Scalar meson contributions to a μ from hadronic light-by-light scattering

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Scalar meson contributions to a µ from hadronic

light-by-light scattering

M. Knecht, S. Narison, A. Rabemananjara, D. Rabetiarivony

To cite this version:

M. Knecht, S. Narison, A. Rabemananjara, D. Rabetiarivony.

Scalar meson contributions to

a µ from hadronic light-by-light scattering.

Physics Letters B, Elsevier, 2018, 787, pp.111-123.

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Scalar

meson

contributions

to

a

_μ

from

hadronic

light-by-light

scattering

M. Knecht

a

,

S. Narison

b,

∗

,1

,

A. Rabemananjara

c

,

D. Rabetiarivony

c,2

a_{CentredePhysiqueThéoriqueUMR7332,CNRS/Aix-MarseilleUniv./Univ.duSudToulon-Var,CNRSLuminyCase907,13288MarseilleCedex 9,France} b_{LaboratoireParticulesetUniversdeMontpellier,CNRS-IN2P3,Case070,PlaceEugèneBataillon,34095Montpellier,France}

c_{InstituteofHigh-EnergyPhysics(iHEPMAD),UniversityofAntananarivo,Madagascar}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received14August2018

Receivedinrevisedform11October2018 Accepted23October2018

Availableonline26October2018 Editor:A.Ringwald

Keywords:

Anomalousmagneticmoment Muon

Scalarmesons Radiativewidth Nonperturbativeeffects

Usinganeffective

σ

/f0(500)resonance,whichdescribesthe

π π

→

π π

and

γ γ

→

π π

scatteringdata,

weevaluateitscontributionandtheonesoftheotherscalarmesonstothehadroniclight-by-light(HLbL)

scatteringcomponentoftheanomalousmagneticmomentaμofthemuon.Weobtaintheconservative

rangeofvalues:Salblμ|S −(4.51±4.12)×10−11,whichisdominatedbythe

σ

/f0(500)contribution

(50%∼98%),andwhere thelargeerrorisduetotheuncertaintiesontheparametrisationofthe form

factors.Consideringournewresult,weupdatethesumofthedifferenttheoreticalcontributionstoaμ

withinthestandardmodel,whichwethencomparetoexperiment.Thiscomparisongives(aexpμ −aSMμ)=

+(312.1±64.6)×10−11_{,wherethe theoreticalerrorsfrom HLbLare dominatedbythe scalarmeson}

contributions.

©2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The anomalous magnetic moments a (



≡

e

,

μ

) of the light

charged leptons, electron and muon, are among the most accu-rately measured observablesin particlephysics. The relative pre-cision achievedby the latest experiments to date is of0.28 ppb inthecaseoftheelectron [1,2], and0.54 ppm inthecaseofthe muon[3].AnongoingexperimentatFermilab[4–6],andaplanned experimentatJ-PARC[7],aimatreducingtheexperimental uncer-taintyonaμ to thelevelof0.14 ppm, andthere isalsoroom for futureimprovementson theprecision ofae.The confrontationof

theseveryaccuratemeasurementswithequallyprecisepredictions fromthestandardmodelthenprovidesastringenttest ofthe lat-ter,and,astheexperimentalprecisionisfurtherincreasing,opens up the possibility ofindirectly revealingphysics degreesof free-domthatevengobeyondit.

Fromthislast pointofview,thepresentsituationremains un-conclusiveinthecaseofthemuon(inthecaseoftheelectron,the

*

Correspondingauthor.

E-mailaddresses:marc.knecht@cpt.univ-mrs.fr(M. Knecht),snarison@yahoo.fr

(S. Narison),achrisrab@gmail.com(A. Rabemananjara),rd.bidds@gmail.com

(D. Rabetiarivony).

1 _{MadagascarconsultantoftheAbdusSalamInternationalCentreforTheoretical}

Physics(ICTP),viaBeirut6,34014Trieste,Italy.

2 _PhDstudent.

measured value of ae agreed with the predicted value obtained

from themeasurement of the fine-structure constant of Ref. [8]; however,themorerecentdeterminationof

α

[9] nowresultsina tensionatthelevelof2

.

5 standarddeviationsbetweentheoryand experiment). Indeed,the lateststandard model evaluationsof aμ

(Ref. [10] providesarecentoverview,aswell asreferencesto the literature;see alsoSection 10 atthe endofthis article)reveal a discrepancybetweentheoryandexperiment,whichhoweverisat thelevelof

∼

3

.

5 standarddeviationsonly.Itistherefores manda-tory, as the experimental precision increases, to also reduce the theoreticaluncertaintiesintheevaluationofaμ.

Presently,thelimitationinthetheoreticalprecisionofaμ isdue tothecontributionsfromthestronginteractions, whichare dom-inated by the low-energy, non perturbative, regime of quantum chromodynamics(QCD).Thepresentworkisdevotedtoahadronic contribution arising at order

O(

α

3

₎

_{, and currently refered to as}

hadroniclight-by-light (HLbL), seeFig. 1.More precisely, we will be concerned witha particular contribution to HLbL, dueto the exchange of the 0++ scalar states

σ

/

f0

(

500

)

, a0

(

980

)

, f0

(

980

)

,

f0

(

1370

)

, and f0

(

1500

)

. In earlier evaluations of the HLbL part

of aμ, some of these states were either treated in the frame-workoftheextended Nambu–Jona-Lasiniomodel[11,12],orthey weresimplyomittedaltogether[13,14].Morerecently,inRef. [15] the contributions from the

σ

/

f0

(

500

)

and a0

(

980

)

scalars have

been reconsidered in the framework of the linearized Nambu– Jona-Lasiniomodel.InRef. [16],thecontributionfromthea0

(

980

)

,

https://doi.org/10.1016/j.physletb.2018.10.048

Fig. 1. Light-by-lightHadronscatteringcontributiontoal.Thewavylinesrepresent

photon.Thecrosscorresponds totheinsertionoftheelectromagneticcurrent.The shadedboxrepresentshadronssubgraphs.

Fig. 2. Scalarmesonexchange(dottedlines)toLight-by-lightscatteringcontribution toaμ.Thewavylinesrepresentphoton.Theshadedblobrepresentsformfactors. ThefirstandseconddiagramscontributetothefunctionT1,andthethirdtothe

functionT2definedinEq. (3.11).

f0

(

980

)

, f0

(

1370

)

stateswereevaluatedassingle-mesonexchange

termswithphenomenologicalformfactors,seeFig.2.Finally,the contribution fromthe lightest scalar, the

σ

/

f0

(

500

)

is contained

inthedispersiveevaluationofthecontributiontoHLbLfrom two-pionintermediatestateswith

π π

rescatteringofRefs. [17,18].

Theapproach consideredhereforthetreatment ofthe contri-bution from scalar states to HLbL has, to some extent, overlaps with both of the last two of these more recent approaches. It restsonasetofcoupled-channeldispersionrelationsforthe pro-cesses

γ γ

→

π π

,

KK ,

¯

where thestrong S-matrixamplitudes for

π π

→

π π

,

KK are

¯

represented by an analytic K-matrix model, firstintroduced inRef. [19],andgraduallyimprovedover timein Refs. [20–22],asmoreprecise dataon

π π

scatteringandonthe reactions

π π

→

γ γ

became available. The details of the model willnot bediscussed here,asthey areamply documentedinthe quoted references. The interest for our present purposes of the analysis of the data within this K-matrix framework is twofold. First, it contributes to our knowledge of the two-photon widths of some of the scalar states, which we will need asinput. Sec-ond, through the fit to data of the K-matrix description of

π π

scattering, it provides information on the mass and the total hadronic width of the

σ

/

f0

(

500

)

resonance, which will also be

needed.

Therestofthisarticleisorganizedasfollows.Section2briefly recalls the basic formalism describing the hadronic light-by-light contribution to the anomalous magnetic moment of a charged lepton. This is then specialized to the contribution due to the exchange of a narrow-width scalar state (Section 3). Some rele-vant properties of the vertex function involved are discussed in Section 4, where a vector-meson-dominance (VMD) representa-tion satisfying its leading short-distance behaviour is also given. Three sections are devoted to a review of the properties (mass andwidth)ofthe f0

/

σ

scalar,comingeitherfromsumrules

(Sec-tion 6) or fromphenomenology (Section7). In Section 7we fur-thermoredescribehowourformalismalsoallowstohandlebroad resonances like

σ

/

f0

(

500

)

or f0

(

1370

)

. The values of the mass

andofthewidthofthe

σ

/

f0

(

500

)

retainedforthepresentstudy

aregiveninthelast ofthesethree sections(Section 8). The two-photon widths of the remaining scalar mesons are discussed in Section 9.Ourresultsconcerning the contributionsof thescalars to HLbL are presented and discussed in Section 10. Finally, we summarizethepresentexperimentalandtheoreticalsituation con-cerningthestandard-modelevaluationoftheanomalousmagnetic

moment ofthe muon (Section11) andend thisarticleby giving ourconclusions(Section12).

2. Hadroniclight-by-lightcontributiontoal

The hadronic light-by-lightcontribution to the muon anoma-lousmagneticmoment,illustratedinFig.1,isequalto[24]:

albl_μ

≡

F2

(

k

=

0

)

=

1

48mtr



(

p

/

+

m

)[

γ

ρ

,

γ

σ

](

p

/

+

m

)

ρσ

(

p

,

p

)



(2.1)

wherek isthemomentumoftheexternalphoton,whilem and p

denotethemuonmassandmomentum.Furthermore[p

=

p

+

k]



ρσ

(

p

,

p

)

≡ −

ie6



_d4_q 1

(

2

π

)

4



_d4_q 2

(

2

π

)

4 1 q2 1q22

(

q1

+

q2

−

k

)

2

×

1

(

p

−

q1

)

2

−

m2 1

(

p

−

q1

−

q2

)

2

−

m2

×

γ

μ

(

/

p

−

q

/

1

+

m

)

γ

ν

(

p

/



−

/

q1

−

/

q2

+

m

)

γ

λ

×

∂

∂

kρ



μνλσ

(

q1

,

q2

,

k

−

q1

−

q2

) ,

(2.2)

withq1

,

q2

,

q3 themomentaorthevirtualphotonsand



μνλρ

(

q1

,

q2

,

q3

)

=



d4x1



d4x2



d4x3ei(q1·x1+q2·x2+q3·x3)

× 

0

|

T

{

jμ

(

x1

)

jν

(

x2

)

jλ

(

x3

)

jρ

(

0

)} |

0

(2.3)

the fourth-rank light quark vacuum polarization tensor, jμ the

electromagneticcurrentand

|

0

theQCDvacuum.

In practice,the computationofalbl_{μ involves}the limitk

≡

p

−

p

→

0 ofanexpressionofthetype:

F

(

p

,

p

)

= −

ie6



d4q1

(

2

π

)

4



d4q2

(

2

π

)

4

J

μνρσ τ

₍

_p

_,

_p

_;

_q 1

,

q2

)

×

F

μνρσ τ

(

−

q1

,

q2

+

q1

+

k

,

−

q2

,

−

k

),

(2.4) where

J

μνρσ τ

₍

_p

_,

_p

_;

_q 1

,

q2

)

=

1

(

p

+

q1

)

2

−

m2 1

(

p

−

q2

)

2

−

m2 1 q2 1q22

(

q1

+

q2

+

k

)

2

×

1 48mtr

[(

/

p

+

m

)

[

γ

σ

_,

_γ

τ

_](

_/

_p

₊

_m

₎

_γ

μ

₍

_/

_p

₊

_q

_/

1

+

m

)

×

γ

ν

(

p

/

−

q

/

2

+

m

)

γ

ρ

].

(2.5)

Thistensorhasthesymmetryproperty

J

μνρσ τ

(

p

,

p

;

q1

,

q2

)

=

J

ρνμτ σ

₍

_p

_,

_p

_;

₋

_q2

_,

₋

_q1

₎

_{, while, due to Lorentz invariance,}

F(

p

,

p

)

dependsonthemomenta p and p throughtheir invari-ants only. For on-shell leptons, p2

₌

_p2

₌

_m2_{, this amounts to}

F(

p

,

p

)

≡

F(

k2

)

=

F(

p

,

p

)

. 3. Scalarmesoncontributionstoalbl_μ

Let us focus on the contribution to albl

 due to the exchange

of a 0++ scalar meson S. We first discuss the situation where the width of this scalar meson is small enough so that its ef-fects can be neglected. As a look to Table 1 shows, thiswill be thecaseforS

=

a0

(

980

),

f0

(

980

),

f0

(

1500

)

.Thecircumstances

un-derwhichthequitebroad

σ

/

f0

(

500

)

resonance,andpossiblyalso

the f0

(

1370

)

state,canbetreatedinasimilarmannerwillbe

Table 1

Thescalarstatesweconsidertogetherwiththeestimatesoraveragesforthemass andwidth,asgivenbythe2018EditionoftheReviewofParticlePhysics[25].In thecasesoftheσ/f0(500)and f0(1370)states,therangesrepresenttheestimates

oftheBreit–Wignermassesandwidths.

Scalar Mass [MeV] Width [MeV]

σ/f0(500) 400–550 400–700 a0(980) 980(20) 50–100 f0(990) 990(20) 10–100 f0(1370) 1200–1500 200–500 a0(1450) 1474(19) 265(13) f0(1500) 1504(6) 109(7) The contribution



(μνρσS)

(

q1

,

q2

,

q3

)

due to the exchange of

a scalar one-particle state

|

S

(

pS

)

to the fourth-order

vacuum-polarization tensor



μνρσ

(

q1

,

q2

,

q3

)

(see Fig. 1) is described by

theFeynmandiagramsshowninFig.2.Itinvolvestheformfactors describingthephoton–photon-scalarvertexfunction



S_μν

(

q1

;

q2

)

≡

i



d4x e−iq1·x



₀

|

_T

{

_j μ

(

x

)

jν

(

0

)}|

S

(

pS

)

=

P

(

q2₁

,

q2₂

)

Pμν

(

q1

,

q2

)

+

Q

(

q21

,

q22

)

Qμν

(

q1

,

q2

),

(3.1) where q2

≡

pS

−

q1. This decomposition of



μνS

(

q1

;

q2

)

follows

fromLorentz invariance,invariance underparity, andthe conser-vationofthecurrent jμ

(

x

)

.Thetensors

Pμν

(

q1

,

q2

)

=

q1,νq2,μ

−

η

μν

(

q1

·

q2

),

Qμν

(

q1

,

q2

)

=

q22q1,μq1,ν

+

q21q2,μq2,ν

− (

q1

·

q2

)

q1,μq2,ν

−

q21q22

η

μν

,

(3.2)

aretransverse,

qμ_1,,₂νPμν

(

q1

,

q2

)

=

0

,

qμ1,,2νQμν

(

q1

,

q2

)

=

0

,

(3.3)

andsymmetricunderthesimultaneousexchangesofthemomenta

q1 and q2 and of the Lorentz indices

μ

and

ν

. The two

off-shell scalar-photon–photon transition form factors

P(

q1

,

q2

)

and

Q(

q1

,

q2

)

depend onlyonthe twoindependent invariantsq21 and

q2₂,and,aresymmetricunderpermutationofthemomentaq1 and

q2. Itis importantto point out that theamplitude for thedecay

S

→

γ γ

, which is proportional to

P(

0

,

0

)

M2

S

(



1

·



2

)

[



i denote

the respective photon polarization vectors, which are transverse,

qi

·



j

=

0],providesinformationon

P(

0

,

0

)

only.

Inorderto simplifysubsequentformulas,we willusethe fol-lowingshort-handnotation:

Pμν

(

qi

,

qj

)

≡

Pμν(i,j)

,

Qμν

(

qi

,

qj

)

≡

Qμν(i,j)

,

(3.4) and

P

(

q2_i

,

q2_j

)

≡

P

(i,j)

;

P

[

q2i

, (

qj

+

qk

)

2

] ≡

P

(i,jk)

,

Q

(

q2_i

,

q2_j

)

≡

Q

(i,j)

;

Q

[

q2i

, (

qj

+

qk

)

2

] ≡

Q

(i,jk)

,

P

(

q2_i

,

0

)

≡

P

(i,0)

;

Q

(

q2i

,

0

)

≡

Q

(i,0)

.

(3.5) The contribution albl

μ

|

S to alblμ from the exchange of the scalar S

isthenobtainedupon replacing,inthegeneralformula(2.3),the tensor



μνρσ

(

q1

,

q2

,

q3

,

q4

)

by i



(μνρσS)

(

q1

,

q2

,

q3

,

q4

)

=

D

(1,2) S



P

(1,2)Pμν(1,2)

+

Q

(1,2)Qμν(1,2)



×



P

(3,4)Pρσ(3,4)

+

Q

(3,4)Qρσ(3,4)



+

D

(1,3) S



P

(1,3)Pμρ(1,3)

+

Q

(1,3)Qμρ(1,3)



×



P

(2,4)Pνσ(2,4)

+

Q

(2,4)Qνσ(2,4)



+

D

(1,4) S



P

(1,4)Pμσ(1,4)

+

Q

(1,4)Qμσ(1,4)



×



P

(2,3)Pνρ(2,3)

+

Q

(2,3)Qνρ(2,3)



≡

i





μνρσ(S;P P)

+ 

μνρσ(S;P Q)

+ 

(μνρσS;Q Q)

,

(3.6)

whereqμ₄

≡ −(

q1

+

q2

+

q3

)

μ.Thescalar-mesonpropagatorinthe

NarrowWidthApproximation(NWA)reads

D

(i) S

≡

1 q2_i

−

M2_S

;

D

(i,j) S

≡

1

(

qi

+

qj

)

2

−

M2S

,

(3.7)

withi

,

j

=

1

,

..

4.Inthelast line, thefirst(third) termcollectsall the contributions quadratic in the form factor

P

(

Q

), while the second term collects all the contributions involving the products

PQ

ofthetwokindsofformfactors.Correspondingly,weperform thedecompositionalbl_μ

|

S

=

alblμ

|

SP P

+

alblμ

|

P QS

+

alblμ

|

Q QS .

Starting from the representation (3.6), it is a straightforward exercise to insert it into the generalexpression inEq. (2.3), and thentocomputetheprojectiononthePauliformfactorasdefined in Eq. (2.1). Forfurther use, we introduce the tensor

F

μαβ

(

q

)

=

η

μβqα

−

η

μαqβ,andtheamplitude

AP P_S

(

q1

,

q2

,

q3

,

q4

)

≡

D

S(1,2)

P

(1,2)

P

(3,4)

,

(3.8)

andsimilarlyforotherproductsofformfactors P Q

,

Q Q .

Thepartofthescalar-exchangetermthatinvolvestheform fac-tor

P

alonethenreads

albl_μ

|

P P_S

= −

e6



_d4_q 1

(

2

π

)

4



_d4_q 2

(

2

π

)

4

J

μνρσ τ

₍

_p

_,

_p

_;

_q 1

,

q2

)

×

2 AP P_S

(

−

q1

,

q1

+

q2

,

−

q2

,

0

)

F

μνα

(

q1

)(

q1

+

q2

)

α

×

F

ρσ τ

(

q2

)

+

AP PS

(−

q1

,−

q2

,

q1

+

q2

,

0

)

×

F

μρα

(

q1

)

q2α

F

νσ τ

(

q1

+

q2

)



,

(3.9)

wherethesymmetrypropertiesoftheintegrand,andofthe ampli-tude AS

(

q1

,

q2

,

q3

,

q4

)

,aswell as

F

ρσ τ

(

q

)

= −

F

ρτ σ

(

q

)

havebeen

used. Noticing that Q μν

(

q

,

k

)

is quadratic in the components of the momentum kμ, one sees that all of



μνρσ(S;Q Q)

(

q1

,

q2

,

q3

)

and

halfofthe termsin



(μνρσS;P Q)

(

q1

,

q2

,

q3

)

willnotcontribute to the

Pauliformfactoratvanishingmomentumtransfer.Thepartofthe scalar-exchangetermthatinvolvesbothformfactors

P

and

Q

thus reducesto albl_μ

|

P Q_S

= −

e6



d4q1

(

2

π

)

4



d4q2

(

2

π

)

4

J

μνρσ τ

₍

_p

_,

_p

_;

_q 1

,

q2

)

×

2 A_SP Q

(−

q2

,

0

,−

q1

,

q1

+

q2

)

F

ρσ τ

(−

q2

)

×

Q_μν

(

q1

,

q1

+

q2

)

+

ASP Q

(

q1

+

q2

,

0

,

q1

,

q2

)

×

F

νσ τ

(

q1

+

q2

)

Qμρ

(

q1

,

q2

)



,

(3.10) whereas albl μ

|

Q Q

S

=

0.The trace calculation3 leads tothe final

ex-pression

3 _{ThecorrespondingDiractraceshavebeencomputedusingtheFeynCalc}

albl_μ

|

S

= −

e6



_d4_q 1

(

2

π

)

4



_d4_q 2

(

2

π

)

4

×

1 q2 1q22

(

q1

+

q2

)

2 1

(

p

+

q1

)

2

−

m2 1

(

p

−

q2

)

2

−

m2



D

(2) S



P

(1,12)

P

(2)T1P P,S

+

P

(2)

Q

(1,12)T1P Q,S

+

D

(1,2) S



P

(1,2)

P

(12,0)T2P P,S

+

P

(12,0)

Q

(12,0)T₂P Q_,S

,

(3.11) where the amplitudes Ti,S are given in Table 2 and the

func-tions

P

(i,j) and

Q

(i,j) in Eq. (3.5). Letus simply note here that

T₁(P P)

(

q1

,

q2

)

andT1(P Q)

(

q1

,

q2

)

comefromthesumofthetwo

di-agrams

(

a

)

and

(

b

)

of Fig. 2 (they give identical contributions), while T₂(P P)

(

q1

,

q2

)

and T₂(P Q)

(

q1

,

q2

)

represent thecontributions

from diagram

(

c

)

. Apart from the presence of two form factors, the situation, at this level, is similar to the one encountered in thecaseoftheexchangeofapseudoscalarmeson,seeforinstance Ref. [28].

4.



S

μν atshortdistanceandvectormesondominance

Inordertoproceed,someinformationaboutthevertexfunction



_μνS

(

q

,

pS

−

q

)

is required. In particular, the question about the

relative sizes ofthe contributions to albl_μ

|

S comingfrom the two

formfactorsinvolvedinthedescriptionofthematrixelement(3.1) needs to be answered. Inorder tobriefly addressthis issue,one firstnoticesthatatshortdistancesthevertexfunction



S_μν

(

q

,

pS

−

q

)

hasthefollowingbehaviour (in thepresentdiscussionqμ isa spacelikemomentum): lim λ→∞



S μν

(λ

q

,

pS

− λ

q

)

=

1

λ

2



1 q2



2



_μνS;∞

(

q

,

pS

)

+

O



1

λ

2



3

,

(4.1) with



_μνS;∞

(

q

,

pS

)

= (

qμqν

−

q2

η

μν

)

A

+



(

q

·

pS

)

qμpS,ν (4.2)

−

q2pS,μpS,ν

+ (

q

·

pS

)(

qνpS,μ

−

q

·

pS

η

μν

)

B

.

The structure of



μνS;∞

(

q

,

pS

)

follows from the requirements qμ



μνS;∞

(

q

,

pS

)

=

0,qν



μνS;∞

(

q

,

pS

)

=

0,andthecoefficients A and B are combinations of the four independent “decay constants” whichdescribethematrixelements



0

| :

Dρ

¯ψ

Q2

γ

σ

ψ

: (

0

)|

S

(

pS

) , 

0

| : ¯ψ

Q2

M

ψ

: (

0

)|

S

(

pS

) ,



0

| :

Ga_μνG_ρσa

: (

0

)|

S

(

pS

) ,

(4.3) ofthethreegaugeinvariantlocaloperatorsofdimensionfourthat can couple to the 0++ scalar states. Here Q

=

diag

(

2

/

3

,

−

1

/

3

,

−

1

/

3

)

denotes the charge matrix of the light quarks, whereas

M

=

diag

(

mu

,

md

,

ms

)

standsfortheirmassmatrix.Thethird

ma-trixelement,involvingthegluonicoperator

:

Ga

μν Gaρσ :,onlyoccurs

totheextent thatthescalarstate possessesa singletcomponent. For a pure octet state, and in the chiral limit, only one “decay constant”, coming from the first operator, remains, and one has

A

/

B

= −

M2_S

/

2.TheasymptoticbehaviourinEq. (4.2) leadstothe suppressionof

Q(

q1

,

q2

)

withrespectto

P(

q1

,

q2

)

athigh

(space-like)photonvirtualities( Q2

i

= −

q2i):

Q

(

q1

,

q2

)

 −

2

P

(

q1

,

q2

)

Q₁2

+

Q₂2

.

(4.4)

Thisshort-distancebehaviourcanbe reproducedbyasimple vec-tormesondominance(VMD)-typerepresentation,

P

VMD

₍

_q 1

,

q2

)

= −

1 2 B

(

q2 1

+

q22

)

+ (

2 A

+

M2SB

)

(

q2₁

−

M2_V

)(

q2₂

−

M2_V

)

,

Q

VMD

₍

_q 1

,

q2

)

= −

B

(

q2₁

−

M2_V

)(

q2₂

−

M2_V

)

,

(4.5)

whichleadsto:

κ

S

≡ −

M2 S

Q

VMD

(

0

,

0

)

P

VMD

₍

₀

_,

₀

₎

= −

2B M2 S B M2_S

+

2 A

.

(4.6)

Incidentally,similarstatementscanalsobeinferredfromRef. [29], where the octet vector–vector-scalar three-point function



V V S

was studiedin thechirallimit.From theexpressions giventhere, oneobtains M2_S

Q

(

q1

,

q2

)

P

(

q1

,

q2

)

= −



9 5 M4_V F2 π

(

M2K

−

M2π

)

˜

c

−

1 2

+

Q₁2

+

Q₂2 2M2 S



−1

 −

2M2S 2M2_S

+

Q₁2

+

Q₂2

,

(4.7) with[30]

˜

c

=

5 16

π α

2



_

ρ→e+e− Mρ

−

3



ω→e+e− Mω

−

3



φ→e+e− Mφ



 (

4

.

6

±

0

.

8

)

·

10−3

.

(4.8)

Numerically, this would correspond to A

/

B

= −

2M2

S (

κ

S

=

1),

ratherthanto A

/

B

= −

M2

S

/

2,which,asmentioned above,should

holdpreciselyfortheconditionsunderwhichtheanalysiscarried outinRef. [29] isvalid.Thisdiscrepancyillustratesthewell-known [31,32] limitation ofthe simple saturation by a single resonance in each channel, which in generalcannot simultaneously accom-modatethe correctshort-distancebehaviour ofa givencorrelator andof thevarious relatedvertexfunctions. Letusalso pointout that A

/

B

= −

M2_S

/

2 corresponds to

P(

0

,

0

)

=

0, i.e. to a vanish-ing two-photon width. This either means that scalars without a singlet component decay into two photons through quark-mass and/orthroughisospin-violatingeffects,or,morelikely,showsthe limitation oftheVMDpicture,which provides,inthiscase, atoo simplistic description of a more involved situation. The second alternative would then require to go beyond a single-resonance description,asdescribed,forinstance,inRef. [32] forthe photon-transitionform factorofthe pseudoscalarmesons. Followingthis path would, however, lead us too far astray, and in the present studywewillkeepthediscussionwithintheframeworksetbythe VMDdescriptionofthetwoformfactors

P(

q1

,

q2

)

and

Q(

q1

,

q2

)

.

Forlateruse,likeforinstancethederivationofEq. (5.4) below,it is also ofinterest to parameterizethe VMD formfactors directly in terms of

P(

0

,

0

)

, which givesthe two-photon width, and the parameter

κ

S asdefinedbythefirstequalityinEq. (4.6):

Table 2

Expressions,inMinkowskispace,oftheamplitudesdefinedinEq. (3.11).

T1P P,S(q1,q2)= 16 3  q22(p·q1)2+q21(p·q2)2− (q1·q2)(p·q1)(p·q2)+ (p·q1)(p·q2)q22+ (p·q1)(q1·q2)q22− (p·q2)(q1·q2)2−m2q 2 1q22−m2(q1·q2)q22  +8(p·q1)q21q22−8q21(p·q2)(q1·q2), TP P 2,S(q1,q2)= 8 3  q2 2(p·q1)2+q21(p·q2)2−2(q1+q2)2(p·q1)(p·q2)+ (p·q1)(p·q2)q12+ (p·q1)(p·q2)q22+m2(q1·q2)(q1+q2)2  . T1P Q,S(q1,q2)= 16 3  (q1·q2)(p·q1)(p·q2)(q12+q22)+ (p·q1)(p·q2)(q1·q2)2+ (p·q2)(q1·q2)q12q22−q21q22(p·q1)2− q2 1q22(p·q2)2−q22(q1·q2)(p·q1)2−q21(q1·q2)(p·q2)2− (p·q1)2q42− (p·q2)2q41− (p·q1)q21q42− (p·q1)(p·q2)q21q22+m2q21q22(q1+q2)2 −40 3  (p·q1)(q1·q2)q21q22− (p·q2)(q1·q2)2q21  +8q4 1  (p·q2)(q1·q2)−q22(p·q1) , T2P Q,S(q1,q2)= 4 3  2(q1·q2)(p·q1)(p·q2)(q21+q22)−q 2 2(q1+q2)2(p·q1)2−q21(q1+q2)2(p·q2)2−q22(q 2 1+q22)(p·q1)2−q21(q21+q22)(p·q2)2+2m2q 2 1q22(q1+q2)2  .

Wemaydrawtwo conclusions fromtheprecedinganalysis. First, thatasensiblecomparisontobemade, forspace-likephoton vir-tualities, isthus notbetween

P(

q1

,

q2

)

and

Q(

q1

,

q2

)

,butrather

between

P(

q1

,

q2

)

and, say,

−(

2M2S

+

Q12

+

Q22

)

Q(

q1

,

q2

)/

2. At

high photon virtualities, their ratio tends to unity. Second, that

|

P(

0

,

0

)

|

and M2

S

|

Q(

0

,

0

)

|

may well be of comparable sizes. For

instance,withinVMD,weobtain

P

(

0

,

0

)

= −

M2_S

Q

(

0

,

0

)

(4.10)

fromtheanalysisofRef. [29]. 5. Angularintegrals

Thenext stepconsistsintransformingthetwo-loopintegralin Eq. (3.11) into an integration in Euclidian space through the re-placement



d4qi

−→

i

(

2

π

2

)

∞



0 d QiQi3



_d

_

ˆ Qi 2

π

2

,

(5.1) withQ_i2

= −

q2_i,i

=

1

,

2,and d



_Qˆ i

=

d

φ

Qˆid

θ

1Qˆid

θ

2Qˆisin

(θ

1Qˆi

)

sin 2

_(θ

2Qˆi

),



d



_Qˆ i

=

2

π

2

_,

_(5.2)

wherethe orientation of the four-vector Q μ in four-dimensional Euclidianspaceisgivenby theazymuthal angle

φ

_Qˆ andthe two polarangles

θ

_1Qˆ and

θ

_2Qˆ.Sincetheanomalousmagneticmoment isa Lorentz invariant, its value does not depend on the lepton’s four-momentumpμ beyonditsmass-shellcondition p2

=

m2.One may thus average, in Euclidian space, over the directions of the four-vector P (theEuclidiancounterpartofp,i.e.P2

_{= −}

_m2₎

albl_μ

|

S

=

1 2

π

2



d



_Pˆalbl_μ

|

S

.

(5.3)

Thisallowstoobtainarepresentationofa_μlbl

|

P P_S +P Q asanintegral overthreevariables, Q1, Q2,andtheanglebetweenthetwo

Eu-clidianloopmomenta[33].Actually,intheVMDrepresentationof Eq. (4.5),theformfactorsbelongtothegeneralclassdiscussedin Ref. [28],forwhichonecanactuallyperformtheangularintegrals directly,withouthavingtoaverageoverthedirectionofthelepton four-momentumfirst.WithinthisVMDapproximationoftheform factors,theanomalousmagneticmomentthenreads

albl_μ

|

VMD_S

=



_α

π



3

[

P

(

0

,

0

)]

2 ∞



0 d Q1 ∞



0 d Q2 M4_V

(

Q2 1

+

M2V

)(

Q22

+

M2V

)





w₁P P

(

MV

)

− 

wP P₂

(

MV

)



+

κ

S 2

Q₁2

+

Q₂2 M2_S





w₁P P

(

MV

)

− 

w2P P

(

MV

)

−

M2V M2_S



w₁₂P P

(

MV

)

−

2



w1P Q

(

MV

)

−

2



w2P Q

(

MV

)



+

κ

S2 4



Q₁2Q₂2 M4_S



w P P 1

(

MV

)

− ˜

w12P P

(

MV

)

−

2 Q 2 2 M2 S



w₁P Q

(

MV

)

−

2 Q₁2

+

Q₂2 M2 S



w₂P Q

(

MV

)



≡



α

π



3

[

P

(

0

,

0

)]

2



I

p

+

κ

S

I

pq

+

κ

S2

I

q

,

(5.4)

where

κ

S wasdefinedinEq.4.6,andwith



w₁P P_,2,P Q

(

M

)

≡

w₁P P_,2,P Q

(

M

)

−

w₁P P_,2,P Q

(

0

),

(5.5) w₁₂P P

(

MV

)

=

w1P P

(

MV

)

−

w2P P

(

MV

),

(5.6)

˜

w₁₂P P

(

MV

)

=

Q₂2M2_V M4 S wP P₁

(

MV

)

−

Q₁2

+

Q₂2 M2 S M2_V M2 S w₂P P

(

MV

).

The dimensionlessdensities (the overallsignhas beenchosen such that thesedensities are positive)occurring inthese expres-sions canbe found inTable 3.Theyare obtaineduponusing the angular integrals given in [28]. Some of their combinations are plottedin Figs. 3,4,and 5.Generically, they are peakedin a re-gionaround Q1

∼

Q2

∼

500MeV,andaresuppressedforsmaller

orlargervaluesoftheEuclidianloopmomenta. 6. I

=

0 scalarmesonsfromgluoniumsumrules

Theevaluationofalbl

μ

|

VMDS asgiveninEq. (5.4),requiresasinput

values for the masses and the two-photon widths of the vari-ous scalarresonances we wantto include.Forthe narrowstates, this information can be gathered from the review [25] or from other sources, which will be described in Section 9. In this sec-tion,wereviewtheinformationprovidedbyvariousQCDspectral sumrulesandsome low-energytheoremsonthemass,aswellas onthehadronicandtwo-photonwidths,ofthelightestscalar me-son

σ

/

f0

(

500

)

,the f0

(

1350

)

and f0

(

1504

)

interpretedasgluonia

states.

Fig. 3. The weight functions: a):wP P

1 and b):w2P Pas function of Q1and Q2. We have used MV=Mρ=775 MeV and MS=960 MeV.

Fig. 4. The same as in Fig.3but for PQ.

Fig. 5. The same as in Fig.3but for the combinations wP P

Thenatureofthe isoscalarI

=

0 scalarstatesremains unclear as it goes beyond the usual octet quark model description due totheir U

(

1

)

component.Afour-quark descriptionofthesestates havebeenproposed within thebagmodel[34] and studied phe-nomenologically in e.g. Refs. [35,36]. However, its singlet nature has also motivated their interpretation as gluonia candidates as initiatedinRef. [37] andcontinuedinRefs. [38–42].4_Recent anal-ysisofthe

π π

and

γ γ

scatteringdataindicatesaneventuallarge gluon component of the

σ

/

f0

(

500

)

and f0

(

990

)

states [19–23]

whilerecentdataanalysisfromcentralproductions[47] showsthe gluonium nature of the f0

(

1350

)

decaying into

π

+

π

− and into

the specific 4

π

0 _{states via two virtual}

_σ

_/

_f

0

(

500

)

states as

ex-pectedifitisa gluonium[40,41].The

σ

/

f0

(

500

)

areobservedin

thegluoniagolden J

/ψ

and

ϒ

→

π π γ

radiativedecaysbutoften interpretedasS-wavebackgroundsduetoitslargewidth(seee.g. BESIII[48] andBABAR[49]).TheglueballnatureoftheG

(

1

.

5

−

1

.

6

)

hasbeenalsofoundby GAMS fewyears ago[50] onits decayto

η



η

andonthevalueofthebranchingratio

η



η

/

ηη

expectedfora high-massgluonium[40,41].

•

The

σ

/

f0massfromQCDspectralsumrules

Thesingletnatureofthe

σ

/

f0hasmotivatedtoconsiderthatit

maycontainalargegluoncomponent[39–41],whichmayexplain itslarge masscompared tothepion.Thisproperty isencodedin thetraceoftheQCDenergymomentumtensor:

θ

μμ

=

1 4

β(

α

s

)

G a μνGμνa

+



1

+

γ

m

(

α

s

)



u,d,s mq

¯ψ

q

ψ

q

,

(6.1)

where

β(

α

s

)

≡ β

1

(

α

s

/

π

)

+ · · ·

and

γ

m

(

α

s

)

≡

γ

1

(

α

s

/

π

)

+ · · ·

are

theQCD

β

-functionandquarkmassanomalousdimension:

−β

1

=

(

1

/

2

)(

11

−

2nf

/

3

)

,

γ

1

=

2 for SU

(

3

)

c

×

SU

(

nf

)

. A QCD spectral

sumrule(QSSR) [51,52]5 _{analysisofthecorresponding two-point} correlatorinthechirallimit(mq

=

0):

ψ

g

(

q2

)

=

i



d4x



0

|

T

θ

μμ

(

x

)θ

μμ

(

0

)|

0

(6.2)

fromthesubtractedandunsubtractedLaplacesumrules:

L

0

(

τ

)

=

∞



0 dtetτ1

π

Im

ψ

g

(

t

)

L

−1

(

τ

)

= −ψ

g

(

0

)

+

∞



0 dt t e tτ 1

π

Im

ψ

g

(

t

)

(6.3)

leadstothepredictions

Mσ

≈ (

0

.

95–1

.

10

)

GeV and MG

≈ (

1

.

5–1

.

6

)

GeV (6.4)

forthemassesofthe

σ

/

f0 andscalargluoniumstates.

•

σ

/

f0hadronicwidthfromvertexsumrules

The

σ

hadronicwidthcanbeestimatedfromthevertex func-tion:

V



q2

≡ (

q1

−

q2

)

2

= 

π

|θ

μμ

|

π

,

(6.5)

whichobeysaoncesubtracteddispersionrelation[40,41]:

4 _{Forrecentreviewsontheexperimentalsearchesandonthetheoreticalstudies}

ofgluonia,seee.g.Refs. [43–46].

5 _{Forreviews,seethetextbooksinRefs. [53,54] andreviewsinRefs. [55,56].}

V

(

q2

)

=

V

(

0

)

+

q2 ∞



4m2 π dt t 1 t

−

q2

₋

_i

_

1

π

ImV

(

t

)

(6.6)

Fromthelow-energyconstraints:

V

(

0

)

=

O

(

m2_π

)

→

0

,

V

(

0

)

=

1

,

(6.7) onecanderivethelow-energysumrules:

1 4



S≡σ,... gSπ π

√

2 fS

=

0

,

1 4



S≡σ,... gSπ π

√

2 fS M2_S

=

1

,

(6.8)

where fS isthescalardecayconstantnormalizedas



0

|

4

θ

μμ

|

S

=

√

2 fSM2S

,

(6.9)

with[41]:

fσ



1 GeV

,

fσ



0

.

6 GeV

,

fG



0

.

4 GeV

,

(6.10)

forMσ



1GeV,Mσ



1

.

3GeV and MG



1

.

5GeV.Thefirstsum

rulerequirestheexistenceoftworesonances,

σ

/

f0 anditsradial

excitation

σ

, coupled strongly to

π π

.6 _{Solving the second sum} rulegives,inthechirallimit,

|

gσ π+π−

|  |

gσK+K−

|  (

4–5

)

GeV

,

(6.11)

which suggests an universal coupling of the

σ

/

f0 to Goldstone

boson pairs asconfirmed fromthe

π π

and K K scatterings

¯

data analysis[22,23].Thisresultleadstothehadronicwidth:



σ→π π

≡

|

gσ π+π−

|

2 16

π

Mσ



1

−

4m 2 π M2 σ



1/2

≈

0

.

7 GeV

.

(6.12) Thislargewidthinto

π π

isatypicalOZI-violationexpectedtobe duetolargenon-perturbativeeffectsintheregionbelow1GeV.Its valuecomparesquitewellwiththewidthoftheso-calledon-shell

σ

/

f0 massobtainedinRef. [20–22] (seealsothenextsubsection).

•

σ

/

f0

→

γ γ

widthfromsomelow-energytheorems

Weintroducethegaugeinvariantscalarmesoncouplingto

γ γ

throughtheinteractionLagrangianandrelatedcoupling:

L

int

=

gSγ γ 2 FμνF μν

_,

_P

₍

₀

_,

₀

₎

_{≡ ˜}

_g Sγ γ

=



2 e2



gSγ γ

,

(6.13)

where Fμν is thephotonfield strength.Inmomentum space,the correspondinginteractionreads7

L

int

=

2gSγ γPμν

(

q1q2

)

×



₁μ



2ν

,

(6.14)

where



μ_i are the photon polarizations. With thisnormalization, thedecaywidthreads



= |

gSγ γ

|

2 M3_S 8

π



1 2



=

π

4

α

2_M3 S

|˜

gSγ γ

|

2

,

(6.15)

where 1/2 is the statisticalfactor for the two-photon state. One can for instance estimate the

σ γ γ

coupling by identifying the

6 _{TheG(1600)isfoundtocoupleweaklytoπ πandmightbeidentifiedwiththe}

gluoniumstateobtainedinthelatticequenchedapproximation(forarecentreview ofdifferentlatticeresults,seee.g.[43]).

7 _{Weusethenormalizationandstructurein[57] foron-shellphotons.However,a}

Table 3

ExpressionsoftheweightfunctionsdefinedinEq. (5.6) afterangularintegrationintheEuclidianspace[Dm1≡ (P+Q1)2+m2,Dm2≡ (P−Q2)2+m2].

wP P 1 (M)= −  _d 1 2π2  _d 2 2π2 π2_Q 1Q2 Dm1Dm2 T1ES;P P(Q1,Q2) (Q2 2+M2S)[(Q1+Q2)2+M2] = −2 3 π2_Q 1Q2 Q22+MS2  1+Q2 2 2m2 l +Q22  Q12−Q22−M2   Q12−Q22−M2−4m2l  IM1 −  2Q12−Q22−M2+ Q12Q22 2m2 l _Rm1−1 2m2 l −Q22  2+Q2 2 2m2 l  Rm2−1 2m2 l −Q12−Q22+M2  I7M+Q22  3Q12−Q22−M2−4m2l+ Q12Q22 2m2 l Rm1−1 2m2 l Rm2−1 2m2 l +Q12+Q22+M2   2Q12−Q22−M2 Rm1−1 2m2 l IM7 −2Q22  Q12−M2 Rm2−1 2m2 l IM7  , wP Q1 (M)= −  d1 2π2  d2 2π2 π2_Q 1Q2 Dm1Dm2 1 M2 S T1ES;P Q(Q1,Q2) (Q2 2+M 2 S)[(Q1+Q2)2+M2] = −1 3 π2_Q 1Q2 MS2Q22+MS2  Q12+Q22−M2+2 Q12Q22 m2l −4Q12Q22M2  Q12−2Q22−M2−4m2l  IM 1 −Q12  Q12 −3Q22−5M2+2 Q12Q22 m2 l Rm1−1 2m2 l −4Q22  2Q12+ Q12Q22 2m2 l  Rm2−1 2m2 l −Q12−Q22+M2   Q12−Q22−M2  IM 7 +4Q12Q22  2Q12−Q22−M2−4m2l+ Q12Q22 2m2 l  Rm1−1 2m2 l Rm2−1 2m2 l +Q12   Q12−Q22 2 −4M2Q12−8Q22M2 −5M4Rm1−1 2m2 l IM7 +8M2Q12Q22 Rm2−1 2m2 l I7M  , w2P P(M)= +  _d 1 2π2  _d 2 2π2 π2_Q 1Q2 Dm1Dm2 T2ES;P P(Q1,Q2) [(Q1+Q2)2+M2S][(Q1+Q2)2+M2] ≡w˜2P P(M)− ˜wP P2 (MS) M2 S−M2 , wP Q2 (M)= −  _d 1 2π2  _d 2 2π2 π2_Q 1Q2 Dm1Dm2 1 M2 S T2ES;P Q(Q1,Q2) [(Q1+Q2)2+M2S][(Q1+Q2)2+M2] ≡w˜ P Q 2 (M)− ˜w P Q 2 (MS) M2 S(M2S−M2) , with ˜ wP P 2 (M)= 2 3π 2_Q 1Q2  2M2_Q 12Q22+ml2Q12+ml2Q22+ml2M2  IM 1 + Q12 2 Rm1−1 2m2 l +Q22 2 Rm2−1 2m2 l +M2_IM 7 −Q12Q22+2m2lM 2Rm1−1 2m2 l Rm2−1 2m2 l −Q12 2  Q12−Q22+3M2 Rm1−1 2m2 l IM7 − Q22 2  Q22−Q12+3M2 Rm2−1 2m2 l IM7  , ˜ wP Q2 (M)= − 1 3π 2_Q 1Q2  −M2_+2M2_Q 12Q22(Q12+Q22+4m2l)I1M+ Q12 2  Q12+3Q22+M2 _Rm1−1 2ml2 +Q22 2 (Q2 2₊ 3Q12+M2) Rm2−1 2m2 l +M2(Q12+Q22+M2)IM7 −2Q12Q22  Q12+Q22+4ml2 Rm1−1 2m2 l Rm2−1 2m2 l −Q12 2 × (Q14−Q24+2M2Q12+4M2Q22+M4) Rm1−1 2m2 l IM 7 − Q22 2 (Q2 4_−Q 14+2M2Q22+4M2Q12+M4) Rm2−1 2m2 l IM 7  , and IM 1 = 1 m2 lQ12Q22 ln[1−ZM Q1Q2Z ml P Q1Z ml P Q2], I M 7 = ZM Q1Q2 Q1Q2, Rmi≡    1+4m 2 l Q2 i , Zml P Qi= Qi 2P(1−Rmi), (Zml P Qi) 2₌Qi P Z ml P Qi−1, Z ml P Q1Z ml P Q2= − Q1Q2 4m2 l (Rm1−1)((Rm2−1)), ZMK L= K2_+L2_+M2_{− (K}2_+L2_+M2₎2_−4K2_L2 2K L ,

Euler–HeisenbergLagrangian derived from gg

→

γ γ

via a quark constituentloop withthe interaction Lagrangian in Eq. (6.13). In thisway,onededucestheconstraint8:

gSγ γ



α

60

√

2 fSM2S



π

−β

1

 

u,d,s Q_q2

/

M4_q

,

(6.16)

where Qq is the quark charge in units of e; Mu,d

≈

Mρ

/

2 and Mφ

≈

Mφ

/

2 areconstituentquarkmasses.Then,oneobtains:

gσ γ γ

≈

gσγ γ

≈

gGγ γ

≈ (

0

.

4–0

.

7

)

α

GeV−1

,

(6.17)

whichleads,forMσ



1GeV,tothe

γ γ

width:



σ→γ γ

 (

0

.

2–0

.

6

)

keV

.

(6.18)

A consistencycheck of the previous resultcan be obtained from the trace anomaly



0

|θ

_μμ

|

γ γ

by matchingthe k2 _{dependence of}

itstwosideswhichleadsto[58–61]:

8 _{Thissumrulehasbeenoriginallyusedby[39] inthecaseofacharmquark}

loopforestimatingthe J/ψ→γ σradiativedecay.

1 4



S=σ··· gSγ γ

√

2 fS

=

α

R 3

π

,

(6.19) where R

≡

3



Q_q2.

7.

σ

/

f0

(500)

mesonfrom

π π

and

γ γ

scattering

The mass and the width of a broad resonance like the

σ

/

f0

(

500

)

stateingeneralturnouttoberatherambiguous

quan-tities. Anon ambiguous definitionis provided by the location of the poleoftheS-matrixamplitude onthesecond Riemannsheet [62].Thedifficultythenliesinrelatingthispoleinthecomplex do-maintothedescription,forinstanceintheformofaBreit–Wigner function,ofthedataonthepositiverealaxis.Thisissuehasbeen quiteextensivelydiscussedinthecontextoftheline-shapesofthe electroweakgaugeandscalarbosons9 _[63–69].

9 _{Theissuewasmainlycentred aroundthenecessitytodefinegauge-invariant}