The nonet of the light scalar tetraquarks: The mesons a(0)(980) and K-0*(800)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

The

nonet

of

the

light

scalar

tetraquarks:

The

mesons

a

₀

(

980

)

and

K

₀

∗

(

800

)

S.S. Agaev

a

_,

_{K. Azizi}

b

,

∗

_,

_{H. Sundu}

c

a_{InstituteforPhysicalProblems,BakuStateUniversity,Az-1148Baku,Azerbaijan} b_{DepartmentofPhysics,Doˇgu ¸sUniversity,Acibadem-Kadiköy,34722Istanbul,Turkey} c_{DepartmentofPhysics,KocaeliUniversity,41380Izmit,Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received10September2018

Receivedinrevisedform15December2018 Accepted26December2018

Availableonline28December2018 Editor: J.-P.Blaizot

The spectroscopic parameters and partial decay widths of the light mesons a0(980) and K₀∗(800) are

calculated by treating them as scalar diquark–antidiquark states. The masses and couplings of the mesons are found in the framework of QCD two-point sum rule approach. The widths of the decay channels

a0(980)→

ηπ

and a0(980)→KK¯, and K₀∗(800)→K+

π

− and K∗

0(800)→K0

π

0are evaluated using QCD sum rules on the light-cone and technical tools of the soft meson approximation. Our results for the mass of the mesons ma0=991+₋2927 MeV and mK∗=767+₋38₂₉MeV, as well as their total width #a0= 62.01 ±14.37 MeV and #K∗

0=401.1 ±87.1 MeV are compared with last experimental data.

©2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

1. Introduction

Theexperimental investigation of thelight scalar mesonsand theoreticalinterpretationofobtaineddataremainsoneof intrigu-ingproblemsinhighenergyphysics.Experimentalinformationon parametersoftheseparticlessuffersfromlargeuncertainties:Their massesandwidths aresometimesknown with

∼

100MeV accu-racy[1].Thestatusofsomeoftheseparticlesisstillunclear,even theirexistenceisunderquestion.

The theoretical interpretations of light scalars alsomeet with well-known troubles. Really, the nonet of scalar particles in the conventional quark–antiquark model of mesons may be realized as 13_P

0 states. The masses of these scalars, in accordance with

various model calculations are higher than 1 GeV. In fact, the isoscalarmesons f0

(

1370

)

and f0

(

1710

)

,theisovectora0

(

1450

)

or

isospinorK∗

0

(

1430

)

stateswereidentifiedasmembersofthe13P0

multiplet. Butmasses of the mesonsfrom the light scalar nonet lie below 1 GeV. Therefore, during a long time the broad scalar resonances f0

(

500

)

andK₀∗

(

800

)

,relativelynarrowstates f0

(

980

)

and a0

(

980

)

are subject of controversial theoretical hypothesis

and suggestions. The main idea behind attempts to explain un-usual features of these states isan assumption about four-quark (diquark–antidiquark or meson–meson) nature of these mesons [2–4].Withinthisscheme quantumnumbers andlow masses,as

*

Correspondingauthor.

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

wellasmasshierarchyinsideofthelightnonetseemreceive rea-sonable explanations.The present-dayphysics ofthelight scalars consists ofdifferent ideas, models andtheories. The comprehen-siveinformationontheseissuescanbefoundinthereviewarticles [5–8].

Thediquark–antidiquarkpictureallowsonetoansweressential questions aboutinternal organization of light scalarmesons, and calculatespectroscopicparametersanddecaywidthofthese parti-cles[2,9,10].Inthismodelthescalarmesonsemergeasthenonet of particles composed of four valence quarks. Within the nonet the SUf

(

3

)

flavor octet andsingletstates maymix tocreate the

physicalmesons f0

(

500

)

and f0

(

980

)

.The situationhereis

simi-larto the well-known mixingphenomenon inthe

η

−

η

′ system ofthepseudoscalarmesons.Theothertwoscalarparticlesa0

(

500

)

and K∗

0

(

800

)

may be identified withthe isotriplet and isospinor

members of the light multiplet. A model of the scalar mesons above andbelow 1GeV was proposed in Ref. [11], inwhich the heavy nonet is the conventional qq nonet mixed with the glue-ball, whereas the light nonet has a four-quark composition with thediquark–antidiquarkormolecule-likestructures.Aninteresting suggestionaboutthestructureofthescalarmesonswasproposed recentlyinRef. [12].Inthispicturenotonlylightmesonsbutalso the heavy onesare collected into two nonetof the scalar parti-cleswith diquark–antidiquarkstructure: Thephysical mesonsare mixturesofthespin-0 diquarks from(3c

,

3f)representationwith spin-1 diquarks from

(

6c

,

3f

)

representation of the color-flavor group.

https://doi.org/10.1016/j.physletb.2018.12.059

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

The diquark–antidiquark model allowed one to calculate pa-rameters of the light scalars and explore their strong and elec-tromagnetic decay channels. To this end, different calculational schemesandmethodswereused.Thus,themassesofthe f0

(

500

)

,

f0

(

980

)

,a0

(

980

)

andK0∗

(

800

)

mesonswerecalculatedinRef. [13]

intheframework oftherelativisticdiquark–antidiquarkapproach andniceagreementswiththe datawerefound. Inthecontext of thefour-quarkBethe–Salpeterequationthesameproblemwas ad-dressedinRef. [14].Thetwo-pseudoscalarandtwo-photondecays ofthemesonsfromthelightscalarnonetwerestudiedinRef. [15]. Intensive investigations of the light scalar mesons were per-formed using QCD sum rules method [16–24]. In these papers apart from the pure diquark–antidiquarks the light scalars were treated also as mixtures of diquark–antidiquarks with different flavor structuresor assuperpositions of diquark–antidiquark and quark–antiquark components. The aforementioned modification were introduced by the authors mainly to achieve an agreement between theoretical predictions and experimental data. The sum ruleswithinclusionofinstantoneffectswereemployedinRef. [23] to evaluate masses of the scalar mesons above 1 GeV. It was demonstrated,that instantoneffectsseparate the mesons’masses fromeachotherremovingthedegeneracyoftheconventionalsum rulespredictions.

In our work [25] we calculated the masses of the mesons

f0

(

500

)

and f0

(

980

)

by considering them as states composed of scalar diquarks. We took into account the mixing of flavor octet and singlet diquark–antidiquarks that create the physical mesons and, at the same time, neglected their possible mixing with tetraquarks built of the spin-1 diquarks. Obtained in this work predictionsforthe massesofthescalar mesonsare in rea-sonable agreement with existing data. The mixing of the flavor octet andsingletdiquark–antidiquarks usedin Ref. [25] to calcu-latespectroscopicparametersofthemesons f0

(

500

)

and f0

(

980

)

hadimportantconsequencesforstudyingoftheir decaychannels. Indeed,withoutoctet-singletmixingthedecaysofdifferentscalar mesons proceed through different mechanisms.For example,the process f0

(

980

)

→

K K is the superallowed Okubo–Zweig–Iizuka

(OZI)decay,whereas f0

(

980

)

→

π π

canproceedduetoonegluon exchange [18].The octet-singlet mixingallows oneto treat all of thelightscalarmesons’decaychannelsemployingtheOZI mecha-nism,andexplaindifferencesintheirpartialwidthsbythemixing parameters.Thedecaysofthe f0

(

500

)

and f0

(

980

)

mesonsinthis

frameworkwereevaluatedinRef. [26].

The presentwork is an extension of our previous studies de-voted to spectroscopy and decay properties of the light scalar mesons[25,26].Wetreatthemasdiquark–antidiquarkstates com-posed of the scalar diquarks by ignoring their possible mixing with spin-1 diquarks. We calculate the spectroscopic parameters of the mesons a0

(

980

)

and K₀∗

(

800

)

, evaluate their partial

de-cay widths and, as a result, total widths of these particles. All investigations are performedusingQCD sum rulemethod: In or-der to calculate the mass and coupling of the mesons we em-ployQCDtwo-point sumruleapproachby includingintoanalysis quark,gluonandmixingvacuumcondensatesuptodimensionten [27,28]. The sum rules for the strong couplings of the vertices

a0

(

980

)

ηπ

0,a0

(

980

)

K+K−,K₀∗

(

800

)

K+

π

− andK∗

0

(

800

)

K0

π

0 are

derivedusinglight-conesumrule(LCSR)method[29] and techni-caltoolsofthesoft-mesonapproximation[30],whichwasadapted in Ref. [31] tostudy tetraquark–meson–meson vertices. This ap-proachwas successfullyapplied to evaluate strongcouplings and widths ofnumerous decaysinvolving tetraquarks[32,33], includ-ingthelightaxial-vectormesona1

(

1420

)

[34].

Thispaperisorganized inthefollowingway:In thesection 2

we calculate the mass and coupling of the mesons a0

(

980

)

and

K∗

0

(

800

)

. In the section 3 we derive the sum rules to evaluate

the strong couplings gaηπ , gaK K, gK∗Kπ and gK∗K0_π0. The

ob-tained results are utilized in Sec. 4 for numerical evaluation of the strong couplings and widths of the decays a0

(

980

)

→

ηπ

0,

a0

(

980

)

→

K+K−, K₀∗

(

800

)

→

K+

π

− and K∗

0

(

800

)

→

K0

π

0,and

totalwidthsofthemesonsa0

(

980

)

and K₀∗

(

800

)

.InSection 5we discussobtainedresultsandpresentourconcludingnotes.

2. Massandcouplingofthemesonsa0

(

980

)

andK₀∗

(

800

)

Themassandcouplingofthemesonsa0

(

980

)

andK₀∗

(

800

)

can

becalculatedwithinQCDtwo-pointsumrulemethod.Weconsider hereindetailsallnecessarystepstofindthemassandcouplingof thea0

(

980

)

mesonandprovideonlyfinal expressionsandresults

forthe K∗

0

(

800

)

meson.

The mass and couplingof a0

(

980

)

can be extractedfrom the

sumruleanalysisofthetwo-pointcorrelationfunction

$(

p

)

=

i

!

d4xeipx

⟨

0

|

T

{

J

(

x

)

J†

(

0

)}|

0

⟩,

(1)

where J

(

x

)

is theinterpolating currentto thea0

(

980

)

meson. In

the diquark–antidiquarkmodelitcan be writteninthe following form J

(

x

)

=

√

ϵ

"

ϵ

2

#$

u_aTC

γ

5sb

% $

ud

γ

5C sTe

%

−

$

dT aC

γ

5sb

% $

d_d

γ

5C sTe

%&

,

(2)

whereC isthecharge conjugationoperator.Herewealsousethe short-hand notation

ϵ

"

ϵ

=

ϵ

abc

ϵ

dec with a

,

b

,

c

,

d and e being the

colorindices.Letusnote that weusetheconventionaltwo-point sumrulesneglecting thepossibleinstantoneffectsinthe correla-tionfunction

$(

p

)

.

Inaccordancewithstandardprescriptionsofthesumrule com-putations the correlation function

$(

p

)

should be found by

em-ploying both the physical parameters of the a0

(

980

)

meson, i.e.

its mass ma0 and coupling fa0 and in terms of the light-quark

propagators,andasaresult, intermsofvariousquark, gluonand mixedvacuumcondensates.Bymatchingtheobtainedresultsand benefiting fromtheassumption onthe quark-hadrondualityit is possibletoextractsumrulesandevaluatethephysicalparameters ofinterest.

In the case under consideration the physical side of the sum ruletakesthesimpleform

$

Phys

(

p

)

=

⟨

0

|

J

|

a0

(

p

)

⟩⟨

a0

(

p

)

|

J †

_|

₀

_⟩

m2

a0

−

p2

+ . . . ,

(3)

becausethea0

(

980

)

mesonistheground-stateparticle:The

con-tributions coming from the excited and continuum states are showninEq. (3) bydots.Toexpress

$

Phys

(

p

)

intermsofthe

pa-rametersma0 and fa0 weintroducethematrixelement

⟨

0

|

J

_|

a0

(

p

)

⟩ =

fa0ma0

,

(4) andget

$

Phys

(

p

)

=

f 2 a0m 2 a0 m2 a0

−

p2

+ . . . .

Effectoftheexcitedstatesandcontinuumonthe

$

Phys

(

p

)

canbe

suppressedbymeansoftheBoreltransformationwhichyields

B

$

Phys

(

p

)

=

f_a2₀m_a2₀e−m2a0/M2

+ . . . ,

(5)

where M2 _{istheBorelparameter.InEq. (5) bydots weagain}

besubtractedfromBoreltransformationof

$

OPE

(

p

)

toderivethe

requiredsumrules.

The

$

OPE

(

p

)

thatconstitutesthesecondpartofthesumrule’s

equalityis obtainedfromEq. (1) using theexplicitexpression for theinterpolatingcurrent J

(

x

)

andcontractingtherelevantquarks

fields.Asaresultfor

$

OPE

(

p

)

wefind

$

OPE

(

p

)

=

i

!

d4xeipx

ϵ

"

ϵϵ

′

"

ϵ

′ 2

'

Tr

#

γ

5

"

Ses′e

(

−

x

)

γ

5

×

Sd_u′d

(

−

x

)

&

Tr

#

γ

5

"

Saau′

(

x

)

γ

5Sbbs ′

(

x

)

&

+ (

u

_↔

d

)

(

.

(6)

Intheexpressionabove

"

Ss(q)(x

)

=

C ST_s(q)

(

x

)

C

,

whereSs(q)

(

x

)

arethesandq

=

u

,

dquarks’propagators

Sab_q

(

x

)

=

i

δ

ab

/

x 2

π

2x4

− δ

ab mq 4

π

2x2

− δ

ab

⟨

qq

_⟩

12

+

i

δ

ab

/

xm₄₈q

⟨

qq

⟩

− δ

ab x 2 192

⟨

qgs

σ

Gq

⟩

−

i gsG αβ ab 32

π

2x2

)

/

x

σ

αβ

+

σ

αβ

/

x

*

+

i

δ

abx 2

_/

_xm q 1152

⟨

qgs

σ

Gq

⟩

−

i

δ

ab x2

_/

_xg2 s

⟨

qq

⟩

2 7776

− δ

ab x4

_⟨

_qq

_⟩⟨

_g2 sG2

⟩

27648

+ ...

(7)

Inthepresentworkwecalculatethecorrelationfunctionbytaking intoaccountnonperturbativetermsuptodimensionten.

The Borel transform of the correlator

_B$

OPE

₍

_p

₎

_{= $}

OPE

₍

_M2

₎

canbe calculated usingeither thespectral density

ρ

(

s

)

whichis

proportionaltoimaginarypartof

$

OPE

(

p

)

orbyapplyingtheBorel

transformationdirectlyto

$

OPE

(

p

)

.Ifnecessary,

$

OPE

(

M2

)

maybe

computedutilizingbothoftheseapproaches. Theseroutine oper-ationswere explainednumerouslyinexisting literature,therefore wedonot concentrateon thesequestionshere. The obtained ex-pressionfor

$

OPE

(

M2

)

hastobeequated toEq. (5),andonealso

hastoperform thecontinuum subtraction. Afterthese manipula-tionswefindthefollowingsumrule

f_a2₀m_a2₀e−ma02/M2

_{= $}

OPE

₍

M2

,

s0

),

(8)

where

$

OPE

(

M2

_,

_s0

₎

_{is nowthecontinuum subtracted correlation}

function. In Eq. (8) s0 is the continuum threshold parameter: It

separatesfrom each other contribution of the ground-state term andeffectsduetoexcited statesandcontinuum. Thesecondsum ruleisderivedbyapplyingoperatord

/

d

(

−

1

/

M2

)

toEq. (8)

f_a2₀m_a4₀e−ma02/M2

_{= "}

_$

OPE

₍

_M2

_,

_s0

_),

₍₉₎

where

"

$

OPE

(

M2

,

s0

)

=

d

/

d

(

−

1

/

M2

)$

OPE

(

M2

,

s0

)

. These two sum

rulescanbeemployedtoevaluatetheparametersma0 and fa0:

m2_a0

₌

$

"

OPE

(

M2

,

s0

)

$

OPE

(

M2

,

s0

)

,

(10) and f_a2₀

=

e m2 a0/M2 m2 a0

$

OPE

(

M2

,

s0

).

(11)

ThesumrulesfortheparametersofthemesonK∗₀

(

800

)

canbe

foundbythesamemanner.Differencesinthiscaseareconnected withtheinterpolatingcurrentof K₀∗

(

800

)

definedby the expres-sion JK∗

(

x

)

=

ϵ

"

ϵ

$

uT_aC

γ

5db

% $

ud

γ

5C seT

%

,

(12)

andwiththematrixelement

⟨

0

|

JK∗

_|

K₀∗

(

p

)

⟩ =

fK∗mK∗

,

(13)

where mK∗ and fK∗ are the mass and coupling of the state

K₀∗

(

800

)

.Thephenomenologicalsideofthesumruleafterevident replacements is givenby Eq. (5), whereas the

$

OPE_K∗

(

p

)

takes the

followingform

$

OPE_K∗

(

p

)

=

i

!

d4xeipx

_ϵ

_"

_ϵϵ

′

_"

_ϵ

′_Tr

#

_γ

₅

_"

_Se′e s

(

−

x

)

×

γ

5Sdu′d

(

−

x

)

&

Tr

#

γ

5

"

Saau′

(

x

)

γ

5S_dbb′

(

x

)

&

.

(14)

Theremainingoperationsarestandardanddonotdifferfromones describedaboveinthecaseofthea0

(

980

)

meson.

Thenumericalcomputationsrequiretospecifyvaluesofvarious parametersthatentertothequarkpropagators,and,asaresult,to thesumrulesforthemassandcoupling.Amongthemthevacuum expectationvaluesof thequark, gluon andmixedlocaloperators areimportantones:

⟨¯

qq

⟩ = −(

0

.

24

±

0

.

01

)

3GeV3

,

⟨¯

ss

⟩ =

0

.

8

⟨¯

qq

⟩,

m2₀

_{= (}

0

.

8

±

0

.

1

)

GeV2

,

⟨

qgs

σ

Gq

⟩ =

m20

⟨

qq

⟩,

⟨

sgs

σ

Gs

⟩ =

m20

⟨¯

ss

⟩,

⟨

α

sG2

π

⟩ = (

0

.

012

±

0

.

004

)

GeV 4

_.

₍₁₅₎

These condensates enter to the propagator of a light quark and havedifferent dimensions. The terms

⟨

qgs

σ

Gq

⟩

,

⟨

sgs

σ

Gs

⟩

shown in Eq. (15) as well as other ones

∼ ⟨

qq

_⟩

2_,

_{∼ ⟨}

_qq

_⟩⟨

_g2

sG2

⟩

are ob-tainedusingthe factorizationhypothesis ofthehigherdimension condensates.However, thefactorizationassumption isnotprecise anditsviolationisstrongerforhigherdimensioncondensates(see Ref. [35]).Fordimension tencondensateseventheorderof mag-nitude of such a violation is unclear. But here we employ this assumption by ignoring possible theoretical uncertainties gener-atedbyitsviolation.

In the present work we neglect the masses of the u and d

quarks,butsetms

̸=

0 anduseincalculationsms

=

128

±

10 MeV. Our expressions depend also on auxiliary parameters M2 and s0

the choiceofwhich hasto satisfystandard restrictions. Thus,we determine the upper limit M2

max of the working window M2

∈

[

M_min2

,

M2_max

]

by requiring fulfillment of the condition imposed onthepolecontribution

PC

=

$(

M2max

,

s0

)

$(

M2 max

,

∞)

>

0

.

10

.

(16)

The lower bound of the Borel parameter M2

min is fixed from

convergenceoftheoperatorproductexpansion(OPE).By quantify-ingthisconstraintwerequirethat acontributionofthelastterm inOPEshouldbearoundof5%,i.e.

$

Dim10

(

M2_min

,

s0

)

$(

M2_min

,

s0

)

≈

0

.

05

,

(17)

has tobe obeyed. Anotherrestriction to M2

min isconnected with

the perturbative contribution to sum rules. In the present work we apply the following criterion:at the lower bound of M2 _the

perturbative contributionhastoconstitute morethan 70% partof thefullresult.

Fig. 1. Themassofthemesona0(980)asafunctionoftheBorelparameterM2_atfixeds

0(leftpanel),andasafunctionofthecontinuumthresholds0atfixedM2(right

panel).

Fig. 2. The coupling fa0 of the a0state as a function of M2at fixed s0(left panel), and of s0at fixed M2(right panel).

Boundariesofs0 arefixedbyanalyzingthepolecontributionto

getitsgreatestaccessiblevalues.Minimaldependenceofextracted quantitiesonM2 _{whilevaryings}

0isanotherconstraintthathasto

beimposedwhenchoosingaregionforthisparameter.Performed analysesleadtothefollowingworkingwindowsforM2 _ands

0:

M2

∈ [

1

.

1

,

1

.

4

]

GeV2

,

s0

∈ [

1

.

7

,

1

.

9

]

GeV2

.

(18)

Intheseregionsallofconstraintsimposedonthecorrelation func-tionaresatisfied.Infact,atM2

max thepolecontributionPCequals

to0.115,whereasatM2_minitamountsto78%oftheresult.Inother words,Eq. (16) determinesonlythelowerlimitforthePC:inthe full interval for M2 _{the pole contribution is large which should}

lead to reliable sum rules’ predictions. At the minimal allowed value ofthe Borelparameter contribution ofDim10 term consti-tutesup to5.5%ofthewholeresult.Andperturbative component ofthecorrelationfunction

$(

M2_min

,

s0

)

formsitsnolessthan0

.

71

part.

InFigs. 1and

2

we depictthe sumrules resultsforthemass and coupling of the a0

(

980

)

state as functions of the Borel and

continuum threshold parameters. It is seen, that predictions for the mass and coupling are rather stable against varying of both

M2 _ands

0.Inthecaseofthemassthestability ofthe resulthas

standard explanation:In fact,the sumrule forthemassma0

de-pends on the ratio ofthe correlation function andits derivative (10), where uncertainties to a great extend cancelrendering the massverystableintheworkingregionsofM2 _ands

0.Thestability

ofthecouplingmaybeattributedtothefactthatinterpolating cur-rent J

(

x

)

contains onlylight diquarks (antidiquarks)

ϵ

abcqaTC

γ

5q′_b

incolortriplet, flavor antisymmetricandspin 0 state,andwhich leadstostablepredictions.

Forma0 and fa0 wefind:

ma0

=

991+₋2927 MeV

,

fa0

= (

1

.

94

±

0

.

04

)

·

10−3GeV4

.

(19)

Thesimilaranalysisofthesumrulesforthemassandcouplingof the K₀∗

(

800

)

mesonallowsustofindtheregionsfortheBoreland continuumthresholdparameters

M2

∈ [

0

.

8

,

1

.

0

]

GeV2

,

s0

∈ [

0

.

9

,

1

.

1

]

GeV2

,

(20)

whichleadtothefollowingpredictions:

mK∗

=

767+₋38₂₉ MeV

,

fK∗

= (

1

.

71

±

0

.

07

)

·

10−3GeV4

.

(21)

The sum rules predictions formK∗ and fK∗ are plotted inFig. 3

as functions of the Borel parameter M2_{. Their stability on M}2

including a region s0

<

1 GeV2 demonstrates correctness of the

performedcalculations.Ourresultforthemassofthea0

(

980

)

me-son is ina nice agreement withthe available experimental data

ma0

=

980

±

20 MeV [1]. The latest measurement of mK∗

per-formedbytheBESCollaboration[36] andextractedfromthedecay

J

/ψ

→

K_S0K_S0

π

+

π

−isequalto

mK∗

=

826

±

49+₋49₃₄ MeV

.

(22)

From the process J

/ψ

_→

K

_±

K0

S

π

∓

π

0 the same collaboration

obtained(see,Ref. [37])

mK∗

=

849

±

77+₋18₁₄ MeV

.

(23)

As is seen, the experimental data are not precise, and the cen-tral values formK∗ are higher than our prediction.Nevertheless,

within the experimental and theoretical errors they are compat-ible witheach other. The mass andcoupling ofthe a0

(

980

)

and

K∗

0

(

800

)

mesonscalculatedinthepresentsection willbeusedas

Fig. 3. The mK∗(left panel) and fK∗(right panel) vs M2at fixed values of the continuum threshold parameter s0.

3. Strongdecaychannelsofthea0

(

980

)

andK0∗

(

800

)

mesons In the light of the obtained results we can determine the kinematically allowed strong decay channels of the a0

(

980

)

and

K∗

0

(

800

)

mesons. In the present paper we restrict ourselves by

studying only S-wave decays of these mesons. It turns out that the dominant S-wave strong decays of a0

(

980

)

are processes

a0

(

980

)

→

ηπ

0 and a0

(

980

)

→

K+K−. For the meson K₀∗

(

800

)

thedecaysK∗

0

(

800

)

→

K+

π

−andK0∗

(

800

)

→

K0

π

0 aredominant

ones.

Thesedecaysproceedthroughrearrangementofthequarksand antiquarksfromthetetraquarkto formtwo conventionalmesons. Mechanismsofthesetransformationsarenotquiteclear,butthere are interesting models to explain these phenomena introducing, forinstance,a repulsive barrierbetween thediquark–antidiquark pair[38]. The light cone sum rule method operates with funda-mentalquark-gluondegreesoffreedomandusesfirstprinciplesof theQCD.Inthisapproachone invokesonlyanassumptiononthe quark-hadrondualitytomatchthephenomenologicaland theoret-icalexpressionsofthesamecorrelationfunctiontoderivethesum rulesforquantitiesofinterest.

Itisinstructivetoconsiderthe modea0

(

980

)

→

ηπ

0 ina de-tailedmanner. In orderto calculatethestrong coupling gaηπ we useQCD LCSR method andstart from analysisof the correlation function

$(

p

,

q

)

=

i

!

d4xeip·x

_⟨

π

0

(

q

)

|

T

{

Jη

₍

_x

₎

_J†

₍

₀

₎

_}|

₀

_⟩,

₍₂₄₎

where J

(

x

)

and J η

(

x

)

are the interpolating currents for the

a0

(

980

)

and

η

mesons,respectively. Theinterpolatingcurrent for

thea0

(

980

)

isgivenbyEq. (2).

The situation with the choice of J η

(

x

)

is more subtle and

deserves some explanations. The systemof pseudoscalar mesons

η

−

η

′ hasacomplicatedstructure.Intheworld oftheexact fla-vor SUf

(

3

)

symmetrythemesons

η

and

η

′canbeinterpreted as

theoctet

η

8 andsinglet

η

1 statesoftheflavorgroup,respectively.

Butinthe realworld, where thissymmetry isbrokenthe physi-cal particlesare mixtures ofthe

η

8 and

η

1 states.Ofcourse,the

mesons

η

and

η

′ _{are predominantlythe}

_η

_{8 and}

_η

_{1 states,}

never-theless themixing phenomenon cannot be ignored. Thismixing can be described using the octet-singlet basis. Alternatively, the samephenomenoncanbe treatedemploying thequark-flavor ba-sis(see,Ref. [39] fordetails)

η

q

=

√

1

2

$

uu

₊

dd

%

,

η

s

=

ss

.

(25)

Thequark-flavorbasisismoreconvenienttodescribethemixingin the

η

−

η

′systemandinvestigatedifferentexclusiveprocesses in-volvingthesemesons[40].Thereasonisthatinthisschemewith

ratherhigh accuracythe state andcouplingmixing aregoverned bythesameangle,whereasinthe

η

1

−

η

8 basisonehasto

intro-ducetwomixinganglesforthedecayconstants.

Inthequark-flavorbasistheinterpolatingcurrentofthe

η

me-soncanbeobtainedthroughmixingfromthebasiccurrents

J_q

(

x

)

=

√

1 2

#

u

(

x

)

i

γ

5u

(

x

)

+

d

(

x

)

i

γ

5d

(

x

)

&

,

J_s

(

x

)

=

s

(

x

)

i

γ

5s

(

x

),

(26) andreads Jη

₍

_x

₎

₌

_J q

(

x

)

cos

ϕ

−

Js

(

x

)

sin

ϕ

,

where

ϕ

isthemixingangle.

Thephenomenologicalside ofthesumruleisobtainedby ex-pressing

$(

p

,

q

)

intermsofthestrongcouplinggaηπ andphysical parametersofthea0

(

980

)

and

η

mesons

$

Phys

(

p

,

q

)

=

⟨

0

|

J η

_|

_η

₍

_p

₎

_⟩

p2

₋

_m2 η

⟨

η

(

p

)

π

0

(

q

)

|

a0

(

p′

)

⟩

×

⟨

a0

(

p′

)

|

J†

|

0

⟩

p′2

₋

_m2 a0

+ . . . ,

(27)

wheremη isthemassof

η

thedotsbeingstoodforcontributions ofexcitedstates.Thematrixelement

⟨

a0

(

p′

)

|

J†

|

0

⟩

hasbeen

intro-ducedin the previous section, andthe vertex

⟨

η

(

p

)

π

0

(

q

)

|

a0

(

p′

)

⟩

canbewrittendowninthefollowingform

⟨

η

(

p

)

π

0

(

q

)

|

a0

(

p′

)

⟩ =

gaηπp

·

p′

,

(28) where gaηπ is the coupling corresponding to the strong vertex

a0

(

980

)

ηπ

0. The last element in Eq. (27)

⟨

0

|

J η

_|

η

(

p

)

⟩

is defined

bytheexpression

⟨

0

|

Jη

_|

_η

₍

_p

₎

_{⟩ = −}

1 2ms

$

hqηcos

ϕ

−

hsηsin

ϕ

%

(29)

anddiffersfromthesimilarmatrixelementsofconventional pseu-doscalarmesons:hererelevantcommentsareinorder.Itisknown thattheaxial-anomalymodifiesthematrixelementsofthe

η

and

η

′mesons.Indeed,forhs(q)

η wehave hs(q) η

=

m2ηfηs(q)

− ⟨

0

|

α

s

π

G A µν

"

GA,µν

|

η

(

p

)⟩,

(30) where

⟨

0

|

αs

π GµνA

"

GA,µν

|

η

(

p

)

⟩

isthe matrixelement appeared due totheU

(

1

)

axial-anomaly.Thequantitieshsη(q)canbeexpressedin termsoftheparametershs,hq andmixingangle

ϕ

whichmodifiesEq. (29)

⟨

0

|

Jη

_|

_η

₍

_p

₎

_{⟩ = −}

Hη

2ms

,

(32)

where we introduce the short-hand notation Hη

_{= (}

hqcos2

ϕ

+

hssin2

ϕ

)

. In calculationswe employ the numerical values of hq andhs(inGeV3)

hq

=

0

.

0016

±

0

.

004

,

hs

=

0

.

087

±

0

.

006 (33) extracted from analysis of experimental data. The same phe-nomenologicalanalysespredict

ϕ

=

39

.

3◦

±

1

.

0◦.

Then thephysical side ofthesum rulecan be recastintothe form

$

Phys

(

p

)

= −

H η_f a0ma0 2ms m2

(

p2

−

m2

)

2

+ ...,

(34) wherem2

_{= (}

_m2 a0

+

m 2 η

)/

2.

In the last equality we take into account that p

₌

p′ _and

q

=

0, whichis requiredwhen considering a vertexcomposed of a tetraquark and two conventional mesons [31]. In the case of vertices containing only ordinary mesons calculation of the cor-respondingstrongcouplingcanbeperformedinthecontextofthe LCSRmethod’sfullversion:thelimitq

₌

0 isknownthereasthe softapproximation.Fortetraquark–meson–mesonverticesthefull LCSRmethodreducestoitssoftapproximation,whichisonlyway to compute the strong couplings. Therefore, we use here techni-caltoolselaborated inthesoftapproximationby bearinginmind that inourcase thisisonly available approachto evaluate gaηπ . In the limit q

=

0 the correlation function

$

Phys

(

p

)

dependson

avariable p2_{,asa resultwe haveto fulfiltheone-variable Borel}

transformationwhichyields

B

$

Phys

(

p

)

= −

H η_f a0ma0m2 2ms e−m2/M2 M2

+ . . . .

(35)

WeproceedbycomputingtheQCDsideofthesumrule.Itis eas-ilyseenthat Jq

(

x

)

doesnotcontributetothecorrelationfunction

$(

p

,

q

)

. Indeed, by substituting the current Jq

(

x

)

into Eq. (24)

and performing contractions of the uu and dd fields from Jq

(

x

)

with relevant parts of J

(

x

)

we get apart from light u

,

d-quark

propagators matrix elements of the local operators s

#

is (here,

#

j

=

1

,

γ

5

,

γλ

,

i

γ

5

γλ

,

σλ

ρ

/

√

2 is the fullset ofDiracmatrices) sandwichedbetweenthe

π

0 _andvacuum

⟨

π

0

|

s

(

0

)#

is

(

0

)

|

0

⟩,

which are identically equal to zero. In other words, only

−

sin

ϕ

Js

(

x

)

component of the

η

meson’s current contributes to

thecorrelationfunction

$(

p

,

q

)

. Aftersomemanipulationsweget

$

OPE

(

p

)

=

sin

ϕ

!

d4xeip·x

_√

ϵ

"

ϵ

2

'#

γ

5

"

Sibs

(

x

)

γ

5

×"

Sei_s

(−

x

)

γ

5

&

αβ

$

⟨

π

0

|

ua_αud_β

|

0

⟩ − ⟨

π

0

|

da_αdd_β

|

0

⟩

%+

,

(36)

where

α

and

β

arespinorindices.

Calculations of the correlation function in accordance with recipesdescribedinaratherdetailedforminRef. [31] revealthat thematrixelementsofthepionwhichcontributesto

$

OPE

(

p

)

are

⟨

0

|

ui

γ

5u

|

π

0

⟩

and

⟨

0

|

di

γ

5d

|

π

0

⟩

given,forexample,intheform

√

2

⟨

0

|

ui

γ

5u

|

π

0

⟩ =

fπ

µ

π

,

µ

π

= −

2

⟨

qq

⟩

f2 π

.

(37)

In Eq. (37) fπ and

⟨

qq

_⟩

are the pion decay constant and the quark vacuum condensate, respectively.Then the Boreltransform of

B$

OPE

(

p

)

= $

OPE

(

M2

)

which is necessary to derive the sum

rulereads

$

OPE

(

M2

)

= −

fπ

µ

π 16

π

2 sin

ϕ

∞

!

4m2 s dsse−s/M2

−

sin

ϕ

,

_f π

µ

π 16

⟨

α

sG2

π

⟩ −

fπ

µ

πms 6

⟨

ss

⟩

-.

(38)

Equating the Borel transforms

B$

Phys

(

p

)

and

$

OPE

(

M2

)

we get

theunsubtractedsumrule.Butthesumruleapplicabletoevaluate

gaηπ can be obtainedonly after subtracting the contributions of excited statesandcontinuum. Inthesoftapproximation an addi-tionalprobleminthisprocedureisconnectedwithcontributionsto

B$

Phys

(

p

)

ofexcitedstates,someofwhichevenafterBorel trans-formation remain unsuppressed [30], and should be removedby applyingtheoperator

P(

M2

_,

_m2

₎

_{(see,Ref. [41])}

P

(

M2

,

m2

)

=

.

1

−

M2 d dM2

/

M2em2/M2

.

(39)

Asaresultwederiveourfinalsumruleforthestrongcoupling

gaηπ

= −

_Hηf2ms a0ma0m2

P

(

M2

,

m2

)$

OPE

(

M2

,

s0

),

(40)

where

$

OPE

(

M2

,

s0

)

isgivenbyEq. (38) wheretheupperlimitof

theintegral

∞

isreplacedbys0.

The decay process a0

(

980

)

→

K+K− is investigated by the same manner. The differences here are connected with the cor-relationfunction

$

K

(

p

,

q

)

=

i

!

d4xeip·x

⟨

K+

(

q

)

|

T

{

JK−

(

x

)

J†

(

0

)

}|

0

⟩,

(41)

withtheinterpolatingcurrent JK−

(

x

)

JK−

₍

_x

₎

₌

_ui

₍

_x

₎

_i

_γ

5si

(

x

),

(42)

andalsothematrixelementofthe K mesons

⟨

0

|

ui

γ

5s

|

K−

(

p

)

⟩ =

fKm 2

K

ms

.

(43)

In Eq. (43) mK and fK arethe K± mesons’massanddecay con-stant, respectively. After relevant replacements the phenomeno-logical side of sum rule is obtained from Eq. (27), whereas for

$

OPE_K

(

p

,

q

)

weget

$

OPE_K

(

p

,

q

)

=

i2

!

d4xeip·x

_√

ϵ

"

ϵ

2

#

γ

5

"

Sibs

(

x

)

γ

5

"

Sdiu

(

−

x

)

γ

5

&

αβ

×⟨

K+

(

q

)

|

ua_α

(

0

)

se

(

0

)

|

0

⟩.

(44)

Thefollowingoperationsarestandardmanipulations,thereforewe writedownonlythefinalsumruleforthestrongcouplingga0K K

gaK K

=

ms ma0fa0m2KfK

"

m2

P

(

M2

,

m

"

2

)$

OPE_K

(

M2

,

s0

),

(45) wherem

"

2

_{= (}

_m2 a0

+

m 2 K

)/

2 and

$

OPE_K

(

M2

,

s0

)

= −

fKm2_K 16

√

2

π

2ms s0

!

4m2 s dsse−s/M2

+

fKm2K 16

√

2ms

⟨

α

sG2

π

⟩ −

fKm2_K 12

√

2

0

2

⟨

uu

_{⟩ − ⟨}

ss

_⟩

1

.

(46)

Fig. 4. The strong coupling ga0ηπ as a function of the Borel parameter M2(left panel), and of the continuum threshold s0(right panel).

Forthestrongcouplings gK∗_{Kπ and} g_K∗K0_π0 weobtain:

g_K∗_Kπ

=

ms mK∗f_K∗m_K2f_Km2₁

P

(

M 2

_,

_m2 1

)$

OPE1

(

M2

,

s0

),

(47) and g_K∗K0_π0

=

ms mK∗fK∗m2_K0fK0m2₂

P

(

M2

,

m2₂

)$

₂OPE

(

M2

,

s0

),

(48) wherem2 1

= (

m2K∗

+

m2K

)/

2 andm22

= (

m2K∗

+

m2_K0

)/

2,respectively.

The correlation functions in Eqs. (47) and (48) are given by the expressions

$

OPE₁

(

M2

,

s0

)

= −

fπ

µ

π 16

π

2 s0

!

m2 s dsse−s/M2

−

fπ₁₆

µ

π

⟨

α

sG2

π

⟩ +

fπ

µ

πms 12

0

2

⟨

uu

⟩ − ⟨

ss

⟩

1

,

(49)

and

$

OPE₂

(

M2

_,

_s0

₎

_{= $}

OPE

1

(

M2

,

s0

)/

√

2.

Sumrulesobtainedforthestrongcouplingsgaηπ ,gaK K,gK∗Kπ andg_K∗K0_π0 willbeusedtodeterminethepartialdecaywidthsof

themesonsa0

(

980

)

andK∗₀

(

800

)

.

4. Numericalanalysis

In numerical computations of the strong couplings for the quarkandgluon condensatesweutilizetheir valuespresentedin Eq. (15).Apartfromtheseparameterswealsoemploythemasses anddecayconstants ofthe

π

andK mesons:forthepionmπ±

=

139

.

57061

±

0

.

00024 MeV, m_π0

=

134

.

9770

±

0

.

0005 MeV and

fπ

=

131MeV andforthe K mesonmK±

=

493

.

677

±

0

.

016MeV,

m_K0

=

497

.

611

±

0

.

013MeV and fK

=

155

.

72MeV

.

WehaveemployedthedifferentworkingregionsfortheBorel parameter M2 _{andcontinuum thresholds}

0 whenconsidering

de-cays of the a0

(

980

)

and K₀∗

(

800

)

mesons: these windows have

beenchosen in accordancewith standard constraintsof thesum rulecomputationsexplainedinthesection 2.Forthestrong cou-plingsgaηπ andgaK K theBorelandcontinuumthreshold parame-tersarevariedwithinthelimits

M2

∈ [

1

.

1

−

1

.

4

]

GeV2

,

s0

∈ [

1

.

9

−

2

.

1

]

GeV2

.

(50)

Thecorrespondingsumrulesleadtothefollowingpredictions(in unitsofGeV−1)

gaηπ

=

5

.

36

±

1

.

41

,

gaK K

=

9

.

10

±

2

.

76

.

(51) Itisknownthat astability ofthe obtainedresults on M2 _and

s0 isoneoftheimportantconstraintsimposedonsumrule

com-putations.Asanexample,inFig.4weplotthecouplinggaηπ asa

functionofM2 _ands

0.Itisevidentthat gaηπ dependsonM2 and

s0, which generates essential part of uncertainties in the

evalu-atedquantities.Itisalsoseenthattheseambiguitiesdonotexceed

∼

30%ofthecentral valueswhichisacceptableforthe sumrules computations.

For the partial decay width of the processes a0

(

980

)

→

ηπ

0

anda0

(

980

)

→

K+K−weget

#

#

a0

(

980

)

→

ηπ

0

&

=

50

.

57

±

13

.

87 MeV

,

#

)

a0

(

980

)

→

K+K−

*

=

11

.

44

±

3

.

76 MeV

.

(52)

The total width of the meson a0

(

980

)

is formed mainly due to

thedecaychannelsa0

(

980

)

→

ηπ

0 anda0

(

980

)

→

K+K−:we

as-sumethat P-wavedecaysdonotmodifyitconsiderably.Therefore it seems reasonable tocompare

#

th.

=

62

.

01

±

14

.

37 MeV which

isthesumoftwopartialdecaywidthswiththeavailable informa-tionon

#

exp.

=

50–100MeV notingafulloverlapoftheseresults.

Aswe havenotedabove,experimental dataforthetotalwidthof thelight scalarmesons sufferfromlargeuncertainties. Therefore, wecanstatethatourtheoreticalpredictiondoesnotcontradictto thepresent-dayexperimentaldata.

ThestrongdecaysofthemesonK₀∗

(

800

)

canbeanalyzedinthe sameway.InthecaseoftheK∗

0

(

800

)

meson’sdecaysweuse

M2

∈ [

0

.

8

−

1

.

0

]

GeV2

,

s0

∈ [

1

.

2

−

1

.

5

]

GeV2

,

(53)

andfindforthestrongcouplings(inGeV−1)

gK∗Kπ

=

19

.

46

±

5

.

64

,

gK∗_K0_π0

=

13

.

47

±

3

.

91

.

(54)

Thepartialdecaywidthsareequalto

#

)

K₀∗

(

800

)

→

K+

π

−

*

=

270

.

39

±

78

.

42 MeV

,

#

#

K₀∗

(

800

)

→

K0

π

0

&

=

130

.

69

±

37

.

91 MeV

.

(55)

Thenusingthesetwodecaymodesforthetotalwidthof K₀∗

(

800

)

weget

#

th.

=

401

.

1

±

87

.

1 MeV.Experimentaldataborrowedfrom

Refs. [36,37] predicts

#

exp.

=

449

±

156+₋14481 MeV and

#

exp.

=

512

±

80+₋92₄₄ MeV, respectively,whichhaveratherimprecise na-ture.Itisseenthatourresultiscompatiblewiththesedata.

5. Discussionandconcludingnotes

Investigation of the scalar mesons a0

(

980

)

and K₀∗

(

800

)

by

modelingthemasdiquark–antidiquarkscarriedout inthepresent workhasallowedustoexplorethesuggestionaboutexoticnature oftheseresonances.Usingthewell-knownQCDsumrulemethod wehavecalculatedtheirmassesandtotalwidths.Tothisend,we haveemployedtheinterpolatingcurrents J

(

x

)

and JK∗

(

x

)

defined byEqs. (2) and(12),respectively.

Ourinvestigationhasdemonstratedthatsinglecurrentscanbe successfullyappliedtointerpolatethelight scalarmesons.Inthis point we do not agree with Ref. [20], in which the authors ex-cluded single interpolating currents as ones that do not lead to reliable predictions. An accuracy of theoretical calculations per-formedin ourwork exceedsan accuracy ofsimilar computations in Ref. [20]. Thus, in our study we have takeninto account not onlytermsuptodimensionteninsteadofeight,butalsousedin calculationsmore precise expression for the quark propagator.It ispossiblethatconclusionmadeRef. [20] isconnectedwiththese circumstances.

Ourresultforthemassofthea0

(

980

)

agreeswithexperimental data.Its totalwidthevaluated usingtwo S-wave dominantstrong decay channels is also in accord with the data, because our re-sultliesentirelyinthe experimentalregion

#

exp.

=

50–100 MeV.

Thesituationwithexperimentalinformationontheparametersof the K∗

0

(

800

)

meson is worse than in the case of a0

(

980

)

. Thus,

availabledataonboththemassandtotalwidthofthisscalar me-son in ratherimprecise andsuffers fromlarge uncertainties. The predictionsobtainedinthepresentworkdonotcontradicttolast experimentalmeasurements,nevertheless reliableconclusionscan bemadeonlyonbasisofamorepreciseexperimentalinformation.

Acknowledgements

K.A. andH.S. thank TUBITAK forthe partial financial support providedunderGrantNo.115F183.

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