍

_p

₎

_|

_Jf †

_|

₀

_

m2_f

−

p2

+ . . . ,

wherethe dots stand for contributions of the higher resonances andcontinuum states.Usingtheinterpolatingcurrentandmatrix elementsofthe f mesonfromEqs.(11) and(6) itisaeasytaskto showthat



Phys_f

(

p

)

=

m 2 f m2_f

−

p2



FHcos2

ϕ

+

FLsin2

ϕ



2

+ . . . .

Aftercalculatingthecorrelationfunction



OPE_f

(

p

)

andapplyingthe Borel transformation in conjunction with continuum subtraction onegetsthesumrule

m2_f



FHcos2

ϕ

+

FLsin2

ϕ



2 e−m2f/M2

= 

f

(

s0

,

M2

),

(16)

where



f

(

s0

,

M2

)

=

B 

OPE_f

(

p

)

istheBoreltransformedand

sub-tractedexpressionof



OPE_f

(

p

)

withM2 _ands

0 beingtheBoreland

continuum threshold parameters, respectively. Thissum rule and anotherone obtainedfromEq.(16) bymeansofthestandard op-eration d

/

d

(

−

1

/

M2

₎

_{can be used to evaluate the mass of the f}

meson.

Thesimilaranalysisfor fyields

m2_f



FHsin2

ϕ

+

FLcos2

ϕ



2 e−m2f/M 2

= 

f

(

s0

,

M2

).

(17)

FromEqs.(16) and(17) itisalsopossibletoextract

FHcos2

ϕ

+

FLsin2

ϕ

2

and

FHsin2

ϕ

+

FLcos2

ϕ

2

forevaluating ofthe cou-plings FH and FL, but they may suffer fromlarge uncertainties: Weinsteadevaluate FH andFLfromsumrulesforthescalar “par-ticles”

|

H



and

|

L



,usingto thisendcorrelationfunctions



H

(

p

)

and



L

(

p

)

givenbyEq. (15),where Jf

(

x

)

and Jf



(

x

)

arereplaced by JH

₍

_x

₎

_{and J}L

₍

_x

₎

_{,respectively.}

4. Numericalresults

Incalculationsweutilizethelightquarkpropagator

Sab_q

(

x

)

=

i

δ

ab

/

x 2

π

2_x4

− δ

ab mq 4

π

2_x2

− δ

ab



qq



12

+

i

δ

ab

/

xmq



qq



48

− δ

ab x2 192



qgs

σ

Gq

 +

i

δ

ab x2

/

xmq 1152



qgs

σ

Gq



−

i gsG αβ ab 32

π

2_x2



/

x

σ

αβ

+

σ

αβ

/

x



−

i

δ

ab x2

/

xg2_s



qq



2 7776

− δ

ab x4

_

_qq

_

_g2 sG2



27648

+ . . . ,

(18)

andtake intoaccount quark,gluonandmixedoperatorsupto di-mensiontwelve.Thevacuumexpectationsvaluesoftheoperators usedinnumericalcomputationsarewellknown:

¯

qq



= −(

0

.

24

±

0

.

01

)

3 GeV3,

¯

ss



=

0

.

8

¯

qq



,



qgs

σ

Gq



=

m20qq



,



sgs

σ

Gs



=

m20¯ss



,



α

sG2

/

π



= (

0

.

012

±

0

.

004

)

GeV4,



g3sG3



= (

0

.

57

±

0

.

29

)

GeV6_,wherem2

0

= (

0

.

8

±

0

.

1

)

GeV2.

TheworkingregionsfortheBorelandcontinuumthreshold pa-rametersarefixedinthefollowingform

M2

= (

1

.

1

−

1

.

3

)

GeV2

,

s0

= (

1

.

4–1

.

6

)

GeV2

,

(19)

thatsatisfystandardrequirementsofsumrulescomputations.For example,atthelowerlimitoftheBorelparameterthesumofthe dimension-10,11and12termsin



LL

(

s0

,

M2

)

−

H H

(

s0

,

M2

)

does

notexceed5% ofallcontributions.Attheupperboundofthe work-ing windowforM2 thepolecontributiontothesamequantity is larger than 0

.

12 of the whole result, which is typical for multi-quark systems. Variation of the auxiliary parameters M2 _{and s}

0

withintheregions(19),aswellasuncertaintiesoftheotherinput parametersgeneratetheoreticalerrorsofsumrulescomputations. The tan 2

ϕ

extracted using Eq. (13), as is seen from Fig. 1 (a), demonstratesamilddependenceonM2.Asaresult,itisnot diffi-culttoestimatethat

ϕ

= −

27◦

.

66

±

1◦

.

24

.

(20)

This value of

ϕ

in the heavy-light basis is equivalent to

θ

=

−

33◦

.

00

±

1◦

.

17 inthesinglet-octetbasis.UsingEq.(20) itisnot difficult toevaluate the mesons’ massesinthe one-angle mixing schemethatread

mf

= (

597

±

81

)

MeV

,

mf

= (

902

±

125

)

MeV. (21) Asisseen,the one-anglemixingscheme,iftakeintoaccount the centralvaluesfromEq.(21),doesnotdescribecorrectlythe exper-imental data: it overshoots the mass of the f0

(

500

)

mesonand,

atthesametime,underestimatesthemassofthe f0

(

980

)

meson.

The agreement can be improved by introducing the two mixing angles

ϕ

H and

ϕ

L. It turnsout that to achieve a niceagreement withtheavailableexperimental dataitisenoughto vary

ϕ

H and

ϕ

L withinthelimits(20):

ϕ

H

= −

28◦

.

87

±

0◦

.

42

,

ϕ

L

= −

27◦

.

66

±

0◦

.

31

.

(22) Formf andmf thesumruleswithtwomixingangles

ϕ

H and

ϕ

L leadtopredictions

282 S.S. Agaev et al. / Physics Letters B 781 (2018) 279–282

mf

= (

518

±

74

)

MeV

,

mf

= (

996

±

130

)

MeV

,

(23) whichare compatiblewithexperimental data.Thetheoretical er-rorsinEq.(23) accumulateuncertainties connectedwith M2 and

s0,aswellasarisingfromotherinputparameters.Thedependence

ofmf andmf on theauxiliary parameters M2 and s0 doesnot

exceed limitsallowed forsuch kindof calculations: In Figs. 1(b) and1(c) weplotmf andmf asfunctionsoftheBorelparameter toconfirmastabilityofcorrespondingsumrules.

In the two-angle mixing scheme the system of the physical particles f

−

fischaracterizedbyfourcouplings(7).After deter-miningthemixingangles

ϕ

H and

ϕ

L thatfixthematrixU

(

ϕ

H,

ϕ

L

)

, quantitieswhichshouldbefoundfromtherelevantsumrulesare onlythecouplingsFH andFL.Aswehavementionedabovetothis endweconsidertwoadditionalsumrulesby treatingbasicstates

|

H



and

|

L



asreal“particles”andobtain

FH

= (

1

.

35

±

0

.

34

)

·

10−3GeV4

,

FL

= (

0

.

68

±

0

.

17

)

·

10−3GeV4

.

(24)

ThecouplingFH calculatedinthepresentworkiscomparablewith onefoundinRef. [16] usingthesameinterpolatingcurrent(8) and vacuum condensates up to dimension six and is given by FH

=

(

1

.

51

±

0

.

14

)

·

10−3GeV4.

The mixing angles

(

ϕ

H

,

ϕ

L

)

, the masses

(

mf

,

mf

)

and the couplings

(

FH

,

FL

)

completethesetofthespectroscopic parame-tersofthe f0

(

500

)

and f0

(

980

)

mesons.

5. Concludingnotes

The investigation performed in the present Letter has al-lowedustocalculatethemassandcouplingsofthe f0

(

500

)

and f0

(

980

)

mesonsbytreating themasthemixturesofthediquark–

antidiquarks

|

H



and

|

L



.Wehavedemonstratedthatbychoosing the heavy-light basis and mixing angles

ϕ

H

= −

28◦

.

87

±

0◦

.

42 and

ϕ

L

= −

27◦

.

66

±

0◦

.

31 a reasonable agreement with exper-imental data can be achieved even information on the f0

(

500

)

meson suffers from large uncertainties [23]. The assumption on structures of the light mesons made in the present work deter-minesalsotheir possibledecaymechanisms. Indeed,it isknown that the dominant decay channels of the f0

(

500

)

and f0

(

980

)

mesons are f0

(

500

)

→

π π

and f0

(

980

)

→

π π

processes. In

ex-perimentsthedecay f0

(

980

)

→

K K wasseen,aswell.Themixing

of the

|

H



and

|

L



diquark–antidiquark states to formthe physi-cal mesons implies that all of thesedecays can run through the superallowedOkubo–Zweig–Iizuka (OZI)mechanism:Withoutthe mixing the decay f0

(

980

)

→

π π

can proceed dueto one gluon

exchange, whereas f0

(

980

)

→

K K is still OZI superallowed

pro-cess[16]. The another problemthat finds its naturalexplanation within the mixing framework is a large difference between the

full widthof themesons f0

(

500

)

and f0

(

980

)

, whichamount to

=

400

−

700 MeV and

=

10

−

100MeV [23], respectively. In fact, the strong couplings gf0(500)π π and gf0(980)π π that

deter-minethewidthofthedominantpartialdecays f0

(

500

)

→

π π

and f0

(

980

)

→

π π

dependonthemixingangle

ϕ

L intheform g2_{f0(500)π π}

∝

1 sin2

ϕ

L

,

g2_{f0(980)π π}

∝

1 cos2

_ϕ

L

.

(25)

Inthemodelunderconsiderationthisdependenceisamainsource thatgeneratesthenumericaldifferencebetweenthepartialwidth ofaforementionedprocesses,andhencebetweenthefullwidthof themesons f0

(

500

)

and f0

(

980

)

.

Analysis of the partial decays of the mesons f0

(

500

)

and f0

(

980

)

,aswellascalculationofthe spectroscopicparameters of

other light scalarmesons deservesfurther detailedinvestigations resultsofwhichwillbepublishedelsewhere.

Acknowledgement

K. A.thanks TÜBITAKforthepartialfinancialsupportprovided underGrantNo.115F183.

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