The strong decays of the light scalar mesons f0(500) and f0(980)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

The

strong

decays

of

the

light

scalar

mesons

f

₀

(

500

)

and

f

₀

(

980

)

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,41380Izmit,Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received5April2018

Receivedinrevisedform19June2018 Accepted22July2018

Availableonline4August2018 Editor:J.-P.Blaizot

Thepartialwidthofthedecaychannelsf0(500)→

π π

,f0(980)→

π π

and f0(980)→KK are¯ calculated

usingQCDlight-conesum rulesmethodandatechniqueofthesoftmesonapproximation.Thescalar particles are treatedas mixtures ofthe heavy |H= ([su][¯su¯]+ [sd][¯sd¯])/√2 and light |L= [ud][¯ud¯]

scalardiquark–antidiquarkcomponents.Obtainedresultsforthefullwidthofthe f0(500)mesonth.=

434.7±72.3 MeV andforthe f0(980)mesonth.=42.12±6.70 MeV arecomparedwiththe world

averagesfortheseparameters,andareasonableagreementbetweenthemisfound.

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

1. Lightscalarmesonswithmassesm

<

1GeV formafamilyof particles,structureandpropertiesofwhichremainuncleartillnow andgiverisetodifferentmodelsandtheories.Thestandardmodel ofthemesonsandbaryonsthatconsidersmesonsasboundstates ofquarksandantiquarkscouldnotcorrectlydescribethemass hi-erarchyoftheseparticles.Therefore,thescalarsespecially f0

(500)

and f0

(980)

mesonshavealreadybeeninthespotlightof uncon-ventional theories claiming to solve relevant problems. In most of existing models the light scalar mesons are treated as multi-quarkstates:Theseparticles wereconsideredasfour-quark states

q2q

¯

2 [1], or analyzed as meson–meson molecules [2,3]. Experi-mentalinvestigation of thelight scalars also meets with difficul-ties.Theirmassesandwidthsareknownwithlargeuncertainties, whichgenerateadditionalproblemsfortheoreticalstudies.Indeed, for example, the mass and full width of the f0

(500)

meson is

m

=

400–550MeV and



=

400–700MeV [4],respectively.The ex-perimental data of this quality almost do not restrict suggested models. The contemporary physics of the light scalars embraces varietyofideas,modelsandtheories,informationonwhichcanbe foundinthereviews[5–8].

The diquark–antidiquark model of the light scalar mesons [1,

9,10] openednewopportunities fortheir theoreticalstudies.This model was used to calculate the spectroscopic parameters and width of the scalar mesons in the context of various computa-tionalschemes[11–20].Becausewithinsome oftheseapproaches

*

Correspondingauthor.

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

purediquark–antidiquarkstatesdidnotleadtodesiredpredictions forthe parameters ofthemesons differentmixingschemes were introduced to evade emerged discrepancies. In these studies the physical particles were considered as superpositions of diquark– antidiquarkswithdifferentflavorstructures[17],orasmixturesof diquark–antidiquarksandconventionalqq mesons

¯

[18–20].

Recently, a suggestion was made to treat the scalar mesons by groupingthemintotwo nonetswithmassesbelowandabove 1 GeV [21]. In thiswork the possible mixingof the flavor octet andsingletstatesinsideofeachnonet,aswellasmixingbetween statesfromthe differentnonetswas systematically elaborated. In our work [22] we treatedthemesons f0

(500)

and f0

(980)

from the first nonet of the scalar particles by taking intoaccount the mixingofflavoroctetandsingletdiquark–antidiquarksby neglect-ing,atthesametime,theirpossiblemixingwithtetraquarks com-posedofthespin-1diquarks.Tothisend,weusedtheheavy-light basis

|

H

 =

√

1 2



[

su

][

su

] + [

ds

][

ds

]



,

|

L

 = [

ud

][

ud

],

(1)

andintroduced the two-angle mixingschemeto get thephysical mesons



|

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



.

(2)

ForsimplicityinEq.(2),andinwhatfollowsweusethenotations

f

=

f0

(500)

and f

=

f0

(980).

https://doi.org/10.1016/j.physletb.2018.07.042

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

Calculationsperformed inRef. [22] using QCD two-point sum rulesapproachledtothefollowingresultsforthemixingangles

ϕ

H

= −

28◦

.

87

±

0◦

.

42

,

ϕ

L

= −

27◦

.

66

±

0◦

.

31

.

(3) Formassesofthescalarparticlesweobtained

mf

= (

518

±

74

)

MeV

,

mf

= (

996

±

130

)

MeV

,

(4) whichareinreasonableagreementwiththeexperimentaldata.

Apartfromthe massesofthemesonswedefinedalsotheir cou-plings



0

|

Ji

|

f

(p)

 =

Fi_fmf

,



0

|

Ji

|

f

(

p)

 =

Fifmf

,

i

=

H,L, (5) andsuggestedthattheyfollowthepatternofstatemixing



FH_f FL_f FH f FLf



=

U(

ϕ

H,

ϕ

L

)



FH 0 0 FL



.

(6)

Here FH and FL can be formally interpreted as couplingsof the “particles”

|

H



and

|

L



.CalculationsusingQCDtwo-pointsumrules allowedustoevaluatethemandfind

FH

= (

1

.

35

±

0

.

34

)

·

10−3GeV4

,

FL

= (

0

.

68

±

0

.

17

)

·

10−3GeV4

.

(7) InthepresentLetterweextendourinvestigationofthe f0

(500)

and f0

(980)

mesonsbyanalyzingamechanismoftheirstrong de-cays and calculatecorresponding partial widths. To thisend, we use an information on the f

−

f system’s parameters, i.e. on themasses,mixinganglesandcouplingconstants,whichwere ex-tractedfromanalysisofthetwo-point sumrules inRef. [22] and are not subject to any adjustments. In investigations we employ QCDlight-conesumrule(LCSR)method[23] andtechnicaltoolsof thesoft-meson approximation [24]. It is worth notingthat these methodswere adapted inRef. [25] to studystrongvertices com-posedoftetraquarksandtwoconventionalmesons.

2. The dominant strong decay channels of the f0

(500)

and

f0

(980)

mesons aretheprocesses f0

(500)

→

π π

and f0

(980)

→

π π

.Thedecay f0

(980)

→

K K wasalsoobservedandinvestigated in experiments [4]. Suggestion on the structure of these scalar particlesassuperpositionsofthe

|

H



and

|

L



diquark–antidiquark stateshasimportantconsequencesforanalysisoftheirdecays. In-deed,ignoringthemixingphenomenonandassumingthat f0

(500)

and f0

(980)

mesonsarepure

|

L



and

|

H



four-quarkstatesonehas tointroduce different mechanismsto describe decays f0

(980)

→

K K and f0

(980)

→

π π

: Ifthefirstchannelrunsthrough the su-perallowedOkubo–Zweig–Iizuka(OZI)mechanism,thesecondone canproceedsdueto one gluon exchange[14]. The mixingofthe

|

H



and

|

L



statestoformthephysicalparticlesallowsonetotreat all of these strong decays on the same footing using the super-allowed OZI mechanism. It is known that the full width of the mesons f0

(500)

and f0

(980),

whichamountto



=

400–700MeV and



=

10–100 MeV [4], respectively, suffer from large uncer-tainties and differ from each other considerably. In the mixing frameworkthisdifferencefindsitsnaturalexplanation:Asweshall see below the dependence of the strong couplings correspond-ing to the vertices f0

(500)

π π

and f0

(980)

π π

are proportional to1/sin

ϕ

L and1/cos

ϕ

L.Thedependenceofthestrongcouplings onthemixingangle

ϕ

L alongsidewithother parametersthat en-tertosumrulesgeneratesagapinthepartialwidthsofthescalar particles.

Thedecayofthe f0

(500)

mesontoapairofpionscanproceed throughtheprocesses f0

(500)

→

π

+

π

−and f0

(500)

→

π

0

π

0.Let

usconcentrateoninvestigationofthemode f0

(500)

→

π

+

π

−.In ordertocalculatethestrongcouplinggfπ π weemployQCD

light-conesumrulemethodandbeginfromanalysisofthecorrelation function

(p,q)

=

i



d4xeip·x



π

+

(q)

|

T

{

Jπ

(x)

Jf†

(

0

)

}|

0

,

(8)

where Jf

₍

_x

₎

_{and J π}

₍

_x

₎

_{are the interpolating currents for the f} and

π

− mesons, respectively. In the two-mixing angle scheme

Jf

(

x

)

isgivenbytheformula

Jf

(x)

=

JH

(x)

cos

ϕ

H

−

JL

(x)

sin

ϕ

L

.

(9) Here JH

₍

_x

₎

_{and J}L

₍

_x

₎

_{are theinterpolatingcurrentsofthe scalar} mesons’ heavy and light components, respectively. They are de-finedbymeansofthefollowingexpressions

JH

(x)

=



dab

_

dce

√

2



u_aT

(x)C

γ

5sb

(x)

uc

(x)

γ

5C sTe

(x)

+

d_aT

(x)C

γ

5sb

(x)

dc

(x)

γ

5C seT

(x)



,

(10) and JL

(x)

=



dab

_

dce

_uT a

(x)C

γ

5db

(x)

uc

(x)

γ

5Cd T e

(x)

.

(11)

InEqs.(10) and(11) a,b

,

c

,

d

,

e arecolorindices,whereasC isthe chargeconjugation operator.Weinterpolatethepionbymeansof thepseudoscalarcurrent

Jπ

(x)

=

u(x)i

γ

5d(x), (12) withthematrixelementdefinedas



0

|

Jπ

|

π

−

(p)

 =

fπ

μ

π

,

μ

π

= −

2



qq



f2

π

.

(13)

InEq.(13) fπ and



qq



arethepiondecayconstantandthequark vacuumcondensate,respectively.

The required LCSR can be derived after standard operations: One has to calculate the correlation function employing physical parametersoftheinvolvedmesonsandequateittoanexpression of

(

p

,

q

)

obtained interms ofthe quark-gluon degreesof free-dom.Westart fromthephysicalrepresentationofthecorrelation function

(

p

,

q

)

thatisgivenbytheformula



Phys

(p,q)

=



0

|

J π

_|

_π

−

_(p)

_

p2

₋

_m2 π



π

−

(p)

π

+

(q)

|

f

(

p

)





f

(p



₎

_|

_Jf †

_|

₀

_

p2

₋

_m2 f

+ . . . ,

(14)

where p

,

p and q are four-momenta of the f ,

π

− and

π

+

mesons,respectively.Thecontributionoftheexitedstatesand con-tinuumisdenotedinEq.(14) bydots.Thematrixelement ofthe pion that enters to this expression is well known. The element



f

(

p

)

|

Jf †

|

0



canbefoundbytakingintoaccountthestructureof thecurrent Jf

₍

_x

₎

_{andthefactthatonlyits lightcomponent} con-tributes to this matrix element



f

(

p

)

|

Jf †

|

0



=

FLmfsin2

ϕ

L.We define the matrix element corresponding to the strongvertex in thefollowingmanner



π

−

(p)

π

+

(q)

|

f

(p



)

 =

gfπ πp

·

p

.

(15)

When applying the LCSR method to vertices composed of a

tetraquark and two conventional mesons one has to use a tech-nique of the soft-meson approximation [25]. The reason is that the tetraquarkcontains four valencequarksandcontraction with two quark fields from a meson leads to local matrix elements

oftheremaining light meson. Then theconservation ofthe four-momentumatthevertexrequiresfulfilmentoftheequalityq

=

0 (or p

=

p).Inotherwords,inthecaseofthetetraquark–meson– mesonvertexthesoft-mesonapproximationisonlywayto calcu-latethecorrespondingcorrelationfunction.Forverticesof conven-tionalmesonsthe correlationfunction canbe expressedinterms of a meson’s distribution amplitudes. This is the full LCSR ap-proachwithin ofwhich one mayemploy thesoftapproximation, aswell.Forour purposesa decisivefact isthe observationmade inRef. [24]:thesoft-mesonapproximationandfullLCSRtreatment oftheconventionalmesons’verticesleadsforstrongcouplingsto resultsthatarenumericallyveryclosetoeachother.

In the soft-meson approximation we have to use the one-variableBoreltransformationandsubtract unsuppressedtermsin thephysicalside ofthe sumrules. Weneglectalsothe massone of the final mesons in



OPE

(

p

,

q

=

0) and



OPE_K

(

p

,

q

=

0). De-tailedstudiesofmasseffectsinexclusiveprocessesprovethatthey induceonlytwist-4contributionstophysicalquantitiesunder con-sideration[26].Hence, inthesoft-mesonapproximationthemass effectsarealsosubleadingcorrections.

In order to compute the tetraquark–meson–meson vertex we usetheone-variableBoreltransformation,whichforthe



Phys

(

p

)

(weuse

(

p

)

≡ (

p

,

0))leadstothefollowingresult

B



Phys

(p)

=

gfπ πfπFL

μ

πmfm2sin2

ϕ

L e−m2/M2

M2

+ . . . ,

(16) where m2

= (

m2_f

+

m2_π

)/2 and

M2 is the Borel parameter. In Eq. (16) thedotsstandforthecontributionoftheexcitedand con-tinuumstates,amongofwhich thereexist termsthat inthe soft limitevenaftertheBoreltransformationremainunsuppressed rel-ativetotheground-state’scontribution[24].Inthecaseunder con-siderationwe are interested onlyintheground-state term there-fore these unsuppressed contributions should be removed from Eq. (16).Butbeforeperforming necessaryoperationswe calculate the



OPE

(

p

)

andfind



OPE

(p)

=

sin

ϕ

L



d4xeip·x



cab

_

cde

_γ

5



Sibd

(x)

γ

5



Sdiu

(

−

x)

γ

5

αβ

× 

π

+

|

ua_α

(

0

)d

e_β

(

0

)

|

0

.

(17)

Computationsof



OPE

₍

_p

₎

_{usingthepionlocalmatrixelements in} accordancewithprescriptionsexplainedinratherdetailedformin Ref. [25],andtheBoreltransformationoftheobtainedresultgive

(M

2

)

=

fπ

μ

π 16

π

2 sin

ϕ

L ∞



0 dse−s/M2s

+ 

α

sG 2

π



sin

ϕ

L fπ

μ

π 16

.

(18)

InordertoperformthecontinuumsubtractioninEq.(18) onehas toremovetheunsuppressedtermsfromthe

B

Phys

₍

_p

₎

_whichcan befulfilledbyapplyingtheoperator[27]

P

(M

2

,m

2

)

=



1

−

M2 d dM2



M2em2/M2

.

(19)

Thenforthestrongcouplinggfπ π weget

gfπ π

=

1 sin

ϕ

L 1 fπFL

μ

πmfm2

P

(M

2

,m

2

)

(M

2

,

s0

),

(20) where



(M

2

,

s0

)

=

fπ

μ

π 16

π

2 s0



0 dse−s/M2s

+ 

α

sG 2

π



fπ

μ

π 16

.

(21)

The analysisoftheprocess f0

(500)

→

π

0

π

0 doesnotdiffer con-siderably fromcalculations presented above the difference being encodedinthecurrentofthe

π

0 _meson.

3. The decaysofthemeson f0

(980)

to

π π

and K K pair pro-ceedbythesamesuperallowedOZImechanism.Inthecaseofthe process f0

(980)

→

π π

the

|

L



componentof f0

(980)

determines the decays f0

(980)

→

π

+

π

− and f0

(980)

→

π

0

π

0. For these channels a situation doesnot differ from the decays f0

(500)

→

π π

: One needs to replace in Eq.(20) sin

ϕ

L

→ −

cos

ϕ

L

,

mf

→

mf,andsetm2

= (

m2_f

+

m2π

)/2.

Thesemodifications andproperly

chosen parameters M2 and s0 are enough to perform numerical analysis ofthedecay channels f0

(980)

→

π

+

π

− and f0

(980)

→

π

0

_π

0_{,andfindtheirpartialwidths.}

Investigation ofthe strongdecays f0

(980)

→

K K actually im-pliesanalysisofthefollowingtwodecaymodes f0

(980)

→

K+K− and f0

(980)

→

K0K

0

.Naturally,allofthesechannelsrunthrough decaysofthe f0

(980)

meson’sheavy component

|

H



.Letus con-sider in some details theprocess f0

(980)

→

K+K−. The correla-tionfunctionnecessarytostudythisdecayis



K

(p,q)

=

i



d4xeip·x



K+

(q)

|

T

{

JK

(x)

Jf†

(

0

)

}|

0

,

(22)

wheretheinterpolatingcurrentforthe f0

(980)

mesonis

Jf

(x)

=

JH

(x)

sin

ϕ

H

+

JL

(x)

cos

ϕ

L

.

(23) ForK mesonsweusethepseudoscalarcurrent

JK

(x)

=

u(x)i

γ

5s(x), (24) withthematrixelement



0

|

JK

|

K−

(

p)

 =

fKm

2 K ms

+

mu

.

(25)

Skipping detailsofcalculationsthatare similartoonespresented above we write down final expressions:Thus, for



OPE_K

(

p

,

q

)

we get



OPE_K

(p,

q)

= −

sin

ϕ

H



d4xeip·x



abc

_

dec

√

2

×

γ

5



Sias

(x)

γ

5



Seiu

(

−

x)

γ

5

αβ



K +

_|

_ub α

(

0

)s

dβ

|

0

.

(26)

Thefinalexpressionforthestrongcoupling gfK K is

gfK K

= −

1 sin

ϕ

H ms fKFHm2Kmfm2

P

(M

2

,m

2

)



K

(M

2

,s

0

),

(27) wherem2

_{= (}

_m2 f

+

m2K

)/2 and





K

(M

2

,

s0

)

=

fKm2_K 16

√

2ms

π

2 s0



0 dse−s/M2s

−



2



uu

 − 

ss



12

√

2 fKm 2 K

+ 

α

sG2

π



fKm2K 16

√

2ms

.

(28)

The strong couplings gfK K and gfK0_K0 provide necessary infor-mationforcomputingthe f0

(980)

→

K+K−and f0

(980)

→

K0K

0

Fig. 1. The dependence of the strong coupling gfK K on the Borel parameter M2at fixed s0(left panel), and on the continuum threshold s0at fixed M2(right panel).

4. In calculations we utilize the light quark propagator (see, Ref. [22]) and use for the quark and gluon condensates the following values:

¯

qq



= −(

0.24

±

0.01)3 GeV3,

¯

ss



=

0.8

¯

qq



,



α

sG2

/

π



= (

0.012

±

0.004)GeV4.Apartfromtheseparameterswe alsoemploythemassesofthelightquarksmu

=

md

=

0 andms

=

128

±

10 MeV,aswell asthemassesanddecayconstants ofthe

π

andK mesons: forthepion_mπ±

=

139.57061

±

0.00024 MeV,

mπ0

=

134.9770

±

0.0005MeV and fπ

=

131MeV andforthe K mesonmK±

=

493.677

±

0.016MeV,mK0

=

497.611

±

0.013MeV and fK

=

155.72MeV.

For the decays of the f0

(500)

the working windows for the Borel and continuum threshold parameters are fixed within the limits

M2

= (

0

.

7

−

1

.

2

)

GeV2

,

s0

= (

0

.

9

−

1

.

1

)

GeV2

.

(29) Calculationsofthestrongcouplingsleadtothepredictions

gfπ π

=

33

.

94

±

3

.

86 GeV−1

,

|

g_fπ0_π0

| =

32

.

76

±

3

.

56 GeV−1

.

(30)

Asaresult,forthepartialdecaywidthoftheprocesses f0

(500)

→

π

+

π

−and f0

(500)

→

π

0

π

0 wefind





f0

(

500

)

→

π

+

π

−



=

223

.

5

±

53

.

7 MeV

,



f0

(

500

)

→

π

0

π

0

=

211

.

2

±

48

.

4 MeV

.

(31)

The full width of the meson f0

(500)

is formed almost entirely duetothedecaychannel f0

(500)

→

π π

becausethewidthofthe mode f0

(500)

→

γ γ

isverysmall.Itseemsreasonabletocompare



th.

=

434.7

±

72.3 MeV which is the sum of two partial decay

widths(31) with theavailable informationon



=

400–700 MeV notingexistence ofan overlapping regionof theseresults.As we havepointedout,dataforthefullwidthofthelightscalarmesons sufferfromlarge uncertainties. Therefore, we can onlystate that ourtheoreticalpredictioniscompatiblewithexperimentaldata.

Thestrongdecaysofthe f0

(980)

mesoncanbeanalyzedinthe same manner. The differences between the channels f0

(500)

→

π π

and f0

(980)

→

π π

appear dueto the spectroscopic param-eters of the involved mesons, and regions chosen for the Borel parameter and continuum threshold. In the case of the f0

(980)

meson’sdecaysweuse

M2

= (

1

.

1–1

.

5

)

GeV2

,

s0

= (

1

.

3–1

.

5

)

GeV2

.

(32) Thenforthecouplingsandpartialdecaywidthswefind

gfπ π

=

3

.

02

±

0

.

35 GeV−1

,

gfπ0_π0

=

3

.

75

±

0

.

45 GeV−1

,

|

gfK K

| =

4

.

29

±

0

.

75 GeV−1

,

|

gfK0_K0

| =

4

.

97

±

0

.

98 GeV−1

,

(33) and





f0

(

980

)

→

π

+

π

−



=

14

.

36

±

3

.

31 MeV

,



f0

(

980

)

→

π

0

π

0

=

22

.

19

±

5

.

64 MeV

,





f0

(

980

)

→

K+K−



=

3

.

98

±

1

.

04 MeV

,



f0

(

980

)

→

K0K 0

=

1

.

59

±

0

.

47 MeV

.

(34)

Incalculationswe haveutilized thedifferentworkingregions for theBorelparameterM2_{andcontinuumthresholds}

0.Wehave cho-sen these regions using standard requirements of the sum rules computations.Itisknownthatastabilityoftheobtainedresultson

M2 ands0 isone ofthe importantconstraintsimposed on these auxiliary parameters. We demonstrate in Fig. 1 as a sample the variation ofthecoupling

|

gfK K

|

onthe M2 ands0. Onecan see that

|

gfK K

|

dependson M2 ands0,whichisamainsourceof un-certainties of the evaluated quantities. It is also clear that these ambiguities are less than 30% of the central values which is ac-ceptableforthesumrulecomputations.

It is remarkable that there are valuable experimental infor-mation andindependent theoretical predictions for the coupling

gfK K. It was extracted from different processes, and calculated by means of numerous methods. Thus, from analysis of the ra-diative decay

φ

→

f0

γ

the CMD-2andSNDcollaborationsfound

gfK K

=

4.3

±

0.5 GeV and5.6

±

0.8 GeV [28,29],respectively.The KLOE Collaboration used the same process and from two differ-entfits extractedthefollowing valuesgfK K

=

4.0

±

0.2 GeV and 5.9

±

0.1 GeV [30].OurresultforgfK K canbeeasilyconvertedto aformsuitableforcomparisonwiththeseexperimentaldata,and isequalto4.12

±

0.72 GeV.Asisseen,ourpredictionforthestrong couplinggfK K isinareasonableagreementwiththis experimen-talinformation.Atthesametime,itovershootsexperimentaldata extracted from other processes such as D_s+

→

π

−

π

+

π

+ decay and pp interactions, where the coupling gfK K was found equal to 0.5

±

0.6 GeV and2.2

±

0.2 GeV (see, Refs. [31] and[32]), re-spectively.

The theoretical predictions for gfK K appear to vary within wide limitsanddepend onamodel acceptedfor f0

(980)

andon methods used in investigations. For example, in Ref. [33] it was foundequalto gfK K

=

3.8 GeV,whereas inRef. [34] the authors predicted 6.2

≤

gfK K

≤

7.8 GeV. The latter estimation was ob-tainedinthecontextofthefullLCSRmethodbymodeling f0

(980)

asascalarmesonwitha

¯

ss component.As itwasemphasizedby theauthors,their resultislargerthanprevious determinations.It isalsolargerthanourpredictionfor gfK K thereasonbeing con-nectedpresumablywithamixingfactorofthe

¯

ss component ne-glected incomputations.Information onother theoretical studies andreferencestocorrespondingarticlescanbefoundinRef. [34]. UsingresultspresentedinEq.(34) weareabletoevaluatethe width of the decays



[ f0

(980)

→

π π

]

=

36.55

±

6.54 MeV and





f0

(980)

→

K K



=

5.57

±

1.48MeV.Byneglectingthe contribu-tion



[ f0

(980)

→

γ γ

] forthefullwidthofthemeson f0

(980)

we find



th.

=

42.12

±

6.70 MeV,whichisinaccord withthe

experi-mentaldata.

5. Thepartialandfullwidthsofthescalarmesons f0

(500)

and

f0

(980)

obtained in the present work by treating them as the mixtures of the different diquark–antidiquark components seem arein reasonableagreement withexistingexperimental data. Be-cause there are great discrepancies between results of different experiments, we compare our predictions with the world aver-age for these parameters presented by the Particle Data Group in Ref. [4]. Thus, the full widthof the f0

(500)

meson is slightly larger than the lower bound of the experimental data: There is smalloverlapregionbetweenthetheoreticalandexperimental re-sults. For the f0

(980)

meson we have found



th.

∈ 

exp., which

is in a nice agreement with the data. Another parameter R

=

(

π π

)/

[(

π π

)

+ (

K K

)

]

=

0.87+₋0₀._.06₀₈ provides aninformationon partialdecaywidthsofthemeson f0

(980)

andonitsstrangeand non-strange components. The prediction for R agrees with the

upperlimitforthisparameterfromRef. [4].

As isseen, the modelof the light scalarmesons f0

(500)

and

f0

(980)

based on the mixing of the diquark–antidiquark states leads to the results that are in agreement with the world aver-ages fortheir full widths.Nevertheless, some effects whichhave beenneglectedinthepresentinvestigation,namelypossible mix-ing with the mesons from the second (heavier) scalar nonet, as wellas f0

(980)

−

a0

(980)

mixingmayimproveourpredictions.

The strange and non-strange quark contents of the f0

(500)

and f0

(980)

mesons also need additional investigations. In fact, the model accepted here implies that both the mesons f0

(500)

and f0

(980)

have the strange andnon-strange components. The existence of sizeable non-strange content in the f0

(980)

me-son does not contradict to experimental measurements. But the strangecomponentofthe f0

(500)

meson,asitwaspointedoutin Ref. [35],may causedifficultiesin interpretationof existing data. In fact,in Ref. [35] the f0

(500)

and f0

(980)

mesonswere mod-eledasmixturesofstrangess andnon-strange

(

uu

+

dd

)/

√

2 parts. Inthismodeltheratio



[

D+_s

→

f0

(500)

π

+

]/[

D+s

→

f0

(980)

π

+

]

depends on the mixing angle that has to be extracted from ex-perimental measurements. But the E791 Collaboration did not observe a contribution of the process D_s+

→

f0

(500)

π

+ to the decay D+_s

→

π

−

π

+

π

+ [31], which contradicts to the theoreti-cal assumptionon thestrange component ofthemeson f0

(500).

Thisexperiment predictedforthe strong coupling gfK K

=

0.5

±

0.6 GeV, which contradicts also to all other measurements. The modelusedinthepresentworkdiffersfromtheframework intro-ducedin Ref. [35]. Therefore, to clarify a situation with f0

(500)

meson’s strange component the decays D+_s

→

f0

(500)

π

+ and

D+_s

→

f0

(980)

π

+ should be studied within thisnew model.For comparison to theoretical predictions more precise experimental dataarerequired,aswell.

There are no doubts, that the light scalar mesons as unusual particles deserve further detailed theoretical and experimental studies.

Acknowledgement

K.A. and H.S. thank TUBITAK for the partial financial support providedunderGrantNo.115F183.

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