Advances in QCD sum rule calculations

0.14 MB·English

by  Dmitri Melikhov

Advances in QCD sum-rule calculations
Dmitri Melikhov
InstituteforHighEnergyPhysics,AustrianAcademyofSciences,Nikolsdorfergasse18,A-1050Vienna,Austria
D.V.SkobeltsynInstituteofNuclearPhysics,M.V.LomonosovMoscowStateUniversity,Moscow,Russia
5
1
Abstract. WereviewtherecentprogressintheapplicationsofQCDsumrulestohadronpropertieswiththeemphasisonthe
0
followingselectedproblems:(i)developmentofnewalgorithmsfortheextractionofground-stateparametersfromtwo-point
2
correlators;(ii)formfactorsatlargemomentumtransfersfromthree-pointvacuum correlationfunctions;(iii)propertiesof

n exotictetraquarkhadronsfromcorrelationfunctionsoffour-quarkcurrents.
a Keywords: QCD,operatorproductexpansion,QCDsumrules
J
PACS: 11.55.Hx,12.38.Lg,14.40.Nd,03.65.Ge

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2
INTRODUCTION
]
h
p Themethodofsumrulesis35yearsold.Inspiteofthisrespectableage,themethodisbeingpermanentlyenrichedby
- newideasandnewcalculationsandremainsoneofthewidelyusedandcompetitivetoolsbothforthedeterminations
p
ofthefundamentalQCD parameters(e.g.,quarkmassesanda )andforthecalculationofhadronproperties.Inthis
e s
h talk we review the recentprogressin the applicationsof QCD sum rules to hadronpropertieswith the emphasison
[ the selected topics: (i) sum rules for two-pointvacuum correlationfunctionsand leptonic decay constants of heavy

mesons;(ii)sumrulesforthree-pointvacuumcorrelationfunctions,formfactorsandthree-mesoncouplings;(iii)sum
1
v rulesforexotictetraquarkstates.
9 QCD sum rules [1] (see also [2, 3] for further references) is one of the main analytic methods for the study of
1 hadronpropertiesfromthefield-theoreticGreenfunctions(correlators)infullQCD.Thecorrelatorsarecalculatedby
3 meansoftheWilsonianoperatorproductexpansion(OPE)whichprovidestherigorousframeworkfortheseparation
6
of long and shortdistances, in QCD being dominatedby nonperturbativeandperturbativephysics, respectively[4].
0
TheOPEclearlyidentifies,e.g.,theoriginofchiralsymmetrybreakingandtheemergenceofhadronmasses,leadsto
.
1 factorizationofcomplicatedamplitudesofhadroninteractionsatlargemomentumtransfers.
0
QCDsumrulesprovidehadronamplitudeswhichsatisfyallrigorouspropertiesimposedbyperturbativeQCDand,
5 •
atthesametime,containnonperturbativecontributionsdeterminedinauniqueway.AsanOPE-basedmethod,QCD
1
sum rules are formulatedin the Euclideanregion.However,by combiningOPE with the knowledgeof the analytic
:
v structureoftheGreenfunctionsandresummationschemes,theanalyticcontinuationtotheMinkowskispacemaybe
i
X performed.In thisrespectQCD sum rulesmayhavea broaderrangeof applicabilitythanlattice QCD. Lastbutnot
least,asananalyticmethod,QCDsumrulesprovidephysicsinsightsinthehadronstructure,whicharenoteasytoget
r
a fromthenumericalresultsoflatticeQCD.
The method of QCD sum rules favourably compares with other analytic methods, such as effective theories or
•
functionalmethods:the methodof sum rulesis based onthe WilsonianOPE in fullQCD andthereforeinvolvesno
otherimplicitassumptionsoftenpresentinotheranalyticmethod.
a.OPEandthesumruleforthecorrelator
ThebasicobjectinthemethodofQCDsumrules–aswellasinlatticeQCD–isthevacuum-to-vacuumcorrelator,i.e.,
thevacuumaverageoftheT-productofquarkandgluoncurrents.InlatticeQCD,onefindsthiscorrelatornumerically
atlargevaluesoftheEuclideantimet .InthemethodofQCDsumrules,onecalculatesthecorrelatoranalyticallyas
theTaylorexpansionint .Technically,oneconsidersaso-calledBorelizedcorrelator,i.e.appliestheBoreltransform
totheFeynmandiagrams,writtenasspectralrepresentationsintheenergyvariables.TheinverseBorelmassparameter
isrelatedtot .TheOPEprovidestheanalyticdoubleexpansionofthiscorrelatorinformofaperturbativelycalculable
powerseriesinthestrongcouplingconstanta andinpowersoft ;the“powercorrections”—termsinvolvingpowers
s
oft —aregivenviacondensates,expectationvaluesofgauge-invariantoperatorsoverthephysicalvacuuminQCD;
thesecondensatesdescribeinanunambiguouswaynonperturbativeQCDcontributions.
Alternatively,onemayderivearepresentationfortheBorelizedcorrelatorintermsoftheintermediatehadronstates.
ThetworepresentationsfortheBorelizedcorrelator—byOPEandbysumoverhadronstates—constitutethetwo
sidesoftheQCDsumrule.
b.Isolatingtheground-statecontributionfromtheBorelizedcorrelator
Atlarget ,theground-statedominatesthecorrelatorwhichthusfullydeterminestheground-stateparameters.Inthe
regionofsmallandintermediatet ,wherethetruncatedOPEgivesagooddescriptionofthecorrelator,excitedhadronic
statesgivesizeablecontributions.Inordertogetridoftheexcitedstatesandtoisolatetheground-statecontribution
from the correlator, one invokes the idea of quark–hadronduality [5–7]: the excited states are dual to high-energy
partsofFeynmandiagramsofperturbativeQCD.Theground-statecontributionisthenequaltothe“dualcorrelator”
–thecorrelatorinwhichthespectralintegralsforperturbationtheorydiagramsarecutatacertaineffectivecontinuum
thresholds ,orsimply“effectivethreshold”.Theeffectivecontinuumthresholddiffersfromthephysicalcontinuum
eff
threshold determined by masses of low-lying hadrons. Obviously, apart from a truncated OPE for a correlator, the
effective continuum threshold is a crucial ingredient of every sum-rule extraction of ground-state parameters; this
quantitygovernstheaccuracyofthequark–hadrondualityanddeterminestolargeextentthenumericalvalueofthe
extractedparametersof the boundstate. The truncatedOPE itself cannotprovideprecise valuesof the ground-state
parameters. Therefore, the method of QCD sum rules provides hadron parameters with some uncertainty which is
referredtoassystematicuncertainty[8].
Understandingthepropertiesoftheeffectivecontinuumthresholdandfindingacriterionforfixingthisquantityis
thekeytoobtainingreliablehadronparametersfromsumrules.
1. TWO-POINTCORRELATIONFUNCTION ANDTHE OPE
Let us start with the simplest object – the two-point correlation function;the perturbativeexpansion for this object
is known to a higher accuracy compared to more complicated correlators. Because of that, the formulation and
applicationoftheappropriateandreliablealgorithmsfortheextractionofthehadronparametersfromthiscorrelator
isbecomingincreasinglyimportant.
The two-pointfunction,i.e. the vacuumaverageof the T-productof two interpolatingquarkcurrentsis the basic
objectforthesum-rulecalculationofthedecayconstantsoftheheavy-lightmesonssuchasB,B ,D,D ortheirvector
s s
analogues.Forinstance,forheavy-lightpseudoscalarcurrents j =m q¯ig b(herem isthescale-dependentMSmass
5 b 5 b
oftheheavyquarkandM willdenoteitspolemass;thelight-quarkmassisneglected)oneobtains
b
P (p2)=i d4xeipx W T j (x)j†(0) W (1)
5 5
Z D (cid:12) (cid:16) (cid:17)(cid:12) E
TheWilsonOPEfortheT-productandforthecorrelation(cid:12)functionhasthe(cid:12)followingform:
(cid:12) (cid:12)
T j (x)j†(0) =C (x2,m )1ˆ+(cid:229) C (x2,m ):Oˆ(x=0,m ): (2)
5 5 0 n
n
(cid:16) (cid:17)
and
P (p2)=P (p2,m )+(cid:229) Cn W :Oˆ(x=0,m ): W (3)
pert (p2 M2)nh | | i
n − b
Here the physicalQCD vacuum W is a complicatedobjectwhich differsfrom perturbativeQCD vacuum 0 . The
| i | i
propertiesof the physicalvacuum are characterized by the condensates – the nonzeroexpectation values of gauge-
invariantoperatorsoverthisphysicalvacuum:
W :Oˆ(0,m ): W =0. (4)
h | | i6
Thenumericalestimatesforthecondensatesmaybefoundin[2,3].Hereweonlylisttherecentdeterminationsofthe
lowest-dimensioncondensateswhichclaimanextremelyhighaccuracy:
a
hW |q¯q(2GeV|W iMS=(282±2MeV)3[9], hW |psGamn Ga,mn |W i=0.013±0.0016GeV4[10]. (5)
Thetwo-pointfunctionsatisfiesthedispersionrepresentation(whichrequiressubtractionsnotshownhere)
ds
P (p2)= r (s), (6)
s p2
Z −
and may be calculated both using OPE (which gives it in the form P (p2)) and using the sum over the hadron
OPE
intermediatestates(whichgivesitintheformP (p2)).Thesumruleisthestatementthatbothformsrepresentthe
hadr
samequantityandthusshouldbeequaltoeachother
P (p2)=P (p2). (7)
OPE hadr
Thespectraldensitiesforthetworepresentationsread
r (s)= r (s,m )+(cid:229) C d (n)(s M2) W O (m )W , r (s)= f2M4d (s M2)+r (s) (8)
OPE pert n − b h | n | i hadr B B − B cont
(cid:20) n (cid:21)
HereM denotestheheavy-mesonmass, f isitsdecayconstantdefinedas
B B
0 j B = f M2. (9)
h | 5| i B B
ThetruncatedOPEserieshasquarkandgluonsingularitiesanddoesnothavethehadronones;therefore,comparison
ofthetruncatedOPEandthehadronrepresentationin(7)maybedoneintheregionof p2 farfromhadronthresholds
andresonances.
Performingthe Boreltransformwhichservesseveralpurposes(suppressingthe contributionofthe excitedstates,
killingthesubtractiontermsinthedispersionrepresentationforP (p2),improvingtheconvergenceoftheperturbative
expansion[1])onearrivesattheBorelimageofthetwo-pointfunction
¥
P hadr(t )= dsexp(−st )r hadr(s)= fB2MB4e−MB2t + dse−st r hadr(s), (10)
Z spZhys
where s =(M +M )2 is the physical continuum threshold, determined by the masses of hadrons which may
phys B P
∗
appearastheintermediatestates,and
¥
P (t )= dsexp( st )r (s)= dse st r (s,m )+P (t ,m ), (11)
OPE OPE − pert power
−
Z Z
m2
b
wherepowercorrectionsP (t ,m )aregivenviathecondensatesandradiativecorrectionstothem.
power
Thesumrulenowtakestheform
P (t )=P (t ). (12)
OPE hadr
Recall that the hadron (i.e. full-QCD) representation P (t ) is an infinite sum of the exponential terms, whereas
hadr
power corrections in P (t ) contain polynomials in t multiplied by exp( M2t ). Therefore the truncated OPE
providesagooddescriptiOoPnEofP (t )at“nottoolarge”valuesoft .Thisdeter−minbesthechoiceoftheBorelwindow–
hadr
theworkingt -rangewheretheOPEgivesanaccuratedescriptionoftheexactcorrelator(i.e.,allhigher-orderradiative
andpowercorrectionsare undercontrol)andat thesame time the groundstate givesa “sizable”contributionto the
correlator.
The best-known 3-loop calculations of the perturbative spectral density [11] have been performed in form of an
expansionintermsoftheMSstrongcouplinga (m )andthepolemassM :
s b
a (m ) a (m ) 2
r (s,m )=r (0)(s,M2)+ s r (1)(s,M2)+ s r (2)(s,M2,m )+ . (13)
pert b p b p b ···
(cid:18) (cid:19)
An alternative option [12] is to reorganizethe perturbative expansion in terms of the running MS mass, m (n ), by
b
substitutingM inthespectraldensitiesr (i)(s,M2)viaitsperturbativeexpansionintermsoftherunningmassm (n )
b b b
a (n ) a (n ) 2
M =m (n ) 1+ s r + s r +... . (14)
b b p 1 p 2
(cid:18) (cid:19) !
Asnoticedin[12,13],twodifferentscales, m andn ,naturallyemergewhenreorganizingtheperturbativeexpansion
fromthepoleb-quarkmasstotherunningb-quarkmass.Inourdiscussionwedonotdistinguishbetweenthesescales,
butinpracticalcalculationsthescaleshavebeentreatedindependently.
Advanced algorithms foranisolationofthe ground-state contribution
Thehadronrepresentationcontainsthesumoverallhadronintermediatestates,whereasweareprimarilyinterested
in the ground state contribution. To exclude the excited-state contributions, one adopts the duality Ansatz: all con-
tributionsofexcitedstatesarecounterbalancedbytheperturbativecontributionaboveaneffectivecontinuumthresh-
old, s (t ,m )whichdiffersfromthephysicalcontinuumthreshold.Applyingthedualityassumptionyields:
eff
seff(t,m )
fB2MB4e−MB2t = dse−st r pert(s,m )+P power(t ,m )≡P dual(t ,seff(t ,m )). (15)
Z
m2
b
TherhsisthedualcorrelatorP (t ,s (t ))(weshallnotexplicitlywritem asanargumentofs butthisdependence
dual eff eff
shouldbekeptinmind).Obviously,eveniftheQCDinputsr (s,m )andP (t ,m )areknown,theextractionof
pert power
thedecayconstantrequiress (t ,m ). Letusemphasize,thattheeffectivethresholdshouldbethe functionoft and
eff
m :(i)onecaneasilycheckthats shoulddependont inorderthet -dependencesofther.h.s.andthel.h.s.of(15)
eff
matcheachother;(ii)sincethetruncatedOPEisusedinther.h.s.of(15),theeffectivethresholdalsodependsonthe
choiceofthescalem .
Inearlyapplicationsofthemethodofsumrules,itwascommontousetheapproximations (t )=const;thevalue
eff
ofthisconstanthasbeenfixedbyrequiringthemaximalstability(i.e.theleastunphysicaldependenceofthehadron
observableontheBorelparametert ).Thisprocedureprovedtoworkreasonablywell,althoughitdidnotallowone
toprobetheuncertaintyoftheextractedhadronparameterinducedbyusingtheapproximationofaconstanteffective
continuumthreshold.
ItshouldbeemphasizedthateveniftheOPEforthecorrelationfunctionisknownwithveryhighaccuracyinthe
Borelwindow,thehadronparameterscanstillbedeterminedwithsomeuncertaintywhichreflectsthelimitedintrinsic
accuracyofthemethodofsumrules.Werefertothecorrespondinguncertaintyastothesystematicuncertainty.The
latterisrelatedtotheadoptedprescriptionforfixingtheeffectivecontinuumthresholds (t ).
eff
As the accuracy of the OPE for the correlation functionshas increased, one faced the acute necessity to provide
more accurate and reliable proceduresfor the extraction of hadron parameters: gaining control over the systematic
uncertaintieshasbecomemandatory[8].
Theresultsof[14] demonstratedthatinthosecases wherethe bound-statemassM isknown,onecanuse itand
B
improvetheaccuracyofthedecayconstant.WeintroducethedualinvariantmassM andthedualdecayconstant
dual
f
dual
d
Md2ual(t )≡−dt logP dual(t ,seff(t )), fd2ual(t )≡MB−4eMB2t P dual(t ,seff(t )). (16)
ThedeviationofM (t )fromM measuresthecontaminationofthedualcorrelatorbyexcitedstates.
dual B
Startingwithanytrialfunctionfors (t )andminimizingthedeviationofM fromM inthet -windowyieldsa
eff dual B
variationalsolutionfors (t ).Assoonasthelatterisfound,onereadilyobtainsthecorrespondingdecayconstant f
eff B
from(15).
Weconsiderpolynomialsint andobtaintheirparametersbyminimizingthesquareddifferencebetweenM2 and
dual
M2 inthet -window:
B
c 2 1 (cid:229)N M2 (t ) M2 2. (17)
≡ N dual i − B
i=1
(cid:2) (cid:3)
Asshowninseveralexactlysolvablemodels,thebandoftheestiamtesfor f correspondingtothevariationalsolutions
B
for linear, quadratic,and cubic trial s (t ), providesa realistic estimate for the systematic uncertaintyof the decay
eff
constant[15,16].
The resulting f obtained according to the procedure described above is sensitive to the input values of all the
B
OPE parameters (quark masses, a , the condensates) which are known with some uncertainties thus yielding the
s
OPE-relateduncertaintyof f . To obtain the latter, one assumesthe Gaussian distributionsfor the OPE parameters
B
mentionedabove.Moreover,becauseof the truncationof the OPE series, the decay constantsexhibitan unphysical
dependenceontheprecisevalueoftherenormalizationscalesm .Apriori,anychoiceofthescaleisequivalentlygood;
therefore,weaverageoverthescaleinsomeintervalsassumingtheuniformdistributionofm .
Another simple algorithm for fixing the t -dependent effective threshold in the Borel sum rule has been recently
•adoptedin [17]:foreachvalueof t the authorscalculatedM (t ) neglectingthe t -dependenceof s (t ) andthen
dual eff
easilyobtains bysolvingtheequationM (t )=M .Obviously,theresultingeffectivethresholdsdodependont ;
eff dual B
neglectingtheirt -dependencewhilecalculatingthedualmassleadstosomeintrinsicinconsistencies.Followingour
oldidea,we tested thealgorithmof [17] in aquantum-mechanicalpotentialmodelforthecase of apotentialwhich
containstheconfiningandtheCoulombparts[15].Thisanalysisshowsthatinquantummechanicsthealgorithmwith
thevariationalsolutionsdescribedaboveprovidesmorereliableandaccurateestimatesforthedecayconstantsofthe
heavy-lightmesonscomparedwiththealgorithmof[17].
Aninterestingapproachtotheextractionoftheground-stateparameterswithinthefinite-energysumrulehasbeen
•
formulatedand applied to the decay constants of heavy-lightand heavy-heavymesons in [18]. We have also tested
thisalgorithminthepotentialmodel[15].Forthepotential-modelparametersappropriateforforheavy-lightmesons
thealgorithmof[18]wasshowntoprovideratheraccurateestimatesforthedecayconstantssuchthatthe“invisible”
systematicerrorremainsatafewpercentlevelonly.
Charm sector
For the extraction of the decay constants of the charmed pseudoscalar and vector mesons, one makes use of the
best-known three-loop expression for the spectral densities of the two-point functions for pseudoscalar and vector
currents.TheOPEintermsofthepolemassM calculatedin[11]doesnotexhibitaperturbativehierarchy,therefore
b
one rearrange the OPE in terms of the running MS-mass [12]. Then, the perturbative hierarchy of the correlation
functionstarts to dependon m ; this featureallows oneto choosethe rangeof m where the perturbativehierarchyis
visible.Thenegativeeffectofthisrearrangementoftheperturbativeexpansionisthat,becauseofthetruncationofthe
OPEseries,theextracteddecayconstantsacquireanunphysicaldependenceonthescale m .Inthecharmsectorthis
howeverdoesnotleadtoanyseriousproblems.Figure1showsthedependenceofthedecayconstantsofthecharmed
pseudoscalar and vector mesons for the centralvalues of all other OPE parametersafter applyingthe algorithmfor
fixing the effective thresholds described above. One can see a weak m -dependence of the decay constants of the
pseudoscalarmesonsmesons,whereasforvectormesonsthism -dependenceismorepronounced.Averagingoverthe
OPEparametersintheirrespectiveintervalsandoverthescaleintherange1 m [GeV] 3onearrivesatthefollowing
≤ ≤
results[19]
f =(208.3 7.3 5 )MeV, f =(246.0 15.7 5 )MeV
D ± OPE± syst Ds ± OPE± syst
f =(252.2 22.3 4 )MeV, f =(305.5 26.8 5 )MeV. (18)
D∗ ± OPE± syst D∗s ± OPE± syst
Fortheratiowereported f /f =1.221 0.080 0.008 ,whichcomparesnicelywiththelatticeQCDresult
D D OPE syst
∗ ± ±
f /f =1.20 0.02.Theresultsforthecharmedmesonsfromothersum-ruleanalyses[17,20]agreewellwitheach
D D
∗ ±
otherandwiththeresultsfromlatticeQCD[21].
fD,D*@MeVD fDs3,D4s0*@MeVD
280 320 fDs*
260 fD*=252.2±18.7MeV fD* 300 fDs*=305.5±10.8MeV
240
280
220
fD=208.3±3.8MeV 260 f
200 f fDs=246.0±0.9MeV Ds
D 240
Μ@GeVD Μ@GeVD
1.25 1.5 1.75 2 2.25 2.5 2.75 3 1.25 1.5 1.75 2 2.25 2.5 2.75 3
FIGURE1. DecayconstantsofD,DsD∗andD∗s mesonsdependingonthescalem .
fB,B*@MeVD fB*(cid:144)fB
196 1.02
f
194 B fB=192.6±0.4MeV
1
192
190 0.98
188 fB* fB*=186.4±1.8MeV 0.96
186
184 0.94
Μ@GeVD Μ@GeVD
3 4 5 6 7 3 4 5 6 7
FIGURE2. DecayconstantsofBandB∗mesonsandtheratio fB∗/fBdependingonthescalem .
Beauty sector
Similar to the charmsector, the OPE for pseudoscalarand vector currentscontainingthe b-quark,does notshow
anyperturbativehierarchy;thereisnoreasontoassumethattheunknownhigher-orderperturbativecontributionsare
small.Rearrangingtheperturbativeexpansionintermsoftherunningmassintroducesthedependenceofthescalem
andopensthepossibilitytochoosetheworkingrangeofm inwhichtheperturbativehierarchyisexplicitthusallowing
tohopetheunknownhigherordersdonotcontributesubstantiallytothecorrelationfunction.
Intheb-sectoroneencounterstwointerestingfeaturesofthesum-ruleanalysis:
Thesum-ruleresultsforthebeauty-mesondecayconstantscorrelateverystronglywiththeb-quarkmass[22]
•
d f /f 8d m /m , (19)
B B b b
≈−
m m (m ).Makinguseofm =4.18 0.03GeV[23]leadsto f >210MeV,incleartentionwiththerecentlattice
b b b b B
≡ ±
QCDresultsfor f 190MeV.Combiningoursum-ruleanalysiswith f and f fromlatticeQCDyields[22]
B∼ B Bs
m =(4.247 0.027 0.011 )GeV. (20)
b ± ± (syst)
Thesum-ruleresultsforthedecayconstantscorrespondingtothisvalueoftheb-quarkmassread
f =(192.0 14.3 3.0 )MeV, f =(228.0 19.4 4.0 )MeV (21)
B ± OPE± syst Bs ± OPE± syst
For the decay constant of B , one observesan unexpectedlystrong m -dependence[24]: Averaging over the scale
∗
r•ange3<m [GeV]<6leadsto
f /f =0.923 0.059, f /f =0.932 0.047.
B∗ B ± B∗s Bs ±
Taking into account only low-scale results for 2.5< m [GeV] < 3.5, yields f /f = 0.994 0.01. The sum-rule
B B
∗ ±
analysis[17]alsogivesindicationsthat f /f 1(sesTableIIof[17]).Surprisingly,theQCDsum-ruleprediction
B B
∗ ≤
for f /f is below the correspondingresults from lattice QCD, which seem to favour a value slightly aboveunity
B B
∗
[21,25].Clearly,suchtensioncallsforfurtherdetailedinvestigations.
m -dependence ofthe physicalquantities
The heavy-light correltors are known with an impressive three-loop accuracy and are therefore rather weakly
sensitivetothevariationsofthescale.Nevertheless,thedualcorrelatorofthevectorcurrentswhichincludesthelow-
energyregionoftheFeynmandiagramsonlyand,respectively,thevector-mesondecayconstantsarerathersensitive
tothechoiceofthescale.Inmanycasesthisscale-dependenceisthemainsourseoftheOPEuncertaintyinthedecay
constants.We shouldmentionthatinsomepublicationsthe m -dependenceistreatedina specificway[20]:onejust
choosesonescaleatwhichthedecayconstanthas,e.g.,anextremuminm ,andprovidestheresultsforthisveryscale
assigning no theoretical uncertainty to the scale fixing. This of course reduces strongly the total uncertainty of the
decayconstantobtainedwiththesum-ruletechniquebutfromourpointofviewsuchatreatmentisnotjustified:the
(unphysical) m -dependenceis an effect of the truncation of the OPE series and thus reflects an essential feature of
QCD. Any of the scale for which a reasonable perturbativehierarchyis seen, may be used for the determinationof
the hadronparameter;the unpleasant m -dependenceof the sum-ruleresults should be thus properlyreflected in the
theoreticaluncertaintyofthehadronparameterobtainedusingaQCDsumrule.
SUMRULES FORTHREE-POINT VACUUMCORRELATION FUNCTIONS
Let us now discuss the calculation of the meson elastic and transition form factors from the three-point vacuum
correlationfunctions[26,27].Thebasicobjectinthiscasehastheform
G (p2,p2,q2)= W T(j(x)j(0)j(y))W exp( ipx)exp( ipy)dxdy. (22)
′ ′
h | | i − −
Z
Thethree-pointGreenfunctioninfullQCDcontainsthedoublepolerelatedtothemesonsinthe p2 and p2-channels
′
in the timelike region. The residue in this double pole is the form factor of interest. The Green function in the
spacelike region may be calculated using the same method as the two-point function, i.e. by performing the OPE.
OnerepresentstheGreenfunctionG (p2,p2,q2)asadoublespectralintegralin p2and p2,performsthedoubleBorel
′ ′
transform p2 t and p2 t (which,similar to thetwo-pointfunction,kills thesubtractiontermsandsuppresses
′ ′
thecontributio→nsoftheexc→itedstates),equatetoeachothertheOPEandthehadronrepresentationsforG (p2,p2,q2),
′
andusedualitypropertytoisolatetheground-statecontribution,thusrelatingthemesonformfactortothelow-energy
regionofthetrianglediagramsofperturbativeQCDandpowercorrectionsgiventhroughthecondensates.Forinstance,
thepionelasticformfactor,inwhichcaseonesetst =t ,hastheform[27]
′
Fp (Q2)fp2= seff(Q2,t)seff(Q2,t)ds1ds2D pert(s1,s2,Q2)e−s1+2s2t + a2s4Gp 2 t +4pa 8s1hq¯qi2t 2 13+Q2t +···,
Z Z (cid:10) (cid:11)
0 0
(cid:0) (cid:1)
D (s ,s ,Q2)=D (0)(s ,s ,Q2)+a D (1)(s ,s ,Q2)+ . (23)
pert 1 2 1 2 s 1 2
···
Anessentialfeatureofthethree-pointsumruleisthattheeffectivethresholdnowdependsontheBorelparametert
andthemomentumtransferQ[28–30];obviously,onefacesaseriousproblemoffindingappropriatealgorithmsfor
fixings (Q2,t ). Itshouldbeunderstoodthattheeffectivecontinuumthresholdforthe formfactordiffersfromthe
eff
effectivethresholdforthedecayconstant.1
For large Q2, power corrections calculated in terms of the local condensates rise as polynomials with Q2, thus
preventingadirectuseofthesumrule(23)atlargeQ2.Thereareessentiallyonlytwopossibilitiestostudytheregion
oflargeQ2 startingwiththevacuumcorrelators:
usenonlocalcondensateswhichareaimedattheresummationofthelocalcondensateeffects[31,32].
•workintheso-calledlocal-duality(LD)limitt =0[31].Aspecificfeatureofthislimitisthatallpowercorrections
van•ishinthislimitanddetailsofnon-perturbativedynamicsarehiddeninonecomplicatedobject–theQ2-dependent
effectivethresholds (Q2).
eff
Asimilartreatmentmaybeperformedfor,e.g.,thep 0 gg transitionformfactor[33–35]forwhichoneobtains
∗
→
thesinglespectralrepresentationintheLDlimit:
s¯eff(Q2)
Fpg (Q2)fp = dss pert(s,Q2) (24)
Z
0
Due to properties of the spectral functions D (s ,s ,Q2) and s (s,Q2), the form factors obey the factorization
pert 1 2 pert
theorems
Fp (Q2) 8pa s(Q2)fp2/Q2, Fpg (Q2) √2fp /Q2, fp =130MeV (25)
→ →
assoonastheeffectivethresholdssatisfy
seff(Q2 ¥ )=s¯eff(Q2 ¥ )=4p 2fp2. (26)
→ →
Remarkably,duetotheQCDfactorizationtheoremsforthehardformfactors,theeffectivethresholdsatQ2 ¥ are
→
giventhroughthedecayconstantsoftheparticipatingmesons.Itshouldbeemphasizedthattheonlyfeatureoftheory
relevantforthispropertyofs (Q2)isfactorizationofhardformfactors.
eff
1 Theeffectivethresholdsforthebaryonformfactorsarestronglysensitivetothechoiceoftheinterpolatingcurrentforaspecificbaryon.
For finite Q2, the effective thresholds s (Q2) and s¯ (Q2) depend on Q2 and differ from each other [36, 37].
eff eff
Nevertheless,settings (Q2)=s (Q2 ¥ )forallnottoosmallQ2 [27]providesanapproximateparameter-
rmeff rmeff
→
free predictionforthe formfactorswhichis becomingincreasinglyaccurateas soonasQ2 increases.The resultsof
[37]giveconvincingevidencesthats (Q2)ands¯ (Q2)areclosetotheirasymptoticvaluesalreadyatrelativelylow
eff eff
valuesQ2 4 8GeV2.
≈ −
Thus,theLDapproximationfortheformfactors—whichrequiresasitscrucialingredienttheknowledgeofO(1)
and O(a ) double spectral densities—is increasingly accurate in the region not too close to zero recoil. The LD
s
approximationisverypromisingfortheapplicationto,e.g.,heavy-to-lightweakformfactors.Astillmissingingredient
hereis the two-loopO(a ) doublespectraldensityof the trianglediagramfordifferentcurrentsand arbitraryquark
s
massesintheloop.Thisisareallychallengingcalculationwhichhoweveropensthepossibilitiesofveryinteresting
applications.Sofartheonlyknownresultscorrespondtoallmasslessquarksintheloop[38]andtoHQET[39,40].
SUM RULES FORTHEEXOTIC POLYQUARKCURRENTS
TheOPEforthecorrelationfunctionsoftheexoticpolyquarkcurrentsinvolving4(or5)quarkfieldsofthetype
D(x)=q¯ (x)Oˆq (x)q¯ (x)Oˆq (x) (27)
1 2 3 4
where Oˆ is an appropriate combination of the Dirac matrices and possibly also of the (covariant) derivatives, have
specific features compared to the OPE for the bilinear currents of the form j(x) = q¯ (x)Oˆq (x) used for usual
1 2
“nonexotic”mesons.Namely,the lowest-orderO(1) contributionto the OPE forany correlatorinvolvingthe exotic
current,e.g. P = 0T(D(x)D(0)0 , is givenby the disconnecteddiagrams.As knownfromthe generalfeatures
DD
h | | i
oftheBethe-SalpeterequationandalsoemphasizedrecentlybyWeinberg[41],thesedisconnecteddiagramsarenot
related to the exotic boundstates. The connecteddiagramsrelevantfor the exotic states emergein the OPE forany
correlatorattheorderO(a )andhigher;thereforefortheanalysisoftheexoticstatestheknowledgeoftheradiative
s
correctionsis mandatory.This makesthe analysisof the exotic states a more technicallyinvolvedproblemthan the
analysisofthenormalhadrons.
Nevertheless,duetothefactthattheobservedexoticstatesarenarrow,theprocedureofextractingtheirparameters
fromtheOPEhasthesamefeaturesandthesamechallengesasforthenormalhadrons.Ourexperienceintheanalysis
oftheusualhadronsprovesthatatruncatedOPEforthecorrelationfunctiondoesnotallowonetostudyatthesame
time both the existence of the isolated groundstate and of its properties. However,if the mass of the narrowbound
stateisknown,themethodofsumrulesallowsonetoobtainreliablepredictionsforitsdecayconstantsandtheform
factors.
Structure oftheexotictetraquark states
Obviously, the exotic tetraquark states may have a rather complicated “internal” structure; two most popular
scenarios of this structure are a confined tetraquark state (i.e. a bound state in a confining potential between the
twocolor-tripletdiquarks)andamolecular“nuclear-physicslike”boundstateinthesystemoftwocolorlessmesons.
However, an important question about the structure of the exotic state—which to large extent determines also
its production mechanism—is not easy to answer [42]: (i) by a combined color-spinor Fierz rearrangement of the
tetraquark interpolating current D(x) one can write it either in diquark-antidiquark or meson-meson form; (ii) the
samequantumnumbersoftheexoticinterpolatingcurrentmaybeobtainedbydifferentcombinationsofitsdiquark-
antidiquarkormeson-mesonbilinearparts.
Thesimplestcharacteristicofausualmesonisitsdecayconstant,i.e.thetransitionamplitudebetweenthevacuum
andthemesoninducedbyitsinterpolatingcurrent;foraheavyquarkoniumstatethedecayconstantisanalogoustoits
wavefunctionattheoriginy (r=0).
Foranexotictetraquarkstateoneshouldconsiderstheconnectedself-energyfunctions
P = 0T(D(x)D(0)0 DD (28)
DD
h | | i≡h i
and study the corresponding sum rules. However, for an exotic state one may obtain a set of the decay constants,
relatedtodifferentstructureoftheinterpolatingcurrentwiththequantumnumbersoftheexotictetraquarkofinterest.
Theanswertothequestionofthedominantstructureofthetetraquarkmaybegivenonlybytheanalysisofalargeset
ofthedecayconstants.
Asthe firststep, oneneedstostudy systematicallytheinterpolatingcurrentsfortetraquarkcurrentswith different
q•uantumnembers.AsthenextsteponecancalculatethesetofP .Becauseofthefactorizationpropertyofthetwo-
DD
pointfunctionofthelocaltetraquarkcurrents[43],theradiativecorrectionstoP aregivenviaradiativecorrections
DD
tothevarioustwo-pointfunctionsofthebilinearquarkcurrents.Forsomeofthesetwo-pointfunctions(namely, VV
h i
and AA )theradiativecorrectionsarewell-known,forsomeofthem(suchas TT ,T isthetensorbilinearcurrent)
h i h i
thesecorrectionsshouldbecalculated.
Then, the set of the sum rules for different two-pointfunctionsP should be studied and only then the answer
DD
•
aboutthestructureoftheobservednarrowexoticcandidatesmaybeobtained.Especiallyinterestingcasesherearethe
narrowchargedtetraquarkZ (4430)(JP =1+ and the width 45 MeV, valence-quarkcontentc¯cu¯d) and X(3872)
−
(JPC=1++X(3872),thewidth<24MeV). ≃
Anotherinterestingpossibility—sofarnotdiscussedintheliterature—isconsideringnonlocalinterpolatingcurrents
fortheexoticmesons.Thenonlocalityoftheinterpolatingcurrentsshouldallowonetoaccessinabetterwaysubtle
detailsofthetetraquarkstructure.
Strong fall-apartdecays ofthe exotictetraquark states
Inthelastdecade,QCDsumruleshavebeenextensivelyappliedtotheanalysisofstrongdecaysofexoticmultiquark
states(seee.g.[44,45]andreferencestherein).ThebasicobjectfortheanalysisofthesedecaysinQCDisthethree-
pointfunctionsofthetype
G (p,p′,q)= 0T(D(0)j(x1)j(x2)0 exp( ip′x1 iqx2)dx1dx2. (29)
h | | i − −
Z
Thiscorrelatorcontainsthetriple-poleintheMinkowskiregion
f f f g
G hadr(p,p′,q)= (p2 MX2)(Mp12M2MX2M)1(Mq22 M2)+··· (30)
− X ′ − 1 − 2
where dots stay for less singular terms. Here g is the three-hadroncoupling which describes the X M M
XM1M2 → 1 2
transition; f , f , and f are the decay constants of the mesons describing the strength of their their interaction
X M1 M2
withtheinterpolatingcurrent X D(0)0 = f and M j (0)0 = f (weomithereallLorentzindicesandfor
X 1,2 1,2 1,2
h | | i h | | i
simplicity neglect the spins of the hadrons and of the interpolating currents). The OPE allows one to calculate the
expansionofthiscorrelatoratthespacelikemomentafarfromthehadronthresholds.Again,theleadingcontribution
ina isgivenbyadisconnecteddiagram(seeFig.3a)whichfactorizesanddoesnotdependonthemomentumofthe
s
exoticcurrentp2 atall:
G (p2,p2,q2)=P (p2)P (q2)+a G (p2,p2,q2) (31)
OPE ′ ′ s connected ′
q q
p p
p’ p’

FIGURE3. (a)ThedisconnectedO(1)diagramwhichdoesnotdependonthevariable p2relevantforthetetraquarkproperties;
(b)Oneofthelowest-orderconnectedO(a s)diagramwhichcontributestothetetraquarkdecayamplitude.
PerformingtheBoreltransform p2 t ,whichcomprisesoneofthestepsofthesum-ruleanalysis,weseethatthe
Borelimageofthedisconnectedleadi→ng-ordercontributionvanishes.2Thereforeanyattempttoextractthetetraquark
decay amplitude from the leading-ordercontribution is inconsistent. Relevant for the exotic-state propertiesare the
O(a )correctionswhicharetechnicallyverydifficult.Thisisadifficultcalculationbutitshouldbedonebeforeone
s
mayhopetogetreliablepredictionsforthetetraquarkproperties.Sofarthesecorrectionshavebeencalculatedonlyfor
thethree-pointfunctionofthebilinearcurrentsintwocases(i)formasslessquarksand(ii)forinfinitelyheavyactive
quarkandamasslessspectator.FortheO(a )correctionstothethree-pointfunctionsG ,involvingonetetraquarkand
s
twobilinearcurrents,noresultsexistintheliterature.
Nevertheless,thecommonfeatureofallpreviouscalculationsofthesedecayswithinQCDsumrules(e.g.[45,46])
wastheattempttostudythetetraquark(andpentaquark)decaysbasingonthefactorizableleading-ordercontribution
whichintrinsicallyhasnorelationshipwiththetetraquarkproperties(whichisclearbothfromthefactorizationprop-
ertyG (p,p,q)=P (p2)P (q2)andfromthelarge-N behaviouroftheQCDdiagramsemphasizedbyWeinberg[41].
′ ′ c
Thereforetheexistinganalysesshouldbestronglyrevisedbycalculatingandtakingintoaccountthenonfactorizable
two-loopO(a )corrections.
s
Nonzeroresultsbasedontheleading-ordercorrelationfunctionmaybeobtainedonlybyatrick.Letusconsidere.g.
thedecayZ y +p . Onemakesuse ofthetetraquarkcurrent j(x)=c¯(x)c(x)u¯(x)d(x) (weagainomitthe Dirac
′ −
→
matricesforsimplicity).Thecorrespondingthree-pointcorrelationfunctionofinterestis
G (p2,p2,q2) = d4xd4yexp( ipx)exp( iqy) 0T(c¯(0)c(0)u¯(0)d(0),c¯(x)c(x),d¯(y)u(y))0 . (32)
′ ′
− − h | | i
Z
A nonzero result for the Borel transform of the disconnected zero-order contribution may be obtained by first
considering the soft-pion limit q 0, i.e. p = p, which gives for the disconnected contribution P (p2)P (0) and
′
thenperformingtheBoreltransfor→m p2 t .However,thedecayrateobtainedinthiswayisnotreallytrustworthy.
→
Wethereforeconcludethatthe“fall-apart”decaymechanismofexotichadronsdiffersfromthedecaymechanism
oftheordinaryhadronsandrequirestheappropriatetreatmentwithinQCDsumrules.Thecalculationoftheradiative
correctionsismandatoryforareliableanalysisofthepropertiesoftheexoticstates.
SUMMARY ANDOUTLOOK
In the recent years, great progress has been seen both in the calculations of the OPE series for various correlation
functionsandinthedirectionofformulatingadvancedalgorithmsfortheextractionoftheindividualhadronparameters
fromthesecorrelators.Wecouldnotdiscussallthedevelopmentsinthistalkbutletustrytomentioninthissummary
theinterestingopenissuestobeaddressedinthefutureanalyses:
Let us recall that combining moment QCD sum rules with experimental/lattice data gives the most accurate
•
estimatesoftheheavy-quarkmasses[47].
Hadronpropertiesfrom2-pointfunctions:
•
a. We have seen a visible progressin developingthe new algorithmsfor extractinggroundstate parametersfrom
the OPE of the correlatorsand gaining controlover the systematic errorsof the decay constants (finite-energy
sum rules, Borel sum rules). Although it seems impossible to predict both masses and decay constants with a
controlledaccuracy,usingthemassofthegroundstateasinput,systematicscanbecontrolled).
b. Wehaveencounteredinterestingpuzzlesintheb-sector:
(i)Theb-quarkmass4.18GeV[23]whenusedintheBorelsumrulesfor f leadstotensionwithlatticeresults
B
for f .
B
(ii) Unexpectedly strong scale-dependence of decay constants of vector mesons and of f /f even using the
B B
O(a 2)correlationfunction. ∗
s
c. Calculationofthedecayconstantsofheavy-quarkoniumstateswithinthemethodofQCDsumrulesisstillnot
fullysettled:TheproblemhereisthattheOPEforthedoubly-heavycorrelationfunctionscontainrelativelysmall
2 Weemphasizethatforthedecayofausualhadron,theO(1)contributionisgivenbyatrianglediagramwhichdependsonallthreevariables
tph2e,pp′e2r,tqu2rbaantidvethQerCefDoriendoefecdopurroseviddoesesthneotdovmaniinsahnutncdoenrtrtihbeutBioonretlottrhaensdfeocramypo2fi→ntetre;sfto.rthedecaysoftheusualhadronstheO(1)contributionof