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QUARK MASSES

J. GASSER and H. LEUTWYLER

Institutfür TheoretischePhysik, UniversitàtBern, Sidlerstrasse5, CH-3012Bern, Switzerland

INORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)87, No. 3 (1982) 77—169. North-HollandPublishingCompany

QUARK MASSES*

J. GASSER and H. LEUTWYLERInstitutfür TheoretischePhysik, UniversitdtBern, Sidlerstrasse5, CH-3012Bern, Switzerland

Received12 March 1982

Contents:

1. Introduction— historicalprelude 79 14. Quarkmass ratiosfrom othersources 1142. Origin of quark and lepton masses—QCD asan effective 15. Light quark massesfrom QCD sum rules 117

low energytheory 83 16. Heavyquarks 1253. Renormalization 84 17. Quarkmassesfrom SU(4) symmetry 1294. Chiral limit 87 18. Quarkmassesfrom grandunified theories 1325. Spontaneousbreakdownof chiral symmetry 88 19. Summary andconclusions 1366. Thevacuumexpectationvalueof c~q 89 Appendices7. Quarkmassesasperturbations 91 A. Othermultiplets: 3 3* 6 and10 1428. Massesof theGoldstonebosons 93 B. Mesonnonets 1469. First ordermassformulaefor othermultiplets 95 C. Improvedchiralperturbationtheory(ICPT) 149

10. Higher ordertermsin thequark massexpansion 98 D. The pion—nucleono-term 15411. Electromagneticcontributions— renormalizedself-energy 101 E. What if (0It~q~0)vanishes? 15712. Energyof thephotoncloud 103 F. A quantum mechanicalmodel for p—u mixing 15813. Quarkmassratiosfrom mesonandbaryon masses 110 References 162

Abstract:We review thecurrentinformationabout theeigenvaluesof thequark massmatrix. Thetheoreticalproblemsinvolved in a determinationof the

runningmassesm~,md, m~,m~andmb from experimentarediscussedwith theaim of gettingreliablenumericalvaluesequippedwith errorbarsthatrepresentaconservativeestimateof theremaininguncertainties.

* Work supportedin part by SchweizerischerNationalfonds.

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J. GasserandH. Leurwyler,Quark masses 79

1. Introduction — historicalprelude

The earliest information about quark massesderived from current algebra, although in thisframework the quark mass problem showedup only implicitly in the symmetry propertiesof theHamiltonian,i.e. in the commutationrules involving the currentsand the energy-momentumtensor.Partial conservationof vectorand axial currentsimplies that the piece of the Hamiltonian that is notsymmetricunderSU(3)x SU(3) is small. The breakingof SU(3) wasidentified as an octet term in theHamiltonianvery early[1]. The structureof chiral symmetrybreakingcould howeveronly be uncoveredonceit was recognizedthat the symmetry generatedby the axial currentsis not realizedas a normalsymmetry of the states, but is spontaneouslybroken by an asymmetric vacuum [2—6].A furtherimportant step that had to be takento understandthe structureof the symmetry breakingtermsconcerns isospin symmetry. For a long time the strong interaction was supposedto be isospinsymmetric;the observedisospin breakingswere attributedto the electromagneticinteraction.Severalpuzzleshoweveroccurredin this picture:the sign of the proton—neutronandK~— K°massdifferences,the decay~ —~3ir andthe phenomenonof p — w mixing. A first steptowardsa solutionof thesepuzzleswas the tadpolemechanism[7], which associatedthe bulk of the electromagneticself-energieswith anoctetoperator.The origin of the tadpoleremainedmysteriousfor quite sometime. Within the quarkmodelanoperatorof thistype is generatedby the electromagneticself-energydiagramsof the u anddquarks.The quark self-energiesarehoweverproportionalto e2m~and e2md—too small to explain thesizeof thetadpoleterm.Theprincipalsourceof isospinbreakingfinally turnedout not to bethe photoncloudthat surroundsthe particlesbut the fact that the u andd quarkshavevery differentmassesevenifelectromagnetismis switchedoff. (The smallnessof thesequark massesmakesit likely that they areentirely due to an electroweakradiativecorrection determinedby a larger fermion mass scale— themassesm~andm~may well turnout to beof ordere2 in a morecompletetheory.This doesnot concernushere;thepoint we are discussingis that otherdegreesof freedomthan light quarksandphotonsmustbe involvedif oneis to understandthe size of m~— md.)

The role of symmetriesand their breaking through mass terms was clarified by Wilson [8] and itgraduallyemergedthat the part of the stronginteraction Hamiltonian that breaksthe symmetriesofcurrent algebrahasall the symmetrypropertiesof a quark massterm.

In QCD thisstructureof theHamiltonianis built in: the only renormalizablesymmetrybreakingtermthat this theory admits is a quark mass term. The parametersthat break SU(Nf) X SU(Nf) are theeigenvaluesof the quark massmatrix: m~,m~,m~ The strengthof isospinbreakingis measuredbym~— m~,thestrengthof SU(3)breakingby m~— ~(rn~+ md); chiral SU(2)X SU(2) is brokenby m~and~1d etc.

On the basisof currentalgebrasymmetriesaloneit is not possibleto determinethe absolutesize ofthe light quark masses,but it is possibleto determinethe two ratios m~:rnd: m~.We will discussthepresentstatusof the subjectin sections13 and14.

In someof theearlyworkson thequarkmodel thequarksweretreatedasveryheavyin orderto explainon kinematicalgroundswhy theboundstatesarenot ionizedin highenergyreactions.QCD indicatesthatthispictureis wrong,that confinementof colour is adynamicalphenomenonthatpersistsevenif thequarkmassesare put equal to zero. In order for a masslesstheory to makesenseand to representa goodapproximationto thisworld it is crucialthatthistheorybreaksconformalinvariance[9,101. It wasrealizedveryearlythatthetraceof theenergy-momentumtensor(which measuresthebreakingof scaleinvariance)containsa largepiecethat is symmetricunderSU(3)x SU(3).The origin of this term howeverremainedmysterious[seee.g. 11]. In QCD the traceof the energy-momentumtensorcontainsananomaly[12—161

80 J. Gasser and H Leutwyler, Quark masses

proportionalto the squareof the gluon field strength:

= +{1+ y(g)}{muüu + mddd+. . .}. (1.1)

The trace anomaly is a consequenceof the fact that even though the coupling constant g isdimensionless,a changein scale requiresa changein the value of g, determinedby the f3-function.Conformal invarianceis brokenab initio by the renormalizationgroup invariant scaleA (dimensionaltransmutation).

Nambu and Jona-Lasinio[3] had pointedout that in a theory basedon spontaneouslybrokenchiralsymmetry the barefermionmassesm

0 neededto producethe properpion mass must be surprisinglysmall, a roughestimateindicating m0 ~ 5 MeV. Variousestimatesof thesize of thesemasseswere givenin early paperson the subject [4,17, 181. Okubo [191analyzedsum rules for the two point functionsinvolving scalar, vector, tensor,axial andpseudoscalardensities,saturatedthem with the lowest lyingphysicalstatesandobtained

~ m~—7MeV, m~—=156MeV. (1.2)

He alsoobservedthat thesevaluesareconsistentwith the assumptionthat the massdifferencebetweenbaryonsis given by the differenceof the quark massestheycontain (additivity rule). Within QCD thesum rules usedby Okubo are divergent and saturationby lowest intermediatestatesdoesnot makesense[20].Onehasto work harderalongthe lines suggestedby Vainshteinetal. [21]to estimatequarkmassesfrom QCD sum rules.

A different frameworkthat allows oneto determinethe light quark masseswasproposedby oneofus [221.Using SU(6)-symmetryto relate the matrix elementsof the vector current to those of thepseudoscalardensity

(01fly,~dIp~)= ~ f,. = (204±11)MeV

(OIü iysdI~)= f~M~ ~ = (131.9±0.1)MeVm~+ m~

we obtainedthe relation

3th(M~-th)f~=(M~-th2)f~ (1.3)

which gives

th ~(m~+ md)= 5.4MeV (1.4)

for the mean mass of u and d. Using SU(3)-symmetryfor the operatormatrix elementsthe modelfurthermoreimplies the relations

M~—4th2M~—(m~+th)22th - m~+th

M~- M~= 2(MK*- M~)= (ms- th)(1 +fjf~) (1.5)

SSSSSSS

J. Gasser and H. Leutwyler, Quark masses 81

which allow one to extracta valuefor the strangequark massin two independentways,with the result

m~=125MeV and m~=153MeV

respectively.Thefirst relationis aslightly modified versionof the Gell-Mann—Oakes—Renner—Glashow—Weinbergformulaexpressingthe ratio M~: M~in termsof SU(2)x SU(2) to SU(3)x SU(3)symmetrybreakingparameters.Thesecondrelationis a modified versionof the additivity rule accordingto whichthe massesof the bound statesp, K*, ~ only differ through the massesof the quarks they contain:

— M,. = 2(m.— th). The reasonfor this additivity rule to emergewithin a symmetry schemeis thefact that the symmetry relationsare consistent both with the chiral limit mquark -~ 0 and with thenonrelativisticsituationfor heavyquarks,for which additivity holdstrivially: up to correctionsof orderv21c2 the energyof the boundstateis given by the sumof the restenergiesof the quarks.

Since SU(6)hasnot beenestablishedas an approximatesymmetryproperty of the boundstatesinQCD it is not clear that the relationsamongthe matrix elementsthat follow from this symmetry arecorrect.

In the earlyseventiesa numberof differentquarkmassestimateswereobtained[23—27]on the basisof deep inelasticscatteringdata, using various hypothesesabout the short distancebehaviourof theelectromagneticcurrent.Thesehypothesesturn out not to be satisfiedin QCD. It doesnot seemto bepossibleto extractmore thancrude upper boundson the quark massesfrom deepinelastic scatteringdataat the presentlyavailableprecision.

In 1975 we reanalyzedthe Cottinghamformula for the proton—neutronmassdifferenceto obtainanestimatefor the isospinbreakingdue to the electromagneticinteraction.Using the SU(6)-valuesfor themeanmassof u andd wethenobtainedthe estimates[28]

m~=4MeV, md=7MeV. (1.6)

At first sight this substantialmassdifferencebetweenthe u andd quarksseemsto be in conflict withisospinsymmetry: why is isospin an almostperfectsymmetryof the strong interactionsif thesemassesdiffer by almosta factortwo? First, the presenceof a scalein the stronginteractionimplies thatthe sizeof the quark mass terms is not to be measuredin comparisonto zero, but in comparisonto typicaleigenvaluesof the chiral Hamiltonian— free u and d quarkswould havevery different energiesat rest,interactingquarksalwayshaveenergiesof the orderof the stronginteractionscaleandit only makesasmall differencewhethertheir massesarezeroor areof the order4 or 7MeV. The sameis true of thehugedifferencebetweenthe massesof u, d ands quarksindicatedin (1.2): as long as the typicalenergyof an s quark is large in comparison to 150MeV, this difference will not produce large flavourasymmetries.Thereis however one place wherethe large differencebetweenm~,md andm~is notshieldedby the stronginteraction.The massof the Goldstonebosonswould vanishif the quark masseswere zero; the ratio M~: M~: M~is a direct measurefor the ratio th : m~of quark masses(seesection8). This relation posesthe secondpuzzle:why doesthe ratio m~:md= 4:7 not lead to a largeisospinviolation of the Goldstoneboson masses?The reasonwhy the strong interaction hides the SU(2)asymmetryin the quark massterm evenbetter than it hidesthe SU(3) asymmetryis the fact that onlythreeof the four statesthat consistmainly of u and d quarks(uu, üd, du, dd) becomemasslessin thelimit m~= md = 0. The fourth stateremainsheavybecauseof the anomalyin the axial currentwhosepresenceis vital for an understandingof the mesonspectrum(if one e.g. ignoresthe anomalyin thecontext of the SU(6) symmetry-schemementionedabove this model goes to pieces). The isospin

82 1 Gasser and H. Leutwyler, Quark masses

violating partof thequarkmassterm,~(m~— md)(flu— dd)doesnot haveanymatrixelementsbetweenthethreeI = 1 Goldstonemodes,it only producestransitionsfrom thesemodesto the I = 0 combinationflu + dd. Thesetransitionsare howeverinhibited by thelargemassdifferenceproducedby the anomaly.[Thepresenceof s quarksallowsthecombinationflu + dd to gopartly into the~-mesonandthustoremainat a relatively low mass.The mixing ir° — ~ doesproducea tiny secondorder effect in M,,~— M,rO

proportionalto (m~— md)21(m.— th) (seee.g. [29])which is howeverdifficult to separatefrom genuinesecondorderperturbations(section12).] Neverthelesstherearesizeableisospinasymmetriese.g. in theg-termsof irN-scattering[30],but thesmallnessof thesetermsmakesit againdifficult toidentify theeffectin the data.

Sincethe quark massesbreakSU(3)X SU(3)onemayattemptto determinetheir valueon the basisof the observedviolations of chiral symmetry. Chiral symmetry breaking effects were analyzedbyseveralauthors[31—40].Theobserveddeviationsfrom chiral symmetryareindeedfoundto beconsistentwith smallquark masses,typical estimatesgiving th -~15 MeV or th <40MeV. The smallnessof theseeffectsof coursemakesit difficult to obtainaccuratevalues.It is howevercrucialthatonedoesnotfind anylargeviolationsof chiral symmetry.In the caseof SU(2)x SU(2)adeviationof the orderof 7% is a verylarge effect [41]. For a recentdiscussionof deviationsfrom the Goldberger—Treimanrelation seee.g.Dominguez [42] and the referencesquoted therein. We briefly comment on schemesthat try toexplain someof the “observeddeviations” in terms of large quark massesof order th 40 MeV inappendixE.

In the meantimethe perturbativepropertiesof QCD had been firmly established.Georgi andPolitzer [43] pointed out that the ratios of running massescoincide with the symmetry breakingparametersof current algebra at large values of the scale ~. (In the MS schemethe ratios arerenormalizationinvariant so that this connectionholds for any value of the running scale.)The scaledependenceof the quark massreflectsthe fact that a barequark is surroundedby acloud of gluonsandquarkpairs; the energyof the cloud containedin a sphereof radiusr increaseswith r. For sufficientlysmall radiusperturbationtheoryprovidesa reliable descriptionof thiscloud (asymptoticfreedom)andhencea reliable expressionfor the running mass at large values of ~ — r1. Unlessthe scale p- ischangedby ordersof magnitudeor is shiftedinto theregion whereperturbationtheory breaksdownthevariation in the running massis rathersmall. [The valueof the effective strongcoupling constanthasrecentlybeenmeasuredin severalindependentwayswith the resulta~(20GeV2)= 0.18±0.03, a valuethat is considerablysmaller than previously claimed [44—48].With this value of a~the perturbativeexpressionsmakesenseat values of p. of the orderof 1 GeV. The changeof the running massfromp. = 3 GeV down to p. = 1 GeV e.g. is less than 25% if A ~ � 150MeV.] In the following we willalways be quoting values for the running mass at the scale p. = 1 GeV (the renormalizationgroupinvariant massis rathersensitiveto the valueof A which is not knownto sufficient accuracy).

Boundstatesof very heavy quarkshavea smalldiameterandit makessenseto usethe perturbativeseparationof self-energyeffects from gluon exchange.Within perturbationtheory the pole massMqdefinedby the start of the cut in the quark propagatorappearsto be a meaningfulquantity that isrelevant for the description of heavy quark bound states(section 16). For light quarks howevernonperturbativeeffects play a crucial role in the bound statemotion. Politzer [49] determinedtheleading nonperturbativecontribution to the effective quark mass M(p2) defined by S(p)=

Z(p2)[M(p2)—pi’.Many attemptswere madeto establisha connectionbetweenthe quark massin the Lagrangianand

the constituentquark massparameterused in phenomenologicalnonrelativisticmodels.Thesemodelsdo give a semiquantitativeaccount of the bound stateproperties,but it has not been possible to

I Gasser and H Leutwyler, Quark masses 83

interprettheir parametersin termsof the massesandof the couplingconstantin the Lagrangianwhichis supposedto bethe basisof thesemodels.Rulesof thumb such as therule that the constituentmasstobe usedin anonrelativisticdescriptionis given by thequarkmassplus a term that comesfrom the scaleof QCD and is of the orderof 300MeV are consistentwith other rules of thumb such as Mproton

~ M. ~ or p.proton ~ We will not discussanyconstituentquark massvaluesbutstick to the massesthat appearin the Lagrangian.

Thefirst determinationof light quark masseswithin theframeworkof QCD was given by Vainshtein,Voloshin, Zakharov,Novikov, Okun and Shifman [211who analyzedthe two point function for theaxial divergence.We will discussthissubject in detail in section15.

This concludesourhistoricalsurvey. Thefollowing sectionsdescribethe currenttheoreticalstatusofthe quarkmasses.

2. Origin of quark andlepton masses— QCD as an effectivelow energy theory

Thegaugetheory of the electroweakinteractionhasmadeit clear that the quark andleptonmassesdo not havethe statusof basicconstantsof nature:the massof, say,theelectrondoesnot occur in theLagrangianof the theory. (Gaugefields can only coupleto conservedcurrents;the weakV-A currentsareconservedonly if the fermionsaremassless.)The electronmassis attributednot to a massterm inthe Lagrangianbut to the fact that the ground stateof the theoryhappensnot to be symmetricwithrespectto the transformationsgeneratedby the weak currents.The electronmoves at less than thevelocity of light, becauseit has to movethrough and interactwith a condensatethat spontaneouslybreaksthe symmetryof the Lagrangian.In the frameworkof gaugetheory the valuesof the leptonandquark massesare determinedby the propertiesof the condensate.Many modelsfor the spontaneousbreakdownof gaugesymmetrieshavebeenstudied,but we still know very little about the electroweakcondensatewe actually live in. It does not seemto be possibleat this time to calculatethe fermionmasses,although someof the relationshipsamongthesemassesthat follow from simple assumptionsabout the symmetrypropertiesof the condensateareconsistentwith the facts(section18).

Evenif the quarkandlepton masseshavelost the statusof basicconstantsof nature,the successofQED showsthat theelectronmassis acrucialphysicalquantity whosevalueallows oneto accountfor alarge body of facts in a very precisemanner.The reasonfor this successof an effective low energytheory like QED is its renormalizability.The predictionsof the theory only dependon the chargeandthe massof the particle.Themechanismresponsiblefor the generationof the electronmassis irrelevantas far as low energymatrix elementssuch as e.g. the magneticmoment are concerned(providedofcoursethat the energyscale involved in that mechanismis large in comparisonwith the electronmass— the largerthe more accuratethe effective low energytheory).

At energiessmall in comparisonwith M~,M~the quarksand leptonsinteractalmost exclusivelywith gluonsandwith photons.Theweak interactions(and all otherdegreesof freedomrelevantfor thestructureof the symmetry breakingcondensate)are frozen in. The degreesof freedom that are notfrozen— quarks, leptons, gluons, photon— interact in a renormalizablemanner.The correspondingeffectivelow energytheory (QED+ QCD) containstwo couplingconstantse andg and themassesme,m,~,m~,m~,m~,m~,m~,mi,,... as free parameters.The theory is expectedto account for the lowenergyphenomenainvolving theseparticles to a very high degreeof accuracycomparableto theprecisionof QED in atomicphysics: as long as the quarkor leptonmassesare not too largethe directeffectsof the weak interactionscan safely be ignoredat low energies.

84 1 Gasser and H. Leutwvler, Quark masses

Sincethechargee is a smallparameterelectromagneticeffectsarewell accountedfor by thefirst fewtermsof an expansionin powersof e2.The first part of thispaperwill be devotedto the zeroorder termin this expansion— electromagneticeffects will be neglected.In this approximationthe effective lowenergytheory mentionedabovereducesto QCD:

£~QCD= — ~-~--~ F~F~+ ~ ~(iØ — mq)qg q=u,d,s,...

D~= 8w + iF~AaI2. (2.1)

The hadronic energylevelsare of courseonly supposedto be reproducedby this Lagrangianup toeffectsof order e2. The massof the protone.g. is expectedto comeout too low by about 1 MeV — wewill discusstheseelectromagneticcorrectionsin sections11 and 12.

For completenesswe mentionthat (2.1) is not the mostgeneralrenormalizableLagrangianinvolvingquarksandgluons.TheeffectiveLagrangianof the strong interactionspresumablyalsocontainsa smallCP violating term proportionalto FF (Such a term arisese.g. if beforediagonalizationthe quarkmassmatrix hasacomplexdeterminant.)We will not discussthe connectionbetweenthe quark massmatrixandthe weakangles(Cabibbo,Kobayashi—Maskawa)and disregardthe small CP violating term in theLagrangianof QCD.

3. Renormalization

The Lagrangian of QCD contains the bare coupling constant g°, the bare quark massesm°~,ms,. . . andsomeultraviolet cutoff A that regularizesthe divergencesencounteredin the expansionin powersof g°.The theory is fully specifiedby the values of the bareconstantsg°,mc,... onceasuitableregularizationprocedureis chosen.In principle, the renormalizationprogramis straightfor-ward: one calculatesquantitiesof physical interest(say eigenvaluesof the HamiltonianHOCD) in termsof thebareparametersat agiven, largevalueof A. Oncea sufficientnumberof physicalquantitieshavebeendetenninedas functionsof the bareparametersoneinverts the result and expressesthe bareparametersin termsof physicalquantities,alwaysworking at somegiven, largevalueof A. Finally, oneusestheseexpressionsto eliminate the bare parametersin all other quantitiesof physical interest.Renormalizabilityguaranteesthat this operationat the sametime alsoeliminatesthe cutoff.

In QED thereis a very natural set of physical quantitiesin terms of which to expressthe bareconstants: the chargeand the mass of an isolated electron. This choice is by no meansof crucialimportance. Even if isolated electronswere not available (i.e. even if electronsoccurred only inpositroniumboundstates)the renormalizationprogramwould go throughjust as well: the massesoftwo suchboundstatelevelscouldserveas physicalparametersin termsof which the bareconstantsofthe theorycouldbe calculated.The crucialpoint in the renormalizationprogramis that one is abletoestablish a quantitative connection between the constantsin the Lagrangian and some physicalquantities.Sinceoneis able to calculatethe positroniumlevelsin terms of the bareconstantsof QEDalmostas well asthe chargeandthemassof anisolatedelectron,the renonnalizationprogramwouldgothrough almostaswell if therewere no measurementson isolatedelectrons.


In QCD confinementpreventsus from observingisolatedquarks— the colour and the mass of anisolatedquark arenot availableas physicalquantitiesin termsof which to expressthe bareconstantsofthe theory.There are various observablesthat one may use instead.The most obviousquantitiesofphysical interest, the massesof the bound states,have recently becomeaccessibleto a quantitativecomparisonwith the parametersof the Lagrangian.In the frameworkof latticecalculationsit appearstobe possibleto calculate the lowest mesonicand baryonic eigenvaluesin units of the inverse latticespacingA = a1 asa function of the barecouplingconstantandof thebarequarkmasses.Invertingtheseresultsoneobtainsvaluesfor the barecouplingconstantandfor thebarequarkmasseswhich seemto beperfectly consistentwith the valuesobtainedfrom othermethods.If the violationsof chiral symmetryproducedby thelatticecanbebroughtundercontrolthenthesemethodsmaywell provideuswith themostaccuratedeterminationof the parametersof QCD.

A different set of observablesis relatedto high energyprocessesor to the boundstatepropertiesofvery heavyquarkswhich playthe role of testchargesprobing the structureof the QCD Lagrangian.Forobservablesof thistype the relationto the parametersappearingin the Lagrangianmaybe establishedwithin perturbationtheory.

Perturbationtheory also fixes the cutoff-dependenceof the bareparameters:in order that a changein the valueof the cutoff A doesnot modify physical quantitiessuch as boundstatemasses,the bareconstantshaveto be adaptedto the changeaccordingto the renormalizationgroup [53—56].As A —+ ~

both the bare coupling constantand the bare quark massestend to zero, roughly g°— (ln A)1,m~— (in A)112.More precisely,the barecouplingconstantandthe barequark massesaregiven by

I 16i~1A2 / A2\$11(130)2 - / A\2v1/SoexPlf3o(go)25 — ~ ~ln~j-~ m — miklnX) (3.1)

whereA andñi, are held fixed as A —~~. The quantity A is the renormalizationgroup invariant scaleand th, is the renormalizationgroup invariant mass.The constants~o, 13i and Yo are the leadingcoefficientsof the f~-and y-functions [57—67]:

-- ~ g5 o~/3(g)— /3O16~.2 $1(16~.2)2 (g )

y(g)= Yo2~ yl(42)2+O(g)

/3o=l1—~N~ Yor~2

f3i102—~N~ y1=W—~N~ (3.2)

whereNf is the numberof quark flavours. In the MS-schemethe renormalizedcouplingconstantg(p.)andthe renonnalizedquarkmassesm~(p.)aredeterminedby the differentialequations

d

p. m(p.)= —y[g(p.)] m~(p.) (3.3)

J. Gasser and H. Leutwyler, Quark masses

Table 1Runningcoupling constantandratio of runningmassto renormalizationgroupinvariant massin theMS schemewith 3 flavours. The lasttwo rowscontainthecorrespondingvaluesof A in theMS schemewith 4 and5 flavours(to two loops, takenfrom BernreutherandWetzel [71]).For details

seetext.

A3=5OMeV A3=100MeV A3=15OMeV A3=200MeV A3=25OMeV A3=300MeV

(GeV) a,1~a) a~yL) a~(j~) a~yL) a~(js) a~l~)

0.75 0.19 0.62 0.25 0.72 0.31 0.81 0.37 0.90 0.46 1.01 0.56 1.141.0 0.18 0.59 0.22 0.67 0.27 0.74 0.31 0.81 0.36 0.88 0.41 0.951.25 0.17 0.57 0.21 0.64 0.24 0.70 0.27 0.75 0.31 0.81 0.35 0.861.5 0.16 0.56 0.19 0.62 0.22 0.67 0.25 0.72 0.28 0.76 0.31 0.812.0 0.15 0.54 0.18 0.59 0.20 0.63 0.22 0.67 0.24 0.70 0.27 0.742.5 0.14 0.52 0.17 0.57 0.19 0.61 0.21 0.64 0.22 0.67 0.24 0.703.0 0.14 0.51 0.16 0.56 0.18 0.59 0.19 0.62 0.21 0.65 0.22 0.673.5 0.13 0.50 0.15 0.55 0.17 0.58 0.19 0.60 0.20 0.63 0.21 0.654.0 0.13 0.50 0.15 0.54 0.16 0.57 0.18 0.59 0.19 0.61 0.20 0.634.5 0.13 0.49 0.14 0.53 0.16 0.56 0.17 0.58 0.18 0.60 0.19 0.625.0 0.12 0.49 0.14 0.52 0.16 0.55 0.17 0.57 0.18 0.59 0.19 0.61

,ñ/m(1) 1.69 1.49 1.35 1.24 1.14 1.05A4 36MeV 76MeV 120MeV 165MeV 211MeV 259MeVA5 21MeV 47MeV 78MeV 111MeV 146MeV 184MeV

with the boundarycondition that g(A), m,(A) coincidewith the bareconstantsg°,m~for sufficientlylargeA (moreprecisely,the differenceg(A)

2— (g°)2tendsto zero, the ratio m(A)/m~tendsto one).The solutionto thesedifferentialequationsreads

)_g(p.)_47rJ1/3llnL~0[(lnL) 34asp. - 4~r 130L1 f3~L L L

2 (.)

m(p.)= rn1(~L)2)~~~0{1—

2/3iyo in L+ 1 + ~

with L = ln(p.2/A2). In the MS-schemethe renormalizedcoupling constantg(p.) dependsonly on theratio p./A and is independentof the quark masses;N~is the total numberof flavours, including heavyquarks.In the low energyregion whichwe will be concernedwith the heavyquarkdegreesof freedomarehoweverfrozen.In this regiona renormalizationprescriptionthat ignoresthe heavyflavours is moreappropriatethan the MS-scheme:the decouplingtheorem [68,69] assertsthat infinitely he~yquarksdecouplefrom all quantitiesof physical interest. In the following we will thereforeusethe MS-schemewith Nf = 3. The behaviourof the effective coupling constantacross the thresholdsfor the heavyflavours was studiedby Ovrut andSchnitzer[70] andby BernreutherandWetzel [71]. Theseauthorsshowthat the valueA

3 = 130MeV in theMS-schemefor Nf = 3 is equivalentto the valueA4 100MeVin the MS-schemefor Nf = 4 andto A5 65 MeV for N1 = 5.

The numericalvalues of the effective couplingconstantandof the ratio of the running massto therenormalizationgroupinvariant massarelisted in table1 for N1 = 3 anddifferent valuesof A3.


4. Chiral limit

To appreciatethe significance of the quark massesin QCD it is important to understandthetheoreticallimit in which theseparametersareput equalto zero.In fact, the massesof the light quarksu, d and s are small in a senseto be specifiedbelow— to put them equalto zero doesnot distort thepropertiesof the theorybeyondrecognition.In section7 we will discusshow to accountfor the actual,nonzerovalues of m~,m~andm~in a perturbationexpansionaroundthe chiral limit m~ md= m~= 0.

The remainingquarksc,b,... arenot light: althoughonemayof coursestudythe theoreticallimit inwhichthesemassesalsovanish,it doesnot seemto bepossibleto recoverthe actualmassvaluesby anexpansionaroundthat limiting case.At low energiesa better approximationis obtainedif the quarksc, b,... areinsteadtreatedasinfinitely heavy. In this limit the degreesof freedomassociatedwith thesequarksfreezeand may be ignoredin the effective low energytheory. For this reasonwe will in thissection ignore the presenceof heavy quarks altogether,and study the limit m~= m~= m~= 0. (Thediscussionwould not changein any significant way if we instead considereda different number offlavours providedall quark massesareput equalto zero.)

In the chiral limit QCD containsonly one parameter:the renormalizationgroup invariant scaleA(notethat in the chiral limit thepresenceof a CF violating term proportionalto FE hasno observableconsequences).The massof say the proton is somepure numbermultiplying A and likewise for allother physicalstatesof the theory: all massvalues of the spectrumare given by somepure numbermultiplying the proton mass.The numbersM~/M~,M4/M~,...are determinedby the theory in aparameterfree manner.In this sensethe chiral limit of QCD may be called a theory without anyadjustableparameters:QCD is of courseunable to predict the value of M~in GeV units, but itdeterminesall dimensionlesshadronicquantitiesin a parameterfreemanner.The elastic crosssectionfor pp scatteringe.g. is somefixed function of the variabless/Mtand t/M~multiplying thesquareof theproton Compton wavelength. It is most remarkablethat a theory that does not contain a singledimensionlessparameteris expectedto provide a good approximationto the hadronicspectrum.Tosolvethis theory andto verify that thesolution indeedclosely resemblesthe world we live in presentsoneof the most fascinatingchallengesin particlephysics.

The Lagrangianof masslessQCD hasa high symmetry.The ninevectorand eight axial currents

V4y~~-q, a=0,1,...,8

A~=qy~ys~q, a=1,...,8 (4.1)

are conserved— the correspondingchargescommutewith the Hamiltonian.The ninth axial current isnot conserved(60123 = 1):

8~(4y4’y

5q)= .~-~--_~~ (4.2)

[SinceFt is itself a divergence,it is possibleto write down a ninth axial currentthat is conserved.Theconsequencesof this conservationlaw are referredto as theU(1)-problem.The conservedcurrentis not


gaugeinvariant and the associatedchargemakes senseonly if the gaugefield strengthvanishesatinfinity. We assumethat the U(1)-problemdoesfind a solutionwithin QCD andwe will in the followingdisregardthe ninth axial current.For a reviewof the U(1)-problemseeCrewther[72].]

5. Spontaneousbreakdown of chiral symmetry

In the chiral limit the symmetry groupof the QCD Hamiltonianis SU(3)x SU(3)x U(1). It is veryeasyto seethat thegroundstateof the theory cannotpossiblybesymmetricwith respectto thisgroupifmasslessQCD is to be a good approximationto thisworld. In fact, considerthe vectorandaxial vectortwo point functions. If the state Ic~) is symmetric under SU(3)x SU(3) then we must have (a, b =

1,...,8)

(~lA~(x)A~(y)I4)= (~IV~(x)V~(y)~4). (5.1)

[To prove this, observethat underSU(3)x SU(3) the combinationsR = V + A, L = V— A transformlike (8, 1) and (1,8) respectively.If k) is invariant then(4IRLI~)must vanish.This implies (5.1).]

Theequality (5.1) requiresthatwheneverthereis a particleof spin0~or 1 (contributingto the righthand side of (5.1)) theremust be a massdegeneratepartnerof spin 0~or 1~(contributingwith equalstrengthto the left hand side of (5.1)). There is no trace of such parity doubling in the spectrumofhadrons.If masslessQCD is a good approximationto this world thenthe vacuumcannotbe symmetricunder SU(3)x SU(3).

Note that the equality (5.1) is true order by order in perturbationtheory. There may be a state

— the dressedFock vacuumof perturbationtheory— which is Poincaréinvariant up to a phaseandsymmetric with respectto SU(3)x SU(3) such that (5.1) holds. This state is not the stateof lowestenergy,however.The state of lowest energy, the physical vacuum 0), must be asymmetricunderSU(3)x SU(3) [73,74].

A spontaneouslybrokensymmetry calls for Goldstonebosons:if somegeneratorT~of SU(3)xSU(3)doesnot leavethe vacuuminvariant,T’~0)~ 0 thenwe musthavea physicalstateT’~l0)with thesameenergyeigenvalueas 0), becauseHQCD commuteswith T~.If T’~is a vectorchargethenthe stateT~0)describesa masslessscalar, if T’~is an axial charge then we get a masslesspseudoscalar[75].

The eight lightest hadrons(IT, K, ~) arepseudoscalars.They are the Goldstoneparticlesassociatedwith the axial charges. [That theseparticlesare not exactly masslessas it would be appropriateforGoldstoneparticlesis attributedto the fact that the QCD Hamiltonianis not really chirally invariant—

the massesof the light quarksbreak the symmetry SU(3)x SU(3). If we were able to vary the quarkmassparametersin the real world thenwe could seehow IT, K and i~indeedbecomemasslessas we letm~,md andm~tendto zero.]

Thevectorchargesarenot brokenspontaneously:(i) thereis no traceof light scalarsin the spectrumof the mesons(the lightestscalarsareheavierthan the proton), (ii) thereis very good evidencefor thecorrespondingsymmetrygroupSU(3) to be a normal symmetryof the spectrum.Hadronsdo occurinnearly degeneratemultiplets that constituterepresentationsof SU(3).

I Gasser and H. Leutwyler, Quark masses 89

6. The vacuum expectationvalue of ~q

The vacuum expectationvalue of the scalar quark operators flu, dd, ~s was identified as aquantitativemeasurefor the asymmetryof the vacuumlong ago [76].In the chiral limit the operatorsflLUR, uLdR,. . . transformlike (3*, 3) underSU(3)Lx SU(3)R.Hencetheir expectationvaluein a chirallyinvariant statevanishes.The physical ground stateis howevernot chirally symmetric andthereis noreasonfor the expectationvalues (0J~LqRI0)to vanish.As a productof two fields of dimension~ theoperators~q are of courseonly definedup to countertermsof dimensionless than or equalto 3. InmasslessQCD ~qis definedup to a multiple of theunit operator.Therequirementthat q~q~transformslike (3*, 3) fixes this multiple uniquely. To seethis, onemaysimply definethe scalarquark densitiesbythe commutatorsof the axial chargeswith the pseudoscalardensities.The commutatorof two quarkbilinearsis howevera very singularobject at short distances— the commutatorin generalexplodesinthe equaltime limit. To showthat the Gell-Mann—Oakes—Rennerrelation mayneverthelessbe derivedwithin QCD we usethe languageof the operatorproductexpansionwhich exhibitsthesesingularitiesexplicitly. For definitenesswe considerthe chargedaxial current and the correspondingpseudoscalardensity

= dy,.~y5u, P = di y5u

andwrite the operatorproductexpansionin the form

A~(x)P~(y)= ~ C~(x— y) O”(y). (6.1)

The coefficient of the unit operator,C~(z),is Lorentz covariant.Furthermore,in the chiral limit whichweareconsideringhere,the axial currentis conserved,i.e. 8~’C~(z)= 0. The generalLorentzcovariantsolutionto this constraintis

= k . zw/(z2— irz°)2 (6.2)

where k is someconstantof dimension3. Applying the sameargumentto the coefficientsof the otherscalaroperatorsflu, dd, .~s,F2,. . . we concludethat they must also have the form (6.2). In fact thescalars üu, dd and gs have the samedimensionas P~.Their expansioncoefficients are thereforecanonicalandmaybe readoff from the equaltimecommutatorin freefield theory:

C~(z)=0. (6.3)

Since QCD is asymptoticallyfree the effectsof the interactionhaveto disappearat short distances.Theoperatorsof higher dimensioncan thereforenot be accompaniedby singular coefficientsof the type(6.2)— the correspondingconstantsk vanish. Finally we observethat by a suitableredefinition of thequark scalarsof the type qq -~ qq + A) we mayabsorb the contributionof the unit operatorto obtainC~(z)= 0. This operationunambiguouslyspecifiesthe quarkscalars.With thisconventionthe operatorproductcontainsasingle scalarcontribution:

90 J. Gasser and H. Leutwyler. Quark masses

Aw(X)P~(Y)=2~4(flu+ Jd)+ nonscalarterms. (6.4)

Note that the convention does not dependon the channel used. The symmetry SU(3)L X SU(3)Rguaranteesthat the constantsk that multiply the unit operatorin operatorproductswith differentflavour or in the productof a vectorcurrentwith a scalardensityarethe same(we do not considerthesinglet currents).

To the vacuumexpectationvalue only the scalar operatorscontributeand we thereforehavetheexactrelation

(0lAw(X)P~(y)I0)= ~ (Olflu + JdlO). (6.5)

The intermediatestatescontributing to the left handside maybe describedby a spectralfunction ofthe form Pw p(p

2). Since current conservationrequiresp2 p(p2)= 0 only the pion contributes:

(0lAwl7T~)= ~f~’,pw,(0IPIir~)= g,,. (6.6)

and we get

f.,,.g,~= —(OIflu + ddlO). (6.7)

Finally, we observethat for nonvanishingquark massesthe equation8A = (m~+ md)P implies the exactrelation

f,rM~= (m~+ md)g~. (6.8)

If bothf,. and(0I~q~0)tendto finite limits as mquark—*0 thepion massmust tendto zeroin proportionto(m~+ md)”2:

M2 1urn * = — (j flu + ddIO). (6.9)mqu.~k-~Om~+ md f,~

The symmetry of the vacuum under SU(3)L±Rimplies that the expectationvalues of flu, dd and ~scoincide in the chiral limit:

(OIfluIO) = (0(ddIO)= (0~~sj0) (6.10)

and(6.9) requiresthem to be negative.Notethat the constantg,,. andthevacuumexpectationvalues(0I~qI0)dependon the renormalization

point. The transformationlaw is contragredientto the transformationlaw for the quark masses;thequantity m~(0IfluI0)is renormalizationgroupinvariant.

If the quarkmassesdo not vanish,the short distanceexpansionfor the productAw(X)P~(y)containsa term proportionalto the unit operator~-~mzwz6~. Sincethis contributionis moresingular than thecoefficient of the operatorflu + dd [see(6.4)] the simple argumentgiven above that unambiguously


identifiesthe vacuumexpectationvalueof the quarkdensitiesin the chiral limit doesnot suffice. [Thecommutator[Aw,~ containsac-numberSchwingerterm.] The spectralfunction p(p2)associatedwith(0IAwP~I0)tends to a constantat high energies(more precisely, it decreaseslogarithmically like therunningquark mass)andit is not a trivial matter to establishan unambiguousconnectionbetweenthevacuum expectationvalues (OI7qJO) and physical quantities(for a recentpaper on the problem seeNovikov et al. [77]).We shall not makeuseof the order parameters(0J4q~0)exceptin the chiral limitanddo not discussthe differencebetween(OJflulO), (OIddIO) and(O~~sI0)in the real world.

7. Quarkmassesas perturbations

In the chiral limit the massspectrumconsistsof mass-degeneratemultiplets of SU(3); the pseudo-scalaroctet is massless.If the massesof the light quarks are turned on, thesemultiplets split. Inparticular, the membersof the pseudoscalaroctet drift away from mass zero as is schematicallyindicatedin fig. 1. In this and in the following sectionswe investigatethe expansionof the hadronicenergy levels M~(A,m~,m~,m~,me,...) around the chiral limit m~= m~= m~= 0. The quark massexpansionis often referredto as chiral perturbationtheory [78,79]. The flavour asymmetriesarecausedby the quark massterm in the Hamiltonian:

“QCD = 110+ H1

H1 = fd3x {m~flu+ mddd+ m~~s}. (7.1)

_______________ ms

Fig. 1. Squareof boundstatemassasa functionof thequark mass. Schematicplot for m,= md at fixed ratio mjm~.

92 J. Gasser and H. Leutwyler, Quark masses

The expansionof the energylevels in powersof the light quark massesamountsto a perturbativeexpansionof QCD in powersof H1. (Note that H0 treatsonly the light quarksas massless,the masstermsof the heavyquarksareretained.)In lowestorder the perturbationshifts the energylevelsby

6E~= ~ . (7.2)

Using covariantlynormalizedplanewaves

(p’, nip, n) = (2IT)3 2p°83(p’— p) (7.3)

this becomes

ôE~=~-(p, nI(m~flu+mddd+m~~s)Ip,n). (7.4)

Sincethe perturbationdoesnot affect the momentumof the statewe have

= 2E~6E~= (p, nI{m~flu+ mddd + ms~s}Ip,n). (7.5)

The expansionof the squareof the mass M~I(A,m~,md, m~,me,.. .) in powers of the quark massesthereforetakesthe form

~ (7.6)

whereA~denotesthe (mass)2of the level in the chiral limit. The expansioncoefficientsB~,B~,B~.arethe matrix elementsof the operatorsflu, dd, ss in the unperturbed,symmetricstate Ip, n)

B~=(p,n~qqIp,n), q=u,d,s. (7.7)

If therewereno heavy quarks,A~would be given by somepurenumbertimes A2, the coefficientsB~,B~, B~wouldbe(renormalizationpointdependent)purenumberstimesA. In thepresenceof heavyquarksthe purenumbersbecomefunctionsof the heavyquark massratios me/A, mb/A,...

If A~doesnot vanish, the expansion(7.6) for M~maybe replacedby the linear massformula

M~= a~+ m~b’~+ mdb,, + ~ +-~

= A,~,’2, b~= ~A~112B~ . (7.8)

As long as A~is different from zerothe linear and quadraticmassformulaeonly differ by effectsoforder (mquark)2 and we have no reasonto prefer one to the other. For the pseudoscalarmesonsthequantity A~,howevervanishes.In this casea quark mass expansionis possible only for M~.Thequantity M,~(A,ma,...) e.g. hasa squareroot singularity at mquark= 0. [In the above derivation it iscrucial that the states j,, n) are normalizedcovariantly, such that the expectationvalue of the scalarmq4q is independentof the momentumof the state.It is not possibleto work in the restframe of theparticlebecausefor the pseudoscalarmesonsthereis no restframein the symmetrylimit aroundwhichwe areexpanding.]In the following we will consistentlybe working with massformulaefor M2.

I Gamer andH. Leutwyler, Quark masses 93

Note that we aredealingwith perturbationsof a degeneratespectrum:(i) In the chiral limit the masslevels constitutedegeneratemultiplets of the flavour group SU(3). (ii) In addition the presenceofmasslessGoldstonebosonsin the unperturbedsystemproducesa continuousspectrumstarting at thecommonmassof the multiplet. (Thesameeffect occursin QED whereit is dueto photonsratherthanpions.)We briefly discussthe problemscausedby thesetwo typesof degeneraciesand first considerpoint (i).

The mesonicboundstatestransformaccordingto the representations~, 3, 3* and 8 andthe baryonmultiplets constitutethe representations~, 3, 3”, 6, 8 and1S2. To apply perturbationtheory we havetochoosethe statesin such a mannerthat the perturbationdoes not producefirst order transitionsbetweendegeneratelevels.Since the perturbationhas~T3= z~Y = 0 it is convenientto usea basis inwhich 7’~and Y are diagonal.In this basismixing within a given multiplet only occursbetweenstatesthat havethesamechargeandthe samehypercharge.The only multiplet which containstwo suchstatesis the octet (IT°/~j,1~°/A).We will discussmixing of theseoctetstatesin sections8 and9.

Evenif two levelsarenot exactlydegeneratein the chiral limit theperturbationwill inducemixing ifthe perturbationis not small in comparisonwith the energyseparationof the unperturbedlevels. Amixing phenomenonof this type occursin all known light mesonmultipletswith the exceptionof theGoldstonebosons: the octetsare nearly degeneratewith the singlets (p — cv = —6 MeV, 6 — S’’ 0,A2— f = 45 MeV, g— 35 MeV). The origin of this degeneracyis connectedwith the OZI rule[80—83]:gluon exchangediagramstreatall flavours alike and hencegive all nineflavour combinationsthe sameboundstatemassin the chiral limit. In the pseudoscalarchannelnonetdegeneracyis lifted bythe annihilationdiagramsresponsiblefor the axial anomaly.In the otherchannelsthe degeneracywillbelifted e.g.if therearenearbyglueballswith the samejPc (TheOZI rule howeveralsosuppressesthecoupling of thesestatesto the ~q multiplets.) The observednonet degeneracyin the 0~and 1~multiplets suggeststhat there are no nearby glueballs with thesequantum numbers(unless theircouplingsarevery small).

In the caseof the qqq boundstatesthe representations~, 8, ~Qrequirethe space/spinwave functionsto havedifferent symmetryproperties.Thereis no reasonto expectthe differentmultiplets to occuratnearly the samemass.

The implicationsof point (ii) aremuch moresubtle: the presenceof masslessparticlesin the chirallimit leadsto infrared divergencesin chiral perturbationtheory [84].They aredueto the fact that in thechiral limit the staten + IT is degeneratewith the staten — thecorrespondingenergydenominatorin theperturbativeexpansionof M~vanishes.A carefultreatmentof the infraredsingularity showsthat thetermsneglectedin (7.6) arein generalof order (mquark)

312in the caseof the Goldstonebosonstheyareof order in ~uark In mquark [85,86]. We will comebackto this point in section10.

8. Massesof the Goldstonebosons

We first considerthe quark massexpansionfor the pseudoscalaroctet. In this casethe constantA~vanishesand we thereforehaveto work with quadraticmassformulae. For the pion the coefficients~ B~+and B~+were determinedin section 6—they are relatedto the vacuumexpectationvalue(OIfluPO) in the chiral limit:

M~*= (m~+ md)B + O(m ~in mq)

B = —y~-(0Jflui0). (8.1)

94 J. Gasser and H. Leutwy/er. Quark masses

Applying the sameanalysisto the kaon we get

M~±= (m~+ m5)B+ O(m~In mq)

M~o= M~o= (md + rn5)B + O(rn~ln mq) (8.2)

with the sameconstantB (recall that in the chiral limit f1. = fK, (OIfluIO) = (OiddiO) = (0Is~sI0)).If rn~� md the states ir° and ~ mix: the perturbationH1 generatestransitions between the

unperturbedoctetstates IT°)oand n)~.The eigenstatesare

IT°)= cos 0 1IT°)o+sin 0 ij)0

In) = —sin 0 IT°)o+ cos 0 n)o

m,J—rn

tg 26 = ~ m.— ; m = ~(m~+ md) (8.3)

andthe physicalmassesaregiven by

M~o= 2thB — ~(m.— th)B sin2 0/cos20

M~= ~(2m.+ th)B + ~(m.— th )B sin20/cos20. (8.4)

The mixing angleis proportionalto the ratio of the SU(2)-breakingmassdifference m~—m~to theSU(3)-breakingmassdifferencem~— th. Since thisratio is a smallnumber,themixing angleis small. Upto termsof order 02 the stateir° is degeneratewith ir and the massof the ~ is given by

M~= ~(2m.+ th)B+ o(~/~2, rn2 in m). (8.5)

The mixing causesthe levelsto repel.Theneutralpion becomessomewhatlighter than ir~:

M~+_M~O=~(mu_~)_B. (8.6)4 m,~—m

The ninth memberof the pseudoscalarnonet, the ~j’, doesnot becomemasslessin the chiral limit. Inthe framework of the perturbationexpansionwith respectto the light quark massesthis statedoesthereforenot play any specialrole. There is of courseno reasonfor the matrix elements(n’IHiIn) and(n’IHsIir°) to vanishand the perturbationwill thereforeinduce mixing between ~j and ‘q’ as well asbetween ir° and ~‘. The mixing angles are of first order in the quark masses (proportional torn

5—~(m~+md) and to m~—m~respectively).In the massspectrumhoweverthe presenceof the ~‘

makesitself felt only in termsproportionalto (mquk)

2 whichwe haveneglectedin the abovediscussion.[Numerically,the contribution of the i~’ is enhancedin comparisonwith other intermediatestatesbecauseit is associatedwith a relatively small energydenominator— the perturbationshiftsM~fromzero to about 1/3 of M~,.It is howeververy questionablewhether the ~p~’mixing angle may beextractedfrom the massspectrumby disregardingall othereffectsof order (mquark)2.]

Si

Si

Si

SSSS


Spontaneouslybrokenchiral symmetryexplainsat oncewhy themassesof thepseudoscalaroctetarenot at all SU(3)-symmetric(M~: M~: M~= 1: 13: 16) whereasthe massesof all othermultiplets deviateby lessthan 20% from their meanvalues: the squaresof the pseudoscalarmassesareproportionaltothe quark masses.There is no reasonwhy the quark massesshouldshowless scatteringthanthe leptonmassesdo— hence it is natural to find that the Goldstone boson massesalso show considerablescattering.

If we ignore higher order correctionsfor the moment,we may extractthe quark massratio rn5: theither from the ratio

M~: M~= (rn. + th) : 2th —* rn5: th = 25.9 (8.7)

or from

M~:M~~= (2rn5+ th):3th —* m5:th = 24.3. (8.8)

The fact that the two determinationsagreeremarkablywell showsthat the Gell-Mann—Okuboformulais well satisfiedby the pseudoscalaroctet.Theasymmetriesof the massvaluesin this multiplet requirethe s-quarkto be about25 times as heavyas the averageof u andd.

The massformulaefor K~andK°alsoindicatethat in order for theK~to belighter thanthe K°theup-quarkmust be lighter thanthe down-quark.To extracta reliablevaluefor the ratio (m~— rnd): (rn~+

md)from the observedvalueof (M~+— M~o): M~.we howeverneedto correctthesemassformulaeforthe energyof the photon cloud which makes a significant contribution to isospin breaking massdifferences.This will be donein sections11 and 12.

9. First order massformulae for other multiplets

The generalform of the quark massexpansionfor anyhadroniclevel wasgiven in section7. To firstorder the expansioninvolves four unknownconstantsA, Bk’, B”, BS for eachoneof the states.We nowexploit the fact that the unperturbedHamiltonian is symmetricunder the flavour group SU(3). Oneconsequenceof this symmetry is that the constantsA~are the samefor all membersof a givenmultiplet. Furthermore,the symmetryrelatesthecoefficientsB~,B~andB~,which standfor the matrixelementsof the operatorsflu, dd and&~s in the variousstatesof the multiplet. Considere.g. the baryonoctet. In this casethe B-coefficientsinvolve threeindependentmatrix elements(F-coupling,D-couplingandsinglet).We usethis propertyto expressall B-coefficientsin termsof theproton matrix elementsofflu, dd and.~s:

Bd = (pI~dIp)

Bs= (pj~s~p) (9.1)

Thefirst order massformulaefor the baryonoctet then read

96 J. Gasser and H. Leutwy/er, Quark masses

M~= A + muB’~+ mdBd+ msBs+ 0(m312)

M~= A + m~Bhl+ mdB + msB5+ 0(m312)

= A + rnuB’’ + mdBs+ msBd+ 0(m312)

M~-= A + m~B5+ mdB’’ + rn5B’~+ 0(m

3t2)

M2~o= A + muBd+ mdB + m5B” + O(m3/2)

M’s-- = A + muBs+rndB”+ m5B’1 + 0(m312). (9.2)

ThestatesZ°andA aremixed accordingto (8.3) with the samemixing angle.Neglectingtermsof order(ma—md)/(ms— th) we obtain

M~o= A + th(BU + Bs)+ m5B’’ + 0(m

3t2)

~ (9.3)

The baryonmassvaluesareleft unchangedif the quark massesand theconstantsA, B’~,B’~,BS aresubjectto the following transformations:

(a) mq*Amq, ~ A3A

(b) mq—* inq+ p., B~~-*B~ A-*A-p.(B”+ B’~+BS)

(c) inq -* Fflq, B~-~ B~+ a-, A—* A — o(m~+ ind + m5). (9.4)

This implies that only 4 combinationsof the 7 constantsrn~,m~,m5, A, B”, B”, BS arerelevantfor thebaryonmasses.The massformulaethereforeimply 4 constraintsamongthe 8 massvalues.Oneof theseconstraintsis the Gell-Mann—Okuboformula, the otherthreeare the Coleman—Glashowrelationsforthe splittingswithin the isospinmultiplets.In the absenceof informationaboutthe valueof theconstantA it is not possibleto extractquark massratios like m5:th from the observedmassesof the baryons.The sameremarkappliesto all othermultiplets whosemassin the chiral limit is not known. The massof the Goldstonebosonsin the chiral limit is known— this is why we do get information aboutthe ratiom~:th from the mass spectrum of the pseudoscalarmesons.What we can extract from the massspectrumof the othermultiplets is the ratio of differencesR = (rn.-- th): (md — m~)which measuresthestrengthof isospin breaking in comparisonwith SU(3)-breaking.We will determinethis ratio fromdifferentmultiplets in section 13 after having analyzedthe size of the correctionsdue to higher orderquark masstermsanddueto the electromagneticinteraction.

In order for the SU(3)splittings within multiplets not to exceed20% of the meanmassthe quarkmassdifferencem. — th which is responsiblefor thesesplittingsmust be small in comparisonwith thescaleof the strong interaction;sincern5 is muchlargerthan th this implies that all light quark massesm~,

1~dand rn~must be small quantitiesin comparisonwith the scaleof QCD. The sameconclusionresultsfrom the observationthat the pseudoscalaroctetcontainsthe 8 lightesthadrons:the quark massterm doesnot shift the Goldstonebosonsfar away from masszero (M~,=

To give the condition “ma, m~,,m5 -~ scaleof QCD” a quantitativeform we shouldspecify what we

meanby the scaleof QCD. Therearevarious quantitiesthat remain finite in the chiral limit andmayserveas massscales:A, f,r’ M,,, M~,(a’)~

2,(PT) In fact, thesescalesare not independent,but are

J. Gassera~H. Leuovyler, Quark masses 97

given by somepure number of order one— determinedby chiral QCD — times A. The scaleA itselfwhichis closestto theLagrangianturnsout to be too small to serveas a standardwith respectto whichto measurethe size of the SU(3) asymmetryparameterm~— th. The quantity f,,. is also too small. A4.seemsto bemore adequatealthoughthereis no particularreasonto preferthis quantity to (a’)~2or toM~.

The eightfold way is an approximatesymmetryof the stronginteractionnot becausethe quarksu, d,s havesimilar masses,but becausetheir massdifferencesaresmall in comparisonto M,. Isospinis anevenbettersymmetry,not becausem~— ind is small in comparisonwith m~,but becausem~— md is verysmall in comparisonto MV,.

The massof the charmedquarkis not asmall quantity in comparisonwith the scaleof QCD: the nc

is considerablyheavierthan the n’ (M~= 10M~,),the charmedbaryonA~(2273)is morethan twice asheavyasthe proton.If the massof thecharmedquark is varied from its actualvaluedown to m

5, theA~loseshalf its weight andmovesto A(1115); if m~is diminishedfurtherandput equalto th, theA~endsup atthe nucleonmass.The differencem~— th must thereforebe largerthan in. — th by a factorof theorder of 8. The difference me,—th is the symmetry breakingparameterof SU(4), the ratio (me—th): (m.— th) measuresthe strengthof SU(4)breakingversusSU(3)breaking.The size of m~— th makesit clear that one cannotget very reliableinformation from SU(4)symmetry (seesection 17 for a briefdiscussionof SU(4)massformulae).

For the multiplets 3, ~‘, ~ and.iffl the first ordermassformulaeanalogousto (9.2) may be found inappendixA.

The quark massexpansionfor the mesonnonetsis complicatedby mixing. Considerfor definitenessthe vectormesonnonet.In the chiral limit thismultiplet consistsof a singletandan octetwith aspacingof orderM0, — = 6MeV. It is not knownwhetherthe singletor the octet is the lower oneof thetwo.When the quark massesareslowly turnedon the levelsundergomixing throughanglesthat aresensitiveto the detailsof the smallpiecesof the interactionthat violatetheOZI-rule, until the strangequark hasbecomeheavyenoughfor the massterm m5gs to dominate.(Note that the quark massterm producesthe splitting Mc. — M,. 240MeV which is large in comparisonwith the original separationof singletandoctet.)The quark mass term dominatesas soonas m~is large in comparisonwith 1M1 — M816MeV. For the physical valueof m5 the state co) containspractically only strangequarks. Within theaccuracyof first order massformulaethe mixing angle definedby (we choosethe phaseof the state~o)suchthat it reducesto ~s)for ideal mixing):

= —cos ~ 8, T = 0)+ sin ~I1) (9.5)

hasthe idealvalue~ = 35.3°(tg ~ = 1I\/2). In (9.5) we haveneglectedthe small componentof the state~)in the direction of the vector 8, T= 1, T3 = 0) which is causedby the isospinviolating piece of thequarkmasstermandis proportionalto m~,— ind. In fact, the transitionmatrix element(~s~qmq~flu— dd)which determinesthe size of the isospin violating componentof ~)is forbiddenby the OZI-rule. Theisospinonecomponentof cD) is thereforevery small.

The isospin breakingpiece of the perturbationdoeshoweverhavea pronouncedeffect on the statesIp°) and cv), because(i) their energyseparationis not large in comparisonwith the size of theperturbation~(m~— ind)(flU— dd) and (ii) the OZI-rule doesnot inhibit transitionsbetweenthesetwostates.The interferencebetweenp°and cv can be measureddirectly in reactionssuch ase~e—* IT~IT

by analyzingthe behaviourof the crosssection in the vicinity of the cv mass.Up to electromagneticcorrectionsthe probability for an cv to decayinto IT~IT is determinedby the ratio (mU—md):(rn5—


th). The reactione~e—~ IT~IT thereforeprovidesuswith an independentdeterminationof this ratio.The relevantformulae for p°—w-mixingare collectedin appendixB. Furthermore,we give a simplequantummechanicalmodel describingthe interferenceof two resonancesin appendixF.

10. Higher order terms in the quark massexpansion

In the precedingtwo sectionswe focusedon the lowest order termsin the expansionof the hadronmassesM~(A.m,~,ind, in

5, ma,...) around the chiral limit of the light quarks, m~= m., = rn~= 0. Asmentionedin section7, the quark massexpansionis not a simple Taylorseries,but containstermsthatarenot analyticin m,~,m~,and in5 [88].We now discussthesenonanalyticcontributionsandthen makesimpleestimatesof thesize of higher order termsin the quarkmassexpansionin orderto determinetheaccuracyto which quark massratioscan be extractedfrom the spectrumof baryonsandmesons.

For definitenesswe first considerthe massof the nucleonanddiscusstheexpansionaroundthe chirallimit of the u andd quarks, holding m5 fixed at the physical value. In this casethe first order massformulareads

M~=A+mUB”+mdB” (10.1)

whereA, B”, B’’ areunknownfunctionsof A, m~,ma,... (in the precedingsectionwe alsoexpandedA, Bu, B” in powersof in5). As shownby Langackerand Pagels[89]the next term in the expansionisnot of order (inquark)

2, but of order (inquark)312. The structureof the leading nonanalytic terms in thequark massexpansionfor the massof any particlemaybe obtainedfrom improvedchiral perturbationtheory,a methoddescribedin appendixC [90,91]. Onereordersthe quark massexpansionby summingleading infrared divergencesto all orders in the quark mass. The net result is that the leadingnonanalytictermsin M~(A,m~,md, ins,. . .) areremovedby subtractingthe lowestorderdiagram of aneffectivechiral Lagrangianwith physicalmassesin the propagators.As long as the effectiveLagrangiangives the particles the propermassesand satisfies the soft-pion theoremsit also exhibits the correctleadingnonanalytictermsin the quark massexpansion.

Improved chiral perturbationtheory tells us that to removethe leading nonanalytic piece in thequark mass expansionwe may proceedin the samemanneras in the case of the electromagneticself-energies:the main contributionto the photoncloud also comesfrom the Born term diagram(seesection 12). We merely have to replacethe electric chargeby the axial chargeanddescribepionemission in terms of derivative coupling 0~i~/f~.The reasonbehindthis simple result is that theeffectivecouplingof pionsis weakat smallmomentumk — theprobability amplitudefor the quarkmassterm to generatemore than one pion is smaller by powers of k/f,~.What counts in the leadingnonanalyticterm is only the coefficient of the leading infrared singularity and henceonly the lowestorderdiagram.

The leadingnonanalytictermin the proton massis removedby subtractingthe lowest orderdiagramdue to pion emission of an effective chiral Lagrangianwith physical massesin the propagators.If

= md= th the quantity MN definedby

‘52

A2 — a.c2 .)gA 11A4 7.4 7.4 (111 .~)\IVI N — ~ N 32IT2f2 t~lVlN, 1~’1N,JV1~) . )


in termsof the physical nucleonandpion massesMN, M1. is free from the leadingnonanalyticterm inm~,ind. The integral J(MN, MN, M1~)is defined in (C.4) andf,. = 132MeV, g.~.= 1.25; expandingthisintegral in powersof M andrewriting the resultin the form (10.1) oneobtains

M~=A+th(Bu+Bd)_~A1l2M~+.... (10.3)8irf~,.

The last term is nonanalyticin the quark masses,becauseit growswith the third power of M,,, i.e. isproportionalto (m~+ flld).

It is not difficult to understandthe origin of the nonanalyticterm. In the chiral limit the nucleonissurroundedby a cloud of virtual pionsthat extendsto infinity. As the quark massesareturnedon thepions becomeheavy and the cloud shrinks, the probability for finding a virtual pion falling off likeexp(—2M,rr). The static model provides us with a simple descriptionof the propertiesof this cloud.Beforewe look at the static model we should perhapsclarify the significanceof a languagethat useswords like mesoncloud, virtual pionsand the like in the contextof QCD. Whatwe aretalking aboutare nonanalytictermsin the quark massexpansionandwhat we areclaiming is that thesenonanalytictermsare the sameas thoseoccurringin an effectiveLagrangian,the simplestversionof which is thestaticmodel.This model describesthe nucleonin termsof a coreanda pion cloud. Themodelprovidesuswith areliableeffectiveinfraredLagrangianto theextentthatthenucleonisheavyin comparisontothepionsandthat the inverseradiusof thecorewhich plays the role of the ultravioletcutoff that occursin arelativistic effectiveLagrangianis largein comparisonwith both the mesonmassand the baryonmassdifferencedueto the quarkmassterm (to M~— M1. in the caseof an expansionin powersof m~,mdatfixed ins andto M~— MN if nonanalytictermsin in5 dueto K andn-mesonemissionareconsidered).Infact, onemayverify that the expressionsgiven for the nonanalytictermsin appendixC on the basisofimprovedchiral perturbationtheory reduceto the mesoncloudenergiesin the staticmodel if onetakesthe limit MN_SOC. [The coredistribution p(x) that occursin the static limit is the Fourier transformofthe axial vectorform factor G(—q

2).] For thisreasonwe discussthe physicsof the nonanalytictermsinthe framework of the static model. (In the numericalwork we haveused the relativistic prescriptionsgiven in appendixC.) TheHamiltonian of the staticmodel is (seee.g. ref. [92]):

H = Jd3x~[4;2+(V. 4i)2+M~çb2]+V2f JdxP(x) r V(~. T)—~MT3. (10.4)

We haveusedthe Goldberger—Treimanrelationto expressthe couplingconstantin termsof g.~.andf,r;the quantity~M standsfor the neutron—protonmassdifference.It is straightforwardto determinethegroundstateenergyto secondorder in the couplingconstant.NeglectingiXM (i.e. putting in. = ind) onefinds

F,. = ~ f d3x d3y V p(x) . V U,) exp{—M~x— yJ}• (10.5)

The expansionof this expressionfor the energyof the pion cloud in powersof M,r producesthe series1 — M.,.r+ ~M~r2— ~M~r3+~... The first term only contributesto the constantA that representsthe(mass)2of the nucleonin the chiral limit. The secondterm disappearsupon integration andthe third


term is proportionalto in,,+ mt,, and is thereforeabsorbedin the constantsB”, Bd One easily checksthat the fourth term indeedgives rise to the nonanalyticcontributiongiven in (10.3).

In the staticmodel the total energyof the pion cloud aroundthe nucleonis E,. = —185 MeV andtheindividual termsin the expansionF,~= E

0 + E1M~+ E2M~~ amount to E0 = —202MeV, E1M~=+26 MeV, E2M~= —13 MeV. (The correspondingrelativistic prescriptiongiven in appendixC givesF,,. = —121 MeV; seeeq. (C.4) with M0—*M~andM~= M~+ E,,..)

It is clear that onemay give similar expressionsfor the leadingnonanalyticterms in the quark massexpansionwith respectto all threelight quark massesm,,, md, in5. The leading contributionto thenucleonmassthencontainstermsproportionalto M~andto M~in additionto the termproportionaltoM~givenin (10.3). TakenatfacevaluetheleadingnonanalytictermproportionaltoM~shiftsthenucleonmassby —235MeV, to becomparedwith E2M~= —13 MeV. If the leadingnonanalyticcontributionwouldindeedaccountfor thosepiecesof thekaoncloud aroundthenucleonthat arenot alreadyincludedin thefirst order massformula, then we would haveto concludethat the validity of the Gell-Mann—Okuboformula, derivedon the basisof first order massformulae,was an accident.

Fortunately,this is not the case.The staticmodel (or improvedchiral perturbationtheory)makesitclear that onecan trust the expansionof themesoncloud energiesin powersof the mesonmassonly ifthe mesonis light in comparisonwith the inverseradiusof thenucleoncore,representedby the Fouriertransformp(x) of the axial vectorform factor. An expansionof the energycontributedto the nucleonmassby thevirtual K-cloud in powersof MK wouldbe admissibleif MK were small in comparisonwiththe scaleset by the axial vector form factor— this is not the case.(Note that what countshereis not thesizeof in5, it is the size of the Goldstonebosonmassesthat containstrangequarks.)The full energyofthe kaoncloud aroundthe nucleononly amountsto EK = —39 MeV in the staticapproximationandtoEK = —24MeV in the relativistic prescriptiongiven in appendixC. The formal expansionof the selfenergyobtainedby retainingthe termsproportionalto M~and disregardingthe higher order termsislarge becauseit fails to reproducethe actual value of the correspondingself-energy diagram. Theterms up to M~.represent a good approximation only in the case of the pion cloud whoseradiusis indeedlargein comparisonwith the scaleof QCD.

If the nonanalytic terms in the quark massexpansionare handledaccording to improved chiralperturbationtheory the correspondingcorrectionsto thefirst ordermassformulaearesmall. [As discussedin appendix C the singlet and octet pieces of the meson cloud energiesmay be absorbedin aredefinitionof the coefficientsA andB”, B”, B

5 if one desires.What countsis the pieceof the cloudenergythat doesnot haveoctet symmetry.]The correctionsareneverthelesssignificant,becausetheyaccountfor the flavour asymmetriesin thebaryonmassescausedby the fact that differentmembersofthe octetget an unequalshareof the mesoncloud energy.Thenucleoncloude.g.consistsmainlyof pionswhereasthe cloud surroundingthe E predominantlycontainskaons— the contributionto the mass,althoughsmall, is ratherasymmetric.

In fact, the correctionsrequired by improved chiral perturbation theory explain why earlierdeterminationsof the ratio of SU(2)versusSU(3) symmetrybreakingparameters,(ind — m~): (in. — th),from the baryon massesdid not give a consistentpicture. The effect that is responsiblefor thesediscrepanciesis quite amusing and we briefly describeit in the framework of the static model. Theenergyof the IT* cloud aroundthe proton is given by

E ——~-~ ~ d3k k2G2(—k2) 106— f~.J (2ir)32cv(k)cv(k)+ z~M ( . )

SiS

I Gasser andH Leutwyler, Quark masses 101

with w(k)= (M~ + k2)”2. If we put z~M=0 this expressioncoincideswith (10.5) (up to a factor 2/3,becausewe are only looking at IT~here). As discussedin appendixC we howeverhaveto use thephysicalmassesin the energydenominator:F,, — E

0 = M,, + cv — M~and takethe neutron—protonmassdifferenceiXM into account.In the correspondingexpressionfor the energyof the IT-cloudsurround-ing the neutron the sign of E~Mis reversed.To lowest order in in,, — mi,, the chargedpion cloudcontributesa negativeamount to the neutron—protonmassdifference:

‘~ 2 r ~3, ,2r’2i ~2

L~E— z.gA~~ U ~ (10 7)f~. J (2ir)~2w(k) w(k)2

The cloud leads to an attraction of the two levels— if the massesare purified from nonanalyticcontributionsthe neutron—protonmassdifferencebecomeslarger. In the staticmodel the shift amountsto ~E = —0.84MeV. We emphasizethat the contributionwe aretalking about is not nonanalyticbyitself — it is proportionalto in,, — m,~,andwe could just aswell removeit by asuitableredefinition of theconstantsB”, B”. That would indeedguaranteethat thereis no differencebetweenM,, — M~andthepurified massdifferenceM,, — M~to first order in m~— md; the effect would then howevershow up e.g.in the mass difference M~— M

1, becausethe contributionof the mesoncloud to the mass operatorcannotbe representedas the sum of a singlet andan octet. To leadingorderthe determinationof theratio (md— m,,):(m5—th) is of coursenot affectedby shuffling of clouds (the procedureonly modifiesthe quark massexpansionof the baryonmasseson the level of (inquark)

2).

There are of coursealso nonanalyticterms in the quark mass expansionof the Goldstonebosonmasses.Since the constantg~vanishesfor this multiplet the leading infrared singularity is onlylogarithmic [93]. The correspondingnonanalyticcontributions in improved chiral perturbationtheoryaregiven in appendixC.

Finally we commenton the terms of order (inquark)2 which the abovediscussionleft untouched.Toestimatethe size of thesetermsonemaysimply comparethe resultsof an analysisof the baryonmassformulaefor M2 with an analysis of the correspondingmassformulaefor M A somewhatdifferentmethod,which alsoworks in the caseof the Goldstonebosons,is to addthe squareof the sum of thequark masseswhich the particle contains[(in, + in

2)2 for mesons,(in, + m

2+ in3)2 for baryons] to the

massformula for M2 andto checkthat this modification doesnot substantiallyaffect the results.We donot claim that this simple recipeaccuratelydescribesthe termsof order (inquark)2 in the quark massexpansion,but we do believethat it providesuswith an estimateof their orderof magnitude(notethatin the limit of very heavyquarksthe boundstatemassdoestendto the sum of the quarkmasses).

11. Electromagnetic contributions — renormalized self-energy

Theprecedingsectionsdealtwith pureQCD, the electromagneticandweak interactionswere turnedoff. In this approximationthe particles IT, K, N, A, 2~,E arestable.Note that even in the absenceofelectromagneticinteractionsthe proton is lighter than the neutronif in,, < ind. (Thereis no reasonwhyin, shouldbe equalto md if e = 0.)

If the electromagneticinteraction is turnedon, the quarksstart emitting andabsorbingphotons.TheIT0 and the ~ becomeunstableagainst the decays IT0—~yy 10_S fly. The cloud of virtual photonssurroundinga boundstateof quarkscontributesto themassof the state.Theorderof magnitudeof this


contributionis given by thefine structureconstanttimesthescaleof the stronginteraction,i.e. by about1 MeV. It turns out that the isospin breakingeffectsproducedby the massdifference in, ~ m~,happenalso to be of this order of magnitude— we do not know why this is so. In order to extractinformationaboutthe size of themassdifferencem,,— ind from the observedmassspectrumit is thereforenecessaryto first correct the data by subtractingthe self energyassociatedwith the photoncloud: to apply themassformulaeestablishedin the precedingsectionswe needto know wherethe hadroniclevelswouldbe found in a world with e = 0.

To lowest order the effective Lagrangiandescribingthe contributionsfrom virtual photonsis givenby

~e.m. _~e2Jd4yD(x_y)Tj~(x)j~(y) (11.1)

whereD(z) = [i4IT2(z2 — iE)]’ is the photonpropagatorandj,~(x)is the electromagneticcurrent

j~(x)=~Qy,.5q.

(Q is the chargematrix with eigenvalues2/3 and—1/3.) The productj’~’(x)j~(y)is too singularat x yfor the aboveintegralto makesenseasit stands.Thebehaviourof thecurrent operatorproductat shortdistancesmaybe readoff from the expansion

Tj’~’(x)j~(y)= ~ C”(x — y) 0”.

Only scalaroperators0” survivethe integral(11.1).Operatorsof dimensionless thanor equalto 4 areaccompaniedwith coefficients C”(x — y) that producenonintcgrablesingularities. In QCD only theoperators~, flu, dd, ss,... and F~F~’~”belongto this class. More explicitly, retaining only scalaroperatorsandwriting z = x — y the expansionreads

Tf’(x)j,~(y)=C’(z))+ ~ C’’(z)rnqqq+ ~ +~. .. (11.2)q~u,d,...

The coefficientsof the operatorsflu, dd,... areproportional to the correspondingquark masses— wehaveexplicitly displayedthis massfactor in (11.2). At shortdistancesthe coefficient C’(z) behaveslike(z

2)3up to logarithms,while C’(z), CShl(z),...as well as C’~(z)areproportionalto (z2)’. The termsomitted do not give rise to divergencesin the integral.Thedivergencesthusonly involve operatorsthatare already present in the QCD Lagrangian— a suitable renormalizationof vacuum energy,quarkmassesandcouplingconstantremovesthem.The renormalizedeffectiveLagrangianmaybe written as

~e.m._~e2Jd4yDA(x_y)Tj~(x)j,(y)E.)+~ ~mqqq~3F~F~” (11.3)

where.DA(z) is a regularizedphotonpropagator.Thecounterterms~E, ~ L~ind,...and~g dependon A in such a manneras to cancelthe divergencesof the integralas A —* 00~As usualthis requirementdoes not fix the counter terms uniquely; to obtain an unambiguousexpression for the effective


electromagneticLagrangian a renonnalizationprescription must be adopted.The splitting of theHamiltonian into a term that representspure QCD anda term that describesthe electromagneticcontribution dependson this prescription.This implies, in particular, that the values of the quarkmasseswhich we are attemptingto extractfrom datawill dependon the renormalizationprescription.

Fortunately,the dependenceof the quark massvalueson the renonnalizationprescriptionadoptedfor the electromagneticcontribution is very weak.To lowestorder in g the counterterm ~inq is givenby

~mq2O~mqln~ (11.4)

wherep. is the renormalizationpoint. In the caseof the up-quark e.g. a changein p. by a factor 2changesthe value of in,, by 1%~of m~,i.e. by less than 0.01MeV; for the s quark the samechangeamountsto 0.25%~of m~,i.e. to lessthan 0.08MeV. It is crucial that the electromagneticself energyofthe quark is proportionalto the quark mass— if the renormalizationambiguity in in,, was 1%~of a typicalhadronicenergyscaleratherthan 1%~of in,, the dependenceon the renormalizationprescriptionwouldbesignificant. To lowestorderin g the renormalizationof the couplingconstantis given by

= 2561~~~~Q2lflh~ (11.5)In thechirallimit thereisno quarkmasscounterterm;only theoperatorF~F~”‘producesadivergenceinthe electromagneticself-energyof boundstates.Since this operatoris proportionalto the traceof the.energy-momentumtensor[compare(1.1)] its matrix elementis known—it is proportionaltothe (mass)

2ofthe state.In the chiral limit the logarithmicdivergencein the electromagneticself-energyof any hadronthereforeamountsto a universalshift in the massscale(‘Fr Q2 = ~+ ~+ ~ =

z~M=2l6IT2 p.2 (11.6)

Thelogarithmis accompaniedby averysmallcoefficient;for theprotone.g.achangein therenormalizationscalep. by a factor 2 shifts the self-energyby only 0.06MeV. In the chiral limit the electromagneticselfenergyof the Goldstonebosonsis finite to lowestorderin e2 (M= 0 implies iXM = 0). Sincethe neutralaxial currentsremainconservedin thepresenceof theelectromagneticinteraction,IT0, K°,K°andn staymassless.Thechargedaxial currentsarenotconservedin QCD; IT andK~’obtainafinite massof ordereM,.(c.f. eq. (12.9)).

12. Energy of the photon cloud

The energyof the photoncloudis the expectationvalueof —f d3x..9~e.m..To evaluatethis expectationvaluewe needto know the matrix element(~= p q)

T(q2, ~) = ~ifd4x e’~(pjTf’(x)j~,(O)Ip) (12.1)


for the particlein question.As shownby Cottingham[94],the imaginarypart of this matrix elementatspacelikevaluesof q is relatedto the crosssectionfor electronscattering.Theinformation containedinthe structurefunctionsfor electronscatteringin fact suffices to determinethe entirematrix elementT[95] provided Tm T does not contain fixed poles [v-independentcontributions,or, more generally,contributionsof the type v”f3,,(q2) with integern]. We do not know of a proof that fixed polesof thissort areabsentin QCD, but we assumethat this is the case,such that the energyof the photoncloudsurroundingthe proton can in principle be worked out from data on elastic and inelastic electron—proton scattering.The divergencesof the electromagneticeffectiveLagrangianof coursealsoshowupin the energyof the photoncloud: the Cottingharnformula must be renormalized[96].The algorithmwhich we set up [97] to expressthe renormalizedelectromagneticself-energyin terms of the crosssectionfor electronscatteringis only valid in the limit g = 0 (canonicalscaling).To our knowledgeananalogousrenormalizedCottinghamformula that only involves the structurefunctionsfor electronscatteringand is consistentwith the high energybehaviourof thesestructurefunctionsin QCD is notavailablein the literature. [Note that if subtractedfixed q2 dispersionrelationsare used,an algorithmallowing oneto calculatethe subtractionT(q2,0) in termsof the structurefunctionsis needed.]

The constructionof a properly renormalizedCottinghamformula is of theoreticalinterestratherthanof practical importancefor the following reason: the difference between the electromagneticself-energiesof particlesbelongingto the sameisospinmultiplet, saybetweenprotonandneutronis finite inthe chiral limit m~= md= 0. The real world is very closeto this limit so that the divergenceandtheassociatedrenormalizationproblemsare associatedwith very small coefficients. The reasonfor thefiniteness of the self-energydifferences in the chiral limit was given in the last section— it is aconsequenceof the fact that the quark mass counterterms~ hind areproportional to the quarkmassesin,,, md respectivelyandhencevanishin the limit in,, = ind = 0. In fact, considerthe differencebetweenthe protonandthe neutronmatrix elementsof Tj~’(x) j~.(y). The correspondingshort distancesexpansioninvolves the matrix elementsof the operators~, flu, dd, ss,F~,.F’”,...The unit operator—the disconnectedpiece— may be disregardedbecauseit amountsto a universal energyshift for allstates,including the vacuum.The differenceof the matrix elementsbelongingto the isoscalaroperatorsss, F~,,F”‘,. . . betweenproton and neutronis proportionalto m~— ind (in the limit m,,= in, isospinisan exact symmetry of QCD, hencethe matrix elementsof isoscalaroperatorsare the samefor allmembersof an isospin multiplet) and thereforevanishesfor m~— = 0. Finally, in the caseof theoperators flu and dd the expansion coefficient is proportional to in,, and m,,, respectively; thecontribution from theseoperatorsto the short distanceexpansionhencealso vanishesin the limitin,, = ind = 0. This shows that all singular terms in the short distance expansionof the differencebetweentheprotonandneutronmatrix elementsof Tje~(x)j

1,(y)vanishin the limit m,,= = 0; in thatlimit the differenceof the electromagneticenergiesof proton andneutronis thereforefinite andmay beworkedout by usingthe unrenormalizedCottinghamformula.

The electromagneticself-energiesof baryonsand mesonswere investigatedby many authors (seeZee [98],Cmi andStichel [99]for a reviewup to 1972).We havereanalyzedthe variouscontributionstothe Cottinghamformula in 1975, usingthe data on inelasticscatteringavailableat that time (seealsoGunion [100]).We found that the only sizeablecontributionto the proton—neutronmassdifferencecomes from the Born term; resonanceregion, Reggeregion and deepinelastic scatteringcontributevery little. The fact that the deepinelasticregion doesnot contributesignificantly is closely relatedtothe observationmadein the last section: the logarithmic divergencewhich is associatedwith thesecontributionshasa tiny coefficient, even if onelooksat the proton self-energyitself ratherthan at theproton—neutrondifference,for which the deepinelasticcontributionsarefurthersuppressed.


The Born term is determined by the electromagneticform factors of the nucleon which we

approximateby the dipole formulawith F~= 0:G~(t)= G(t); GR~(t)= p.~G(t)

G~(t)=~jj~p.”G(t); G~(t)p.”G(t)

G(t) = (1- t/m2)2. (12.2)

Using the valuesin2 = 0.71 GeV2, p.P= 2.79, p.fl = —1.91 onefinds

M°~= 0.63MeV, M~O~= —0.13MeV. (12.3)

Note that for a heavy particle the magneticcontribution is small (of order M2) and the Born termreducesto the electrostaticenergyof a chargedistribution of which GE(—q2)is the Fourier transform:

M7 = ~-~4I ~-~‘- [GE(—q2)]2. (12.4)4IT j q

For a baryonof chargeQ= 1 this amountsto 0.96MeV.We have performed a new determination of the Born terms for the other members of the baryon

octeton thebasisof the availableexperimentalinformationaboutthe magneticmoments.The magneticmoments of A, 1~, 1, S° and E~ are measureddirectly: p.~’= —0.730±0.006,~ = 2.95±0.16,

= —1.80±0.32,p.~°= —1.68±0.08,p.~= —2.61±1.06 (in units of e/2MwhereMis the massof theparticle). The magneticmoment of the 1°then follows from isospin symmetry: p.~°= ~(p.~+ p.~)=0.58±0.18.Approximating the form factors again by the dipole formula and using the centralexperimentalvalues for the magneticmomentswe obtainthe resultsgiven in column 1 of table2. Notethat the10 self-energycontainstwo Born terms:oneis dueto the transition10 —S 10 + y —*1°, the otherto the transition 1°—sA + y —*1°.The secondcontributiondevelopsan imaginarypartwhich manifestsitself in the decay10 -SA + y. Theknown rateof this decayimplies the valuep.~= 2.31 ±0.26 for themagnetictransitionmoment (in units of eI2M

1). The real part of the secondcontribution amounts toMr” = —0.20MeV.

The uncertaintiesin the magneticmomentsdo not affect the self-energiessubstantially:for com-parisoncolumn 2 of table2 containsthe resultsthat oneobtainsif oneusesthe old SU(3)predictions

A 1 n £0 1 n .r ~.- ~o[101]for the magneticmoments(p. =~p., p. =p.”, p. ~ , p. zzz_(p.P+p. )p. , p.~ p.

= —~V3p.”) ratherthan the experimentalvalues. In column 3 we quote the valuesobtainedbyColemanandSchnitzer[102]who usedtwo different form factormodelsto estimatethe uncertaintiesintheir analysis.Within theerrorstheir resultsareconsistentwith theinformationabouttheelectromagneticform factorsavailabletoday.

As mentionedabovethe dataon inelasticelectronscatteringshow that the Born termdominatestheelectromagneticself-energyof the nucleon:the phasespacefactorsin the Cottinghamformulasuppressthe inelasticcontributions.We estimated[97] the size of the contributionsfrom inelastic intermediatestatesto the proton—neutronmassdifferenceto be lessthan 0.3MeV. We think that this estimatealsoapplies to the othermembersof thebaryonoctet:we assumethat the Born termsgiven in column 1 oftable2 representreliablevaluesfor the total electromagneticself-energyto within ±0.3MeV.

106 1 Gasser and H. Leutwy/er, Quark masses

Table 2Estimatesfor thephotoncloud energiesof thebaryons.Column 1 containsthe Born termscalculatedwith themeasuredmagneticmomentsandcolumn 2 givesthecorrespondingvaluescalculatedwith theSU(3) predictionsfor themagneticmoments.Column3 quotesthevaluesgiven by ColemanandSchnitzerandin column 4 we giveestimatesbasedon thesimple quark

model ansatzdescribedin thetext.

Born term Born term Colemanand Quark

exp SU(3) Schnitzer model2 3 4

p 0.63 0.63n —0.13 —0.13

0.70 0.73—0.21 —0.13

0.87 0.94

—0.07 —0.090.79 (1.95

p—n 0.76 1)76 1.1/1.4 1.0±0.4

— 2~ —0.17 —0.21 —0.7/—1,8 —0.3±0.7—0.86 —1.04 —1.25/—1.6 —1.3±0.6

.i~+ ~ — 2~° 1.98 1.92 2.1/2.0 2.4±0.5

A different estimateis basedon the quark model picture of the photon cloud [103—105].Oneseparatesthe self-energyinto a contributioncoming from photonexchangebetweentwo of the threequarksanda contributionthat accountsfor diagramsof the quark self-energytype:

M5 = a(1/r)(02Q3+0301+ Q,Q2)+C(Q~+Q~+Q~) (12.5)

where 0,, 02, 03 arethe quark charges.As discussedin the precedingsection the self-energyof theboundstatemustbe renorn1alized.Thecounterterm doesnot havethe structureof the aboveformula(in the chiral limit the divergentpart of the massshift is flavour independent).We havehoweveralsoseenthat the deepinelasticcontributionswhichproducethis divergencearetiny. Within the accuracyofquarkmodel estimatesit is not a crime to neglecttermsthat arenot of the structure(12.5).

The first term in (12.5) maybe estimatedon the basisof simple modelsfor the wavefunction, whichgive, typically

(1/r) = 330/390MeV (Itoh et al. [124])

(1/r) = 255 MeV (Isgur [125]) (12.6)

for the averageinversedistanceof two quarksin a baryon. In perturbationtheory the secondterm isproportionalto amqua,.k and hencevanishesin the chiral limit. Perturbationtheory doeshowevernotproperly accountfor the chirality changingpiece of the quark propagator;the constantC containsnonperturbativecontributionsof order a times the scaleof QCD. We do not attempt to determineareliable value for the constantC, but will be contentwith a rough estimate.We note that the entireself-energyof the proton andof the1~aregiven by C. TheBorn termestimatesfor the proton and.~

self-energiesare C= 0.63MeV and C= 0.70MeV respectively. In an electrostaticpicture the self-

SiSiSiSiSiSiSi


energyis C= ~allIrl, whereIl/ri is the meaninversedistanceweightedwith the chargedistribution.Fora chargedistribution correspondingto the dipole formula we have C 0.96MeV and hence I lIr~260MeV. We allow the constantC to haveanyvalue in the range0 < C< 1.5MeV. We considerthisestimateverygenerousas the upperlimit would requirethe Born term to representlessthanhalf of theactualvalue of the renormalizedself-energyof the proton (the direct comparisonwith the Cottinghamformula indicates that contributionsfrom intermediatestatesother than the proton are substantiallysmallerthan the Born term).

The estimatesfor (lIr) and C given aboveleadto the valuesgiven in column 4 of table2. Within theerrors that are of the orderof 0.5MeV the simple quark modelestimatesdescribedaboveagreewiththe Born terms.

The literaturecontainsseveralquark modelcalculationsof the photoncloud energythat are moresophisticatedanddo not rely on the ansatz(12.5) but insteadusedetailedquark wave functions(refs.[108—119];for more recentwork see refs. [120—125]and the referencesquotedin thesepapers).Inparticular, the magneticinteractionamong the quarkshasbeenestimated.Note that some of theseestimateslead to largeelectromagneticcontributionsto .~ — 1 andto E°— E. [Therule accordingtowhich the quark magneticmomentsare given by e/2mconstituent with m~0~5,(u)= 350MeV, in ~ =

550MeV imply large SU(3) asymmetriesin the magneticself-energy.]In the following we shall notmakeuseof the quark modelestimates,but usethe experimentalinformation from electronscatteringon protonsandneutrons,extrapolatedto the othermembersof the baryonoctet in what we consideraconservativemanner.

Our final estimatesfor the photon cloud energiesof p—n, 1~— .~ and S°— are given incolumn 2 of table 3: we use the Born terms with the measuredmagneticmomentsand assesstheuncertaintiesdue to inelastic contributions at ±0.3MeV. Subtracting this from the observedmassdifferenceswe obtain the correspondingvaluesin pure QCD (column 3 of table3). In the caseof thecombinationz~M.r= M1+ + M1- — 2M1o the analogousestimateof the electromagneticself-energygives1.98±0.30MeV, whereasthe simple quarkmodeldescribedaboveleadsto 2.4±0.5MeV. The fact thatthese values agree with the observedmass difference (~M1)~’~’= 1.78±0.14MeV is a welcome

Table 3Isospin breaking mass differences. For p — n, — .~ and .~ — the elec-tromagneticself-energiesareestimatedwith theCottinghamformula: thecentralvalue is the Born term, the error bar is an estimate of the size of inelasticcontributions. For K~— K°the electromagneticself-energy is calculated fromDashen’stheorem,for D~— D°we haveuseda simple quark modelansatz.Column3 is thedifferencebetweencolumns1 and2, exceptfor ~ + .~ — 2.~°and~ —

wherecolumn3 is the input: in pure QCD thesemassdifferencesareverysmall, oforder (rn, — md)

2. (For direct estimatesof thecorrespondingelectromagneticself-energiesseetext.)

Experiment Electromagnetic QCD1 2 3

p—n —1.29 0.76±0.30 —2.05±0.30— .~7 —7.98±0.08 —0.17±0.30 —7.81±0.31

—6.4±0.6 —0.86±0.30 —5.5±0.7~ + .1 — 2Z° 1.78±0.14 1.78±0.14 ±0.02

+ — ir° 4.60 4.6±0.1 ±0.1

K~— K° —4.01±0.13 1.27±0.30 —5.28±0.33D~—D° 5.0±0.8 1.7±0.5 3.3±0.9

108 J. Gasser and H. Leutwy/er. Quark masse,s

consistencycheck:in pureQCD this combinationhasa very smallvalue of order (rn,,— ind)2. The order

of magnitudeof ~ in QCD is given by 1° — A mixing which producesthe masssplitting ~—292(M

1 — MA) = —0.015MeV. The contributionsof order (inquark)

2 andthe nonanalytictermsarealsoof this size. (The situationis very similar to the caseof IT~— IT0, seebelow.)

For the Goldstonebosons a nonrelativisticquark model analysisin terms of (1/r~does not makesense;the massdifferencebetweenthe IT~and ITt) would require(1/r~ 1.2GeV which is outrageous.We seeno reasonwhy a model that fails for the pion shouldbe valid for the kaon.

In the chiral limit currentalgebraimplies the low energytheorem [126]:

- dssln1~-[p’~(s)~pA(s)]; (M~o)~= 0 (12.7)

wherep” andp’~arethe vectorandaxial vectorspectralfunctionsnormalizedas

pV(~)f2 6(M~—s)+~ (12.8)

with f,, = (204±11) MeV andf,. = 132MeV. On accountof the secondWeinbergsum rule (which isvalid in masslessQCD)the expression(12.7)is independentof the renormalizationpoint p.. Saturatingthe integral (12.7) with p and A

1 and eliminating the A1 parametersfA,, MA, by using the analogoussaturationof the Weinbergsum rulesone obtains

/f\2 f

2(A4’2~\Y—_.~. A,f21L~L~1 -~° ‘129k ‘r) ~ P,,C J f2_.~2

\Ji~/ Jp J-,r

which gives ~ = (4.9±0.2) MeV for the value of f~quotedabove.The result is very closeto theobservedmass difference M.,,.+ — M~.o= 4.60MeV. [The error bar does not account for the errorcommittedin approximatingthe spectralfunctionsp~”and~A with single narrow resonances.A moreaccurateanalysis of the p-contribution that includesfinite width effects leadsto the value M~+=

6.1±0.8MeV[127].]If the quark massesareallowed to be different from zero,perturbationtheory showsthat the sum

rule (12.7) divergeswith a coefficient proportionalto am2. In fact the operatorproductexpansionforthe difference V~.V,. — A,~A,.containsa term proportional to m~q— the integral occurring in (12.7)thereforeinvolvesadivergenceproportionalto am(Oi4qiO).The formula (12.7) itself is only valid up toterms of order am. The correctionsappearingin a renormalizedversion of this formula are howeverexpectedto be tiny, becausethe massesin,,, ind arevery small in comparisonto the scaleof QCD. [Arenormalizedformula would be of limited practical use, becausethere is no direct experimentalinformation on pA(

5) and it is not easy to improve the rough approximation by a single narrowresonanceusedin (12.9).]

A direct evaluationof the electromagneticself-energyof the pion on the basis of the Cottinghamformula, using theoreticalmodels both for the elastic and inelastic contributions was performedbySocolow [128].Approximatingthe Born term with a p-dominatedform factor F(t) = (1 — tIM~)~onefinds

(M,,.c — M.,.o)B0 fl = 4 3 MeV (12 10)

J Gamer and H Leutwyler Quark masses 109

Accordingto Socolowtheinelasticcontnbutionsincreasethisto 49 MeV to within anerrorestimatedat±1MeV. [Note that the p-dominance model overestimatesthe data [129];a realistic evaluation of theBorn term with the measured form factor would lead to a smaller value.]

In pure QCD the massdifferenceM,~+— M,~°is very small. As mentionedin section8 the first ordermassformulacontainsacontributionof order (inn — m~)/(m,— th):

M,,+ — M,,o = M~+— M~o+~(n~~~~)2 (!!!.i_ 1)M~ (12 11)

The contributionfrom in,, � ind amountsto a tiny shift of 0.11MeV. The higher order terms in thequark massexpansionproduce corrections that are of the samesize: (i) theleadingnonanalyticcorrectiongiven in (C.7) is of order—0.05 to —0.1MeV (for p. = 1 GeV the correction cancelsthe first order term)and (ii) the correctionsof order (inquark)

2 arealso expectedto be of this size(the additivity rule leadstoa correctionof order —(in,, — ind) 12M,r —0.05MeV). We concludethat the first order massformulacannotbe trusted— it only indicates the order of magnitudeof M,,+ — M,~oin pure QCD. Up to thisuncertaintyof the order of 0.1MeV the observedvalue for M,r* — M,,.o must thereforecome fromelectromagnetism:

M~~—M~o=4.6±0.1MeV. (12.12)

This valueis consistentboth with the estimateobtainedfrom the currentalgebrasumrule (where themain uncertainty stems from the fact that the behaviour of the axial vector spectral function is notknown) and with the estimate basedon the Cottingham formula.

For the electromagnetic self-energy of the kaon current algebra implies the low energy theorem[130]:

= (M~+)”, (.M’~o)”= 0. (12.13)

Using the abovevaluefor ~ — Mo this gives

M~+—Mko= 1.27±0.30MeV. (12.14)

Dashen’stheorem is strictly valid only in the chiral limit: in the real world the relations(12.13) aresubjectto correctionsof orderam, (and amqukIn mq,,ark)which aredifficult to evaluate.The errorbargiven in (12.14) representsthe typical order of magnitudeof SU(3)x SU(3) breaking effects. [Theleading infrared singularitiesresponsiblefor the nonanalytic terms of order amquark ln inquark wereworked out by Langackerand Pagels[131].To obtain the properexpressionfor the correspondingnonanalytic termsone should use improved chiral perturbationtheory. It does not make sensetoreplaceall infraredlogarithmsby auniversalnumberinvolving somemeanmesonmass.Onefirst hastoverify that the leading nonanalyticterm indeedrepresentsa good approximationto the quantity inquestion; if this is the case ICPT implies that onehasto weigh the different contributionswith thelogarithm of the physical mass of the correspondingvirtual meson. We did not perform such acalculationfor the electromagneticself-energies.On the basisof theanalysisgiven in appendixC (in thecontext of the correspondingnonanalyticcontributions to the baryon andmesonmass formulae) we

110 I Gasser and H. Leutwyler, Quark masses

expectthat thesecorrectionswill againturn out to be small, althoughthe contributionsobtainedbyusing universallogarithmsare large. Note that the estimatesof the electromagnetickaon self-energybasedon the Cottinghamformula ~ — M~o= 2.9±1.0MeV [128] arealso considerablylargerthanwhat is implied by Dashen’stheorem. This analysis should be repeatedwith the experimentalinformationavailabletoday.]

We see no reasonwhy symmetry breaking should be unusually large in the electromagneticself-energies.In thefollowing we usethe value(12.14) basedon Dashen’stheorem.

For the D-mesonsthe quark modeldescribedabovegives

M~+— M~o= ~a(1/r~— ~C. (12.15)

Using the estimatesgiven by Lane andWeinberg[105]on thebasis of nonrelativisticpotentialmodels[(1/r) = 350 to 450 MeV] and the crudeestimatefor the constantC given aboveone obtains(seealso[106a])

M~+—M~o=1.7±0.5MeV (12.16)

whereasthe Born term with F(t) = (1— t/M~’ is

— ~ = 1.5MeV (12.17)

close to the electrostaticlimit aM0/4= 1.4MeV for a heavy particle; the correspondingmeaninversedistanceweightedwith the chargedistributionis 11/ri = ~M,,, 390MeV. For an analysisof the inelasticcontributionsto the electromagneticself-energyof the D mesonsseeBoal andWright [132].We do notdiscusstheseestimatesin detailbecauseatthepresentlevel of ourunderstandingtheuncertaintiesinvolveddo not allow usto extractan accuratevaluefor theratioR = (m.— iii): (in,, — m~)of quarkmasses.For thesamereasonwedo not discussisospinbreakingmassdifferencesofresonances.In fact,wedo notunderstandthe valuesK~— K = 6.7±1.2MeV andD+* — D~= 2.6±1.8 MeV given in the particletables:althoughthe photoncloud contributionsareexpectedto besmallerthanfor the correspondingpseudoscalarstates[107,106b]the correctedvaluesarenot in our opinionin goodagreementwith themassformula(B.8)andwith therelationanalogousto (14.13).To makeuseof the experimental information on small differencesofthissort it is crucial that oneunderstandstheelectromagneticcorrectionsappliedin the determinationofthepolepositions(seeZidell, Arndt andRoper[133]for adiscussionof theseproblemsin thecaseof the~resonance).

13. Quarkmassratiosfrom mesonand baryon masses

We are now ready to determinethe two ratios m~:md: in. from the mass formulae for thepseudoscalarmesonsandfor the ~?(baryonoctet. We first determinethe valueof the ratio m~:th fromthe massesof IT, K and ~. A comparisonof the first order mass formulaefor IT, K and for IT, 1~

(correctedfor electromagneticeffects according to Dashen’stheorem)gives in5: 01 = 25.9 and 24.3respectively.Therearetwo kindsof correctionsto be discussedin order to assessthereliability of thesevalues: nonanalyticcorrections(terms of order (inquark)

2 In inquark) andcontributionsof order (inquark)2.

I Gasser and H. Leutwyler, Quark masses iii

The leading nonanalyticcorrectionsaccountfor the self-energiesof the mesonclouds [we definetheenergyof the mesoncloud aroundthe pion, ~ as the differencebetweenthephysical massM~.andthe purified mass M,,. given in (C.6): M,,. = M,,. + z~M,,].If the cutoff p. is given a small value(p. = 0.5 GeV) the energiesof the clouds around ~ir,K and i~ are small (SM,.= —1.6MeV, L~MK=

3.4MeV, AM.,, = —8MeV). If p. is taken large (u = 1 GeV), we obtain AM,,. = +1.2MeV, AMK=—21 MeV, AM,, = —22 MeV. For p. = 1 GeV the correctionin the ratio rn5: 01 amountsto about +3units(m,:01 = 28.8 and26.8from IT, K and ir, ~j respectively);for smallervaluesof the cutoff theshift iscorrespondinglysmaller.

Considernow the corrections of order (mq,,~k)2.In the caseof pion andkaon the additivity rule

accordingto whichwe shouldaddthe term (m1 + rn2)

2 to the first ordermassformula for M2 maygiveareasonableestimatefor the orderof magnitudeof thesecorrections.For a strangequark massof order130 to 170MeV the correctiontendsto lower the valueof the ratio m

5:01 by 2 to 3 units.The termsoforder (m,,,,,,~k)

2thus act in the direction oppositeto the leading nonanalyticcontribution— the twocorrectionsroughly cancel.In the caseof the etait is not clearhow to usethe additivity rule. Onemayusethe picture that 2/3 of the time the eta consistsof two strangequarks,1/3 of the time of two u or dquarksandweigh the correction(rn

1+ rn2)2 accordingly.Thisprescriptionlowers the valueof m

5:01 byabout3 to 6 units andthusovercompensatesthe leadingnonanalyticcorrection.The correctionsto thefirst order formula preservethe octet characterof the mass operatorto a remarkabledegreeofaccuracy,the simple modificationof the additivity rule given abovedoesnot. We do not try to improveon thisprescription;insteadweobservethat the leadingnonanalyticcontributionandthe termsof order(mqu~k)

2act in oppositedirectionswith an amplitudeof the sameorder. We concludethat it makessenseto determinethe ratio rn,: 01 from the uncorrectedphysicalmasses(M~D M~P’ 135.0MeV,M~D 492.4MeV, M~D= 497.7MeV, M~~CD 548.8MeV) and to use the size of the leadingnonanalyticcontributionas an estimateof the error bar to be attachedto this ratio. In this mannerweobtain

rn5/mI1 = 25.0±2.5 (13.1)

for theaverageof the two determinations.We nowconsiderthe ratio of the SU(2)breakingpieceto the SU(3)breakingpieceof the quark mass

term, which wenumerically analyzein termsof the ratio

R= . (13.2)ind — in,,

In the languageof the symmetry breaking parametersE3 and 68 that characterizethe Gell-Mann—Oakes—Rennermodel the ratio R standsfor

~ (13.3)

In termsof this ratio the first ordermassformula for K~— K°maybe written in the form

R- M~-M~.M~o—M~÷~ (13.4)

112 I Gasser and H Leutwyler, Quark masses

Using the valuefor (MKO — MK+)0~derivedfrom Dashen’stheoremthis gives

R = 43.4±2.7. (13.5)

We again analyzethe correspondingleading nonanalyticcorrection and the contributions of order(inq,,~k)2. ThenonanalyticcorrectionshiftsR downwardsby 4 units if p. = 1 GeV (by less thanthat if p.is smaller).The phenomenonobservedin the determinationof the ratio ins: 01 occursalso here: thecorrectionof order (inquark)2, estimatedon the basisof the additivity rule, counterbalancesthe leadingnonanalyticcorrectionsuch that wemayagainusethe uncorrectedvalue(13.5) and usethe sizeof theleading nonanalyticcorrectionto estimatethe error bar to be attachedto R. The result is quotedintable 4.

Nextwe considerthe baryonmassdifferencesfor which the first order formulaemaybe written as

RM~-M,.

R- MS-MN— — M

1~

R_2(M5M~+4(M1M~~ 136- M~—-M~-o

providinguswith threeindependentmeasurementsof the ratio R. The threerelevant isosopinbreakingbaryon massdifferences,correctedfor electromagneticself-energies,are given in table 3. With thesevaluesthefirst order massformulaegive

Table 4Measurementsof the ratio R = (rn, — th )/(md— m,).

Column 1 contains the uncertaintiesin the experi-mentaldata and in the calculationof the electromag-netic corrections.Column 2 is our estimateof theuncertaintiesdue to higherorder termsin thequarkmass expansion. (In the case of p — 0 mixing thesetermshavenot beenanalyzed.We have increasedthecorrespondingerrorbar from ±3to ±5in ordernot tobiastheaveragewhichis calculatedby treatingthefivevaluesgiven in column 3 as independentmeasure-

ments).

~R”5 ~RI~0. Result

1 2 3

±2.7 ±3 43±4p—n ±7.5 ±6 51±10

±1.7 ±3 43±4 5

±5.4 ±3 42±6p—wmixing ±3 44±5 5

SS

Average 43.5±2.2aSSSiSi


linear quadraticRN=64±9 RN=76±11R~=49±2 R.~=46±2R2= 44±6 R5 = 38±5 (13.7)

wherewe have given the results both for the linear massformula (13.6) and for the correspondingquadraticversion; within the accuracyof first ordermassformulaethereis no reasonto preferonetothe other.The valuesdiffer by up to a factor 2. It is evident that the first ordermassformulaedo notprovide us with a reliable determinationof R and that we haveto investigatethe correctionsto seewhetherthe observedmassdifferencesareat all consistentwith the frameworkadvocatedhere.

The higher order termsin the quark massexpansionwere discussedin section 10 on the basis ofimproved chiral perturbationtheory.According to this discussionthe leadingasymmetriesdueto thenonanalyticterms in the quark mass_expansionareeliminatedif onereplacesthephysical massesM~ bythe correspondingpurified massesMn definedin (C.5). In the quark massexpansionof M~thereareofcoursestill higher ordertermsthat areneglectedin the first ordermassformulae,but their effectshouldbe smaller, becausethe termsproportionalto (inq,,ark)

3”2 that arepresentin the physicalmasseshavebeeneliminated.Insertingthe purified massesinto the linear massformula (13.6)we obtain

RN=54±8 RN=48±7R.~=45±2 R

1=41±2R5=44±6 R5=41±5 (13.8)

wherethefirst valueis basedon SU(3)symmetricaxial vectorcouplings,thesecondon SU(3) symmetricmeson-baryoncouplingconstants(seeappendixC, table6).

The scatteringof thesevaluesis indeedconsiderablysmallerthan for the uncorrectedmassvalues.This shows that a substantialfraction of the discrepancyin the values of R obtainedfrom the rawphysical massesis due to the effect describedin section 10 in the framework of the static effectiveLagrangian:the energiesof the pion, kaon andetacloudsare ratherdifferent, not only becausetheyhave completelydifferent radii, but also becausethe mass differencesof the correspondingvirtualbaryonsproducesizeabledifferencesin the energiesof the clouds.

Thedifferencebetweenthe quadraticandthe linear massformulaeis a roughestimatefor thesize ofthe uncertaintiesdueto the remaininghigherorder terms.The first orderformulaefor the squareof themassesgenerallyleadto largerdeviationsthan the first order formulaelinear in the masses.We firstlook at the deviationsfrom the Gell-Mann—Okuboformula

2N+2E-3A -1

- E-N (13.9)

Using the raw physical massesin pure QCD the linear formulagives ~i= —0.070, the quadraticoneimplies ~i = +0.093.The correspondingnumbersfor the purified masses[SU(3)-symmetricaxial vectorcouplings]are = —0.084 and ~i = +0.061for the linear and quadraticversion of (13.9) respectively.The deviationsareperfectly consistentwith the size of thecorrectionsto beexpectedfrom higherordertermsin the quark massexpansion.Estimatingthe termsof order (inqu~,.k)

2 with the additivity rule [i.e.writing Ma,, = M~ + (m

1 + in2+ rn3)2 andusing the valuesof Mn ratherthanM~]the quadraticformula

gives zl = —0.005 for m~= 130 MeV and~i= —0.057 for in. = 170MeV: the terms of order (inq,,ark)2


obtainedfrom the additivity rule essentiallytransform the quadraticmass formulae into the cor-respondinglinear ones.

Theseobservationsare confirmed by an analysisof the ratio R. If oneusesthe quadraticmassformulae with the purified masses (symmetric axial vector couplings) one finds (RN, R1,R5)=(63,43,38) insteadof the values(54, 45,44)given aboveon thebasisof linearmassformulae.Correctingthequadraticmassformulaefor termsof order (mquars)

2 with the additivity rule onefinds (58, 43,40) if= 130MeV and (54,43,41) if m

5 = 170MeV. The terms of order (mquk)2 obtained from the

additivity rule againessentiallyconvertthe quadraticmassformulaeinto the linear ones.On the basisof the abovediscussionwe concludethat the valuesof the ratio R extractedfrom 1 andareaffectedvery little by higher order termsin thequark massexpansion,whereasthevalueobtained

from the neutron—protonmassdifferenceis strongly affected,both by the nonanalyticcontributionsandby the termsof order (inquark)2 the valueRN= 76 ±11 that follows from the quadraticmassformula ifoneusesthe physical masses(correctedfor the electromagneticself-energy,of course)is shifted toRN = 63±9if onesubtractsthe nonanalyticcontributions[SU(3)symmetricaxialvectorcoupling] andisfurtherreducedto RN= 54±8if the termsof order (mqu,wk)2 are takenfrom the additivity rule within

5 = 170MeV. In table 4 we quote the values obtainedfrom the linear mass formulae, using thepurified masses[meanvaluesin (13.8)].Columns1 and2 containthe errorbarsdueto theuncertaintiesin the valuesof the massesin pure QCD andin the higher order termsof the quark massexpansionrespectively.The threevalues of R obtainedfrom the spectrumof the baryonoctet are not onlyconsistentwith oneanother,but arealsoin perfectagreementwith the independentdeterminationof Rfrom the mesonspectrum.(For otherrecentestimatesseerefs. [135—144].)

14. Quark massratios from other sources

An independentdirect measurementof the ratio R= (m.— 01): (ind — m~)is basedon p — w mixing[145—152].The effectof anoff-diagonalterm in thepw massmatrix canbe seenin avery cleanmannerin the reactione~e-sIT~IT which exhibits a shouldernearthe mass of the w [147, 153, 154]. Fromtheseandotherexperimentsthe particledatagroupgives

= 0.014 ±0.002.

Using equation(B.12) of appendixB this becomes

M~,’=—2.22±0.16MeV (14.1)

for the off-diagonal elementof the massmatrix (the phaseof M,,,., is observedto becloseto 180°).The

electromagneticinteractionof coursealso producesp—w-transitions.Theprocessp—sy —s w contributes2

M~= ~ F0Fn, = 0.43±0.04MeV (14.2)

where.F~and F5, arethe couplingconstantsof thesemesonsto the electromagnetlccurrent


~0Ij~,ip°)=E~,F,.,M~F0=~3=fp=144±8MeV

(0Ij~iw~= E~F~15, F5, =51±3MeV. (14.3)

Other electromagneticcontributions are estimatedto be negligibly small [147]. The bulk of theexperimentalvaluemust thereforecomefrom the quark massdifferencem~— rn,,. Inserting the abovevalues for M~°—M~into the massformula (B.14) this gives

R=44±3 (14.4)

where the error bar only includesthe errors given above in the values of ~ M~.We do notattemptto estimatethe higher order termsin the quark massexpansionto assessthe accuracyof thefirst order mass formula (B.14). [Comparealso Langacker[152] who obtains R= 48±10from asomewhatdifferentweighting of the samedata.]

Severalauthors(SegréandWeyers[155];Genz [156];Langacker[157]; Ioffe [158];Voloshin [159])

haveshown that the decayscli’ —5 cli + IT0 and i/i’ —* cli + ~ provideinformationaboutsymmetrybreaking:

both decaysareforbiddenif the quarkmassesvanish.The amplitudeof the first reactionis proportionalto inu — m,, (apart from an electromagneticcontribution of order e2), the amplitude for the secondreaction is proportional to in

5 — 01. loffe and Shifman [160] haveshown that the ratio of rates canunambiguouslybe calculatedon the basisof PCACwith the result

r = F~.g,+,,.O~ (P~)~. (14.5),~‘-cç!,+~ 16R p.,,

In fact, it is not evennecessaryto invokePCAC.To showthis we considerthe limit inq,,ark—s0 at fixedratiosrn~:rn,,. rn~.First orderperturbationtheory in thequark massesimplies that the amplitudefor thedecayi/i’ —5 cl + IT

0 is given by

~ = —(i/JIT°outit~rnqJ~/i’) (14.6)

wherethe matrix elementis to be evaluatedwith the unperturbedstates.Since in the unperturbedsystemthe octetof Goldstonebosonsis degenerate,thereis in generalno physical differencebetweenIT° and ~ in that limit. In the presentcaseperturbationtheory showsthat the first order massformula(8.4) unambiguouslyselectsthe limiting states.The matrix elementsof the operatorc~inqmaythen beworked out by usingSU(3) for the octetbasiso(IT°i,o(i~J.The resultis

Tçt,’..g,+,,.oV3cosO+2sinOR t 2O=~-~ (147)Tç,,..q,±,, 2cosOR— \/3 sin 0’ g 2R

This determinesthe ratio of rates for any value of R. SinceR is large, the angle0 is small and weobtainthe resultof loffe andShifman given above.

The averageof the recentdata [161,162] for the branchingratiosis


B(çb’ -s ~/i+ IT°)= (0.10±0.03)%(14.8)

B(~fr’—s~/r+~)=(2.3±0.4)%

which gives

rex,.= 0.043±0.015. (14.9)

The relation (14.5) then implies

R = 28~

less than the value R = 43.5 given in table 4 by about two standarddeviations.The theoreticalpredictionfor r with R = 43.5 is

r,,, = 0.018. (14.10)

[Thispredictionis basedon SU(3)andhenceis subjectto correctionsof order in5 alsothe contribution

of the electromagneticinteraction to the decay i/i’ —s ~ + IT0 is neglected.We see no reasonfor these

correctionsto be large,but to ourknowledgetheir size hasnot beenestimatedin the literature.]A further source of information on symmetry breaking is ~-decay. In particular, the decays

—s IT~IT IT0 and ~ —s IT0IT0IT0 have been identified as symmetry breaking probes long ago (refs.[163,164]; for a review of the early literaturesee [165]).Using PCAC for pions only (rn~,rn,, —sO, rn

5fixed) onefinds good agreementwith the dataon the slopeparameterof theDalitz plot. Also, the directmeasurementsof the ratio B = F,, o,ro,,.o/F,,..,,.,r+ir-,,.o[167,168] which give B = 1.46±0.14 are in goodagreementwith the predictionB = ~ (up to small correctionsfor the differencein Dalitz plot slopesandphasespace)that follows from the isospinpropertiesof the symmetrybreakingquark massterm.(Notehowever that the overall fit of all data on n-decaysprovided by the particle data group impliesB = 1.268±0.060. Thisdisagreeswith the isospinpredictionby morethan 3 standarddeviations.)

The predictionsinvolving PCACfor theeta arenot in good agreementwith the data.In ouropinion,noneof thesepredictionsis morereliablethan theprediction for the decayrate (f,, = f,,.):

2 3r —_~__~2L—11fl ‘U (1411

3 ç2 —

f’s

which disagreeswith the data [166]:

= (324±46)eV (14.12)

by abouta factor two. The absoluterate for this process(which determinesthe other ratesvia theirbranchingratio) is difficult to measure.The valuedecreasedby morethan afactor two acoupleof yearsago [166].On the theoreticalside the theorem(14.11) is only valid in the limit mquarkO; one has toallow for correctionsof order inquark ln inquark as well as for correctionsof orderm~(which may well bedominated by fl~’-mixing,see e.g. refs. [170,171]). The higher order correctionsappear to besubstantial;to our knowledgethe uncertaintiesinvolved in thesecorrectionshavehowevernot been


investigated.In view of both the theoreticalandthe experimentaluncertaintiesit appearsto be difficultto determinesymmetry breakingparametersfrom n-decaysin a reliablemannerandwe do not discussthe subjectfurtherhere.The readeris referredto the literaturewhich maybetracedfrom the papersofNeveuandScherk[169],Dittner,Bondi andEliezer [165],LangackerandPagels[172],Weinberg[173],Crewther[174],Raby [175],DominguezandZepeda[176], RoiesnelandTruong [177].

The measuredmassdifferenceMD~— MDO = 5.0 ±0.8MeV also providesinformation about isospinbreaking. In contrastto the K~—K°mass differencethe electromagneticself-energyhoweverhasthesamesign as thecontributionfrom ind — rn~.Theuncertaintyin the valueof themassdifferencein QCD(seetable3) is thereforeratherlarge: (MD* — MDo)QC~)= 3.3±0.9MeV. Furthermore,thecorrespondingSU(3)-breakingmass differenceMF — MD = 164±60 MeV basedon MF = 2030±60MeV [178] needsexperimentalconfirmation.Using the meanvaluesthe first ordermassformula

rn,,— in,,MD+—MD°= rn —01 (MF—MD) (14.13)

gives R = 50. In view of the largeuncertaintiesboth in (MD+ — MDo)~’~and in MF — MD the error islarge,of order±20.Onemayreplacethe uncertainobservedvalueof themassdifferenceMF — MD by atheoreticalestimate,basedon SU(4)or on otherdynamicalassumptionsto reducethe errorbarsin theresult. It is howeverclear that theD~— D°massdifferencedoesnot provide reliableinformation aboutthe ratio R at this stage.

15. Light quark massesfrom QCD sum rules

The first determinationof the light quark masseswithin the framework of QCD was given byVainshtein,Voloshin, Zakharov,Novikov, Okun and Shifman [179].They consideredthe two-point-function for two axial divergences:

P(q2)= if d4x e~(0iT8A~(x)aA(0)J0) (15.1)

A,.. = fly

1,,y5d.

Asymptotic freedom guaranteesthat the behaviour of P(q2) at large values of q2 is given by

perturbationtheory.To zerothorder in the strongcouplingconstantthe imaginarypartof P(q2) is givenby (s~>th2):

Im P°~(s)= ~— s012 01 = ~(rn,,+ in,,). (15.2)

Using renormalization group argumentsVainshtein et al. then show that the leading asymptoticbehaviourof Im P in QCD is given by the sameexpression,provided 01 is identified with the runningquark mass01(s)definedin (3.3). At sufficiently largevaluesof s this expressionshouldprovidea goodapproximation to the actual imaginary part of P (see below for a more accurate high energyrepresentationthat includesfirst ordercorrections).At low energiesthe imaginarypart doesnot follow


a smoothcurvesuchas (15.2) nor doesit producethe thresholdat s = 4012 characteristicof perturbationtheory.Nonperturbativeeffects insteadgeneratea pole at s = M~,acontinuumstartingat s = 9M~,., abump at s = M~etc. (for experimentalevidenceon IT’ seerefs. [182,183]):

Im P(s) = ITf~M~3(M~,. — s)+~... (15.3)

To confront the information contained in the high energy representation(15.2) with the knownbehaviourof the amplitude in the low energyregion Vainshtein et al. assumethat the asymptoticbehaviour(15.2)is valid for s > So whereSo is low enoughfor the interval0 < s <s0 to be dominatedbythe pion contributionalone.The sumrule

ds {Im P(s)— Im P~(s)}= 0 (15.4)

thenimplies

(15.5)V3 So

With the estimates0 1.5 GeV2this gives

01(so)= 6.5MeV.

A valuefor s0 somewherein the interval betweenM~= 0.02GeV

2 andM~.. 2.3GeV2 appearsto bereasonablein the senseof duality and finite energysum rules. It is howeverclearthat the numericalvalueis very sensitiveto the choiceof s

0 andthat somewhatdifferent valuesof So, say s0= 1.2GeV2 or

So = 1.8GeV2 (01 8 MeV, 01 = 5 MeV) arealso acceptable.The mostextensiveanalysisof the QCD sumrules in this channelwas performedby Becchi,Narison,

de RafaelandYndurain[184]andby Narisonandde Rafael [185].Theseauthorscircumventthe lack ofinformationaboutthe behaviourof the imaginarypart in the intermediateenergyregionby resortingtoinequalities.Since Im P is positive, to saturatewith the pion contributionalonemeansto get a lowerboundon the valueof a convergentdispersionintegral.Furthermore,they improvethe accuracyof thehigh energy representationby including the first order (two loop) correction [186—191].For theabsorptivepart their resultamountsto

Im p~rt(~) = s th(s)~[i + — {in~(s)+ rn2,,(s)— rn,,(s) rn,,(s)}+ . . .] (15.6)

where in(s) is the runningmassto two loop order anda,(s) is the running couplingconstant.In the most recentversionof their work Narisonandde Rafaelusethe Laplacetransformtechnique

[181]. Let

L(M) = ~ Jdse~MSIm P(s). (15.7)

iSiSiSiSSSSSS


The pion contributionrequires

L(M) � L’T(M) = f~,.Ma,,.exp{—M2,,./M2} (15.8)

for all values of the variableM The perturbativeasymptoticrepresentationequivalentto (15.6)reads

L°~’(M)

=~~~M4th(M)2[1+ (~+2YE)a,(M2) 2{rn2(M)+ rn~(M)-rn,,(M) rn,,(M)}+~. .1(15.9)

where y~= 0.577... is Eulersconstant.In addition to using an improved perturbativehigh energyrepresentationthey also include the

leading nonperturbativecorrectionwhich maybe extractedfrom the operatorproductexpansion:

= ~ 01(M)2{~B2)+ T~ff~M~~J (15.10)

where (B2) is the vacuumexpectationvalue of the squareof the gluon field strength,introduced byVainshtein,ZakharovandShifman [180]:

(B2) = (0iF~,,F~”0) = ~-~(0~~ G~,,.G0” 0)— 0.12GeV4.

Narisonandde Rafaelthenstudythe inequality L’~� L”~+ Lnc~1~t,which gives a lower boundon thequark massfor any valueof M For largevalues of M the boundis very weakas it decreaseswith M2.Letting M becomesmallerthe boundbecomesmorestringentuntil onereachesavalueM

0 belowwhichthe boundagainweakens.Picking the “optimal” valueM0 theyarriveat the bounds

9.4±2.2MeV; A =70MeV; M0=600MeVth(1GeV)� 7.2±1.8MeV; A =140MeV; M0=65OMeV (15.11)

5.3±1.2MeV; A=21OMeV; M0=75OMeV.

(Theerror barsare given by Narisonandde Rafaelon the basis of the size of the correctionsto theleading asymptotic behaviour; we have convertedthe boundsgiven for the renormalizationgroupinvariant massinto the correspondingboundsfor the runningmassat 1 GeV accordingto table 1.)

As pointed out by Hubschmidand Mallik [192] the crucial question is whether the asymptoticrepresentationmayindeedbe useddown to valuesof M of the orderof 600or 700MeV. The questionwas studied by Eidelman, Kurdadzeand Vainshtein [193] in the caseof the two point functionassociatedwith the vector current where the behaviourof the imaginary part is known from eke-annihilationdata.In that channelthe analogousasymptoticrepresentationis indeedconsistentwith thedatadown to M-valuesof this orderof magnitude.Saturatingthe Laplacetransformat M = M~,withthe p-mesonaloneoneobtainsan estimatefor the p couplingconstant

f \/eM•(l+1(B)+O()} (15.12)


which is in fair agreementwith the data. Assumingthat the asymptoticrepresentationfor the Laplacetransformof (UI 8A~3A10) also holds at M = M,, and saturatingit with the pion contributionNarisonand deRafaelobtainthe estimate

01 (M0) = {1 + O(a~)}= 10.7MeV (15.13)

which for A = 140MeV is equivalentto 01 (1 GeV) = 10 MeV. Theformula is similar to the old SU(6)result (seesection 1), thenumericalvalueis twice as large.

It is difficult to estimatethe accuracyof the asymptoticrepresentationof the Laplacetransformatvalues of M of the order of M~or lower on the basis of experiencewith the vector channel: thequalitative behaviourof the imaginary part is different in the two cases.For the vector current theimaginarypart tendsto a constantat high energieswhereasfor the axial divergenceit growsessentiallylinearly with s. At any rate the contributionto the Laplacetransformfrom intermediatestatesotherthan the pion arestrongly suppressedat thesevaluesof M such that one shouldexpectthe inequality(15.11)to be almostsaturated.We considerthe valuesgiven in (15.11) as roughestimatesof the quarkmassratherthan asreliablelower bounds.To get a reliableboundone shouldnot extrapolatethe highenergyrepresentationof the Laplacetransformto valuesof M below 1 GeV. If oneassumesthe highenergyrepresentationonly to be valid down to 1 GeV then the boundsof Narison and de Rafael,expressedin termsof the runningmassat 1 GeV become

5.1MeV, A=7OMeVth(1GeV)� 4.7MeV, A =140MeV (15.14)

4.3MeV, A=21OMeV.

Hubschmidand Mallik avoid using the high energyrepresentationdown to small values of MInsteadthey improve the low energyrepresentationof the amplitude by including contributionsfromhigher intermediatestates.They parametrizethesecontributionsin termsof a narrowradial excitationof the pion at M~ 1.5GeV andallow the coupling strengthof thisstateto vary within a ratherbroadrangeof values.Working with finite energyand dispersionsum rules ratherthan with the Laplacetranform theyfind that the model satisfiesthe constraintsimposedby QCD providedthe quark massisin the range

01 (1GeV)=5.7±2.2MeV. (15.15)

[Sincetheywork at relatively largevaluesof q2 the resultis not sensitiveto the valueof A or of (B2).]

We havecarriedout an independentevaluationof the Laplacetransform with the aim of getting aconservativeestimateof the rangeof 01 valuesthat areconsistentwith the QCD sum rule for the axialdivergence.Considerthe Laplacetransform of Im P(s) at M = 1GeV. This quantity gets sizeablecontributions only from the range0< Vs <2.5GeV. Around Vs 2 GeV the asymptotic formula(15.6) should start to becomea reliable representationof the actual imaginary part. [Note that thequantity 2IT Im pPers(5)/3~is almost constantin the range1.5 GeV<Vs<2.5 GeV. If A is of order140MeV then this quantity varies from 1.25 012 (1 GeV) to 012 (1 GeV) as Vs growsfrom 1.5 GeV to2.5GeV.] The problem is to obtain a reliable estimate of the imaginary part in the region belowV

5 � 2 GeV. The continuumstartsat V5 = 3M.,,. — thecontributionof the threepion intermediatestate

SiSi

I Gasser and H. Leulwyler, Quark masses 121

can be workedout from current algebra.In the chiral limit it is given by

ImP3~(s)=3841

3fM~s. (15.16)

The contribution grows linearly with s like the perturbativeimaginary part, howeverwith a muchsmallercoefficient.For 01 (1 GeV)= 5MeV the ratio Im P

3’~/ImpPert is 0.15, for 01 (1 GeV) = 10 MeV itis four_timessmaller. In the real world the threepion contribution is furthersuppressedby phasespace[at Vs = 1 GeV the suppressionstill amountsto a factor of two]. We concludefrom this that theimaginary part doesnot receivesignificant contributionsbefore the thresholdfor resonanceformation(A1~+ M,,. = 0.9GeV, M,,.. — 1.3 GeV) is reached: apart from the one-pion intermediatestate theimaginarypart practically vanishesbelowV~= 1 GeV. The critical region is the range1 GeV <V

5 <

2 GeV. We parametrizethe imaginarypart in termsof IT, IT’ and a continuumcontribution:

Im P(s)= ITf~,,.M~,{ô(s — M~)+ r 5(s— M~.)}+ 0(s— So) Im p~er

t(5). (15.17)

The radial excitation of thepion is weightedwith the ratio r of the wavefunctionsat the origin:

r= (OJfiysdiIT’) 2 (15 18)(OiuysdIIT)

In fig. 2 we show the value of the running quark massat 1 GeV that one obtainsby calculatingtheLaplace transform of (15.17) at M and equating the result with the asymptotic representationLlse~(M)+ L”°”°~’(M)givenin (15.9) and(15.10).The renormalizationgroupinvariant scaleis takenatA = 140MeV with Nf = 3 [ascan beseenfrom (15.14)the precisevalueof A is not essential;the power

1/2 (mU*md)

_________________________________D

>. 8 - G

C

° ~‘ O~6 O~8 ~ - M

GeVFig. 2. Quarkmassfrom QCD sumrules for A = 140MeV; m, andmd aretherunningmassesin theMS schemeat~e= I GeV, M is thevariableintheLaplacetransform.CurvesA andB arelower boundsobtainedby different algebraictreatmentof thefirst ordercorrections: theverticalbarsare an estimateof the errorsdue to higher ordercontributions.CurvesC, D, E, F, G and H areobtainedby using a seriesof modelsfor the

behaviourof the imaginarypart in theregion 1 GeV <V~<2GeV.

122 J. Gasser and H Leutwyler, Quark masses

correction proportionalto (B2) amountsto less than 10% for M> 1 GeV and might just as well bedropped].If the modelfor the imaginary part is a good approximationwe shouldget the samevalueof01 (1 GeV) for all valuesof M for which the asymptoticrepresentationholds. Sincewe do not trust thisrepresentationbelow M = I GeV we haveindicatedthe correspondingquark mass values by dashedlines.

Considerfirst the boundsof Narisonandde Rafael. In our model theseboundscorrespondto s0

r = 0 (curve B). With the accuracyof the perturbativecalculationthereis no distinction betweenanexpressionof the form (1 + a/IT)

112 and (1 — ~aIir). De Rafael andNarisonusea somewhatdifferentalgebraicform for the perturbativeexpressionto obtaina curve like A. We takethe differencebetweenA andB asan estimatefor the errorsdueto higher orderperturbativecontributions.Now considerthecontributions from intermediatestatesother than the pion. To get a rough estimateof the range ofparameters5o, M,r’ and r of interestwe observethat a linear IT-trajectory with the sameslope as thep-trajectoryandwith IT’ as the first spin zerodaughterpredictsM~= M~+ 2(a’)’, i.e. M,,.. = 1.5GeV.If the continuumthresholdis takenhalf way betweenM~and the next recurrencewe get V~

0=

1.8GeV. According to duality andfinite energysumrules (FESR)the IT’ peakshouldbalancethe areabetween~ + M~)andso; this leadsto r 8. Theresultingvalueof the quark massis given in curveC. BetweenM = 1 GeV and M = 2 GeV the quark massindeedstays practically constantat a value01 (1 GeV)= 7.2MeV.

If one keepsthe continuumthresholdfixed, the main uncertaintyin the model is the size of the IT’

peak,measuredby r. Thevalue r 8 suggestedby FESRmay at first sight appearto be unreasonablylarge.In fact, the analogousratio of saythe vectorcouplingconstantsfor cl’ andits radial excitationcl” isknownto besmall (the measureddecayratesinto e~egive r1, = (f~,: f~) 0.5). If oneestimatesr in theframework of nonrelativisticpotentialmodels one also getsnumbersof order one. Why do FESRsuggesta muchlargervaluein the pion channel?The reasonfor thisis most easilyseenif onereplacesthe imaginary part by a sequenceof equally spacednarrow resonances:since at high energiestheimaginarypart growslinearly with s the residuesr. of theseresonanceshaveto grow linearly with n.The valueof r~must increaseby i~r= 601

2(ITf,rM~a’)2 from resonanceto resonance;for 01 = 5MeVthis gives 1~r 3, for 01 = 10 MeV the increaseis four times larger: E~r 12 (01 = 7 MeV amountsto

6). For very heavyquarksthe excitationenergyof the first excitedstate is small comparedto therest energyof the ground state.There is no dramaticincreasein r~for the first couple of excitedstates— the linear increasein the imaginarypart only developsoncethe excitation energyis of the orderof the rest energy. In the pion channel nonrelativistic models fail, becausethe excitation energyM,,. — M,,. is large in comparisonwith the rest energyof the ground stateand it is not clear how tohandlekinematicfactorslike M,,. : M,

1 10.It is certainly legitimate to questionwhetherFESRprovide uswith a good estimatefor the size of

the imaginary partbelow 2 GeV. Any model that saturatesthe sumrule maybe regardedas acceptable.To showthe dependenceofthe quark massvalueon theparametersof the modelwe give severaldifferentsetsof parameterswhich all leadto aconsistentpicturewith 01 beingroughly independentof the pointat which the Laplacetransformis evaluated.The parametersdefining the curvesD, E, F, G, H aregiven in table5. If the imaginarypart is takentoo small in the intermediateenergyregion the sumruledoesnot saturate.If onewants to havenothing but a narrow resonancebelow 2GeV then saturationonly occursfor sizeablevaluesof r — the lower the massof theresonance,the larger r hasto be (modelsE, F, H). CurvesD andGarealmost indistinguishable:it doesnot matterwhethertheimaginary part iscarried only by a narrow resonancewith r = 8 at M~= 1.3 GeV or whether ~ of the resonancecontribution is spreadover the interval from 1.1 to 1.6GeV. We considermodelsD andG as upper

SSSSSSSS


Table 5Valuesof the parameterss

0,M,, and r used in fig. 2. The runningquarkmass at 1 GeV is determinedby comparing the Laplacetransform in the

region 1 GeV <M <2GeV with theasymptoticrepresentation.

Vso (GeV) M,,. (GeV) r ~(m,+ md) (MeV)

A,B w 0C 1.8 1.5 8 7.2D 1.6 1.3 8 8.8E 2 1.65 8 6.2F 2 1.6 12 7.3G 1.1 1.3 2 8.6H 2 1.75 5 5.2

limits for theimaginarypartin theregion 1 GeV< V~<2 GeV(notethatin modelGthecontinuumstartswith full strengthat V~0= 1.1 GeV, a small IT’ peaksittingon top of it). Model H describestheoppositeextreme,forwhichthereis nothingbuta IT’ peakof modestsizein thisregion.Onthebasisof thesemodelswe concludethat ourpresentknowledgeaboutthe behaviourof the two-point function (0JaA~oAIO) inQCD is consistentwith a quark massvaluearound7MeV:

01 (1 GeV)= (7±2)MeV. (15.19)

In the absenceof reliableinformation about the behaviourof the imaginary part in the intermediateenergyregion it is not possible to reducethe error bar substantially.Quark massesof the order of10 MeVor higher would imply that the imaginarypart of the amplitude(0!T3A~oAIO) containsverylarge contributions from the intermediateenergy region below 2GeV. The presenceof such con-tributions would be a most interestingphenomenonby itself as thesecontributionswould presumablyproducelargecorrections toPCAC (for a discussionof PCACviolationsdueto excitedstatesof thepionsee[194—196]andthe referencesquotedin thesepapers).

It is straightforwardto apply the sameanalysisto the K-inesonchannel.TheITEP groupobtainstheestimatein. (1.6GeV) 110MeV which for A in the range100 to 140MeV is equivalentto

in, (1 GeV)= 120MeV. (15.20)

The boundsof Narisonandde Rafaelfor M � 1 GeV are

130MeV, A =70MeVm~(1 GeV)+ 01(1 GeV)� 120MeV, A = 140MeV (15.21)

110MeV, A =210MeV

(the corresponding“optimal” boundsfor the running mass at 1 GeV are 210, 160 and 130MeVrespectively.As discussedabovetheseoptimal boundsareobtainedby usingthe high energyrepresen-tationat low valuesof M; it is difficult to estimatethe errorbarsin thatprocedure).Usingthe estimatesfor the imaginary part given abovewe obtain

in, (1 GeV)= 180±50MeV. (15.22)


(The positionsof the thresholdsand of the radial excitation K’ areexpectedto be at slightly highervaluesof s thanin thepionchannel.Basedon theobservationthatM~— M~isroughlyequaltoM~.—we haveshifted the valuesof s accordingto 5K = Si~~+ M~— M~which leadsto MK’ — M,,. 80 MeV.)

We concludethat the QCD sum rules are consistentwith a strangequark massin the interval130MeV < in

5 < 230MeV. A strangequark massbelow 130MeV is in conflict with thebounds(15.21)which we arguedto be on the safe side, whereasa strangequark masslargerthan 230MeV calls forvery largecontributions from the intermediateenergyregion.The sum rules areperfectly consistentwith the ratio in5: 01 = 25 obtainedfrom the spectrumof the pseudoscalarmesonsin section13.

Onemayobtaininformation aboutthe quark massvaluesfrom other two-pointfunctionsthanthosediscussedabove.Theproblemwith theseotherchannelsis that thereis evenlessinformationaboutthebehaviourof the imaginary part than in the caseof the axial divergencewhereat least the couplingstrengthof the pion andof the kaon areknown.The analysisis thereforenecessarilyrathercrudeanddoesnot permit oneto extractvery reliableinformation aboutthe quarkmassesat this stage.

In arecentpaperMallik [197]hasanalyzedthe QCD sumrules for all two-point functionsassociatedwith the scalar,vector, tensor,axial andpseudoscalarquarkdensities(seealso CraigieandStern [198]).This analysisnot only showsthat quark massestimatesvery similar to thoseproposedby Okubo [19]maybe obtainedwithin QCD, it alsodemonstratesthat someof the SU(6)-relationsamongthe vector,tensorandaxial couplingconstantsthat lead to the value 01 = 5.4MeV (seesection1) indeedfollow ifonesaturatestwo-point function sum rules in the mannerdescribedabove. In view of the fact that onehasto eliminateunknownsby combining saturatedsum rules the numericalresultsof courseinvolvelarger uncertaintiesthan in the case of the sum rule for the axial divergence.Within the errorsassociatedwith the saturationschemethe quark massvalues given by Mallik areperfectly consistentwith the estimatesgiven above.

Finally, for the purposeof illustration ratherthanwith theintention of reviewingthe subjectin anydetail, we briefly indicate how an analysis of the two-point function associatedwith the baryonic“currents” q(x) q(x)q(x) leadsto informationabout the orderparameter(0I~q~0)andhenceaboutthesize of the quark masses[199].For definitenesswe considersomeparity evenspin~field ~(x) with thequantumnumbersof uud, e.g.:

~(x) = y,.. y~ da(x)(u~(x)Cy” u~(x))Cabc (15.23)

where a, b, c are colour indices and C is the charge conjugation matrix with y~= — Cy,~C’.Lorentzinvarianceimplies a spectralrepresentationof the form

(0I~(x)~(y)I0)= ~ J ds{p1(s)i~+p2(5)}~(X - y; s) (15.24)

where~l‘~(z;s) is the positivefrequencypart of the free Green’sfunction for a scalarfield of massV~.In perturbationtheory (masslessquarks)the spectralfunction p2(s) vanishesto all ordersbecausethepropagatorsof thefield d whichentersthe diagramsat x andleavesat y alwayscontainan oddnumberof y-matrices. In the real world p2(s) is obviously different from zero. The proton e.g. producesthecontribution

pi(s) = K ~(s— M~) ~

p2(s) = KM,. 6(s— M~)+.... (15.25)

J. Gasser and H Leutwyler, Quark masses 125

Only aparitydegeneratespectrumcould beconsistentwith p2(5) = 0. In the short distancebehaviourofthe amplitude(0l~(x)~(y)!°) the contributionof the spectralfunction p2(s) is governedby the vacuumexpectationvaluesof theoperatorswith lowestdimensionthat carry chirality; the leadingshort distancecontribution is againproportional to (0I~qI0).It is thereforepossibleto extractinformation aboutthesize of this quantity if oneassumesthe asymptoticrepresentationfor the Laplacetransformsof pi(s)andp2(s)to bevalid at M M~andto be saturatedby the protonintermediatestate.Similarargumentsmayof coursebeapplied to thetwo-point functionsassociatedwith spin 3/2 fields.We quoteoneof theensuingrelationsbetweenbaryonmassesand the orderparameter(0Jqq~0),dueto Ioffe:

(M~)

3 = 40ir2 (i + [—(OJuuJO)]. (15.26)

As pointed out by Chung, Dosch, Kremer and Schall [200], the result is not independentof theinterpolating field oneworks with. We neverthelessfind it most remarkablethat the values of theparameter(0Ic~qI0)which oneextractsin this mannerfrom the baryonspectrumareconsistentwith theestimatesdescribedin the previoussections.With M

4 = 1232MeV the quark mass value that oneobtainsfrom the relation (15.26)is

01 9.5MeV. (15.27)

16. Heavyquarks

As was shownby Novikov et al. [201]the massesof heavyquarksmaybe extractedfrom e~edataby investigating the QCD sum rules for the vacuumpolarization amplitude.Since there are severalexcellentreviewson this subject [201—204,206—208] we will not describethe analysisin detail. There isonly oneminor point which we find useful to add: we wish to ~press the valuesextractedfrom e~edatain termsof the correspondingrunningquark massin theMS scheme.The sum rules areusuallyevaluatedin termsof the massfunction M(p

2) introducedby Georgi andPolitzer [209].Onewrites thequarkpropagatorin theform

S(p)= Z(p2)[M(p2)—fl]1 (16.1)

and determinesthe value of the quantityM(p2) at the spacelikepoint p2= ~ The functionsM(p2) andZ(p2) maybe expressedin termsof the running massm(,u) andrunningcouplingconstantg(~s) orderby order; they dependon thegauge.To lowestorder the explicit expressionreads(seee.g.ref. [210])

M(p2)= mO.~){1+~—~(a+Ab)+Ofrs~)}

Z(p2)= 1—-~-A(a—3b+~)+O(a~)31T

2 1 2 2t ~ 2 2 rn—p

a=~—ln—T+---~(rn—p)ln ~2

b = (m~2p

2){_i — in (16.2)


whereA is the gaugeparameter(Landaugauge:A = 0, Feynmangauge:A = 1). To give a valuefor thequantityM(p

2) at somepointp2 in a suitablegaugeis equivalentto giving avaluefor therunningmassm(p~)at somescale~. In contrastto M(p2) the running massis a gaugeinvariant notion.We comparethe valueof the quantity M(p2) in the Landaugaugeat the spacelikepoint p2 = — m2 with the runningmassm(,a) at the point ~s= in. The equation

m(m)= in (16.3)

[wherein (is) is the running massin the MS schemewith Nf flavours] unambiguouslydefinesa quarkmassparameterin to all ordersin perturbationtheory— we denotethisquantity by m(m).Thequarkscandb aresufficiently heavyfor perturbationtheory to work at ~ = in~or p. = m~,for the light quarksu,d, s thequantity m(m) is an academicnotion,becauseaperturbativecalculationof the function m(j.t) atsmall scalesof order p. = 5 MeV or ,a = 150 MeV doesnot appearto be feasible.[To be precise,weshouldspecifythe numberof flavours to beusedin the MS schemethat definesthe function m(p.).Weexpectthe dependenceon Nf to be very weak (seebelow); for definitenesswe define the quantityin~(m~)in the MS schemewith 3 flavours,whereasinb(inb) refers to the MS schemewith 4 flavours.]

To lowest order the relation betweenthe Georgi—Politzerfunction at p2= — in2 in the Landaugaugeand the runningmassat scale in maybe readoff from (16.2):

M(-m2)= m(m)~l +~ (~- 2 in 2) + O(a~)}. (16.4)

The numericalcoefficient in front of a~is very small; for a, <0.3the differencebetweenM(—m2)andin (in) is lessthanS%~— small in comparisonwith the errorsof the methodused to determinethe valueof M(—in2).

In the caseof the b quark massthe main uncertaintysteinsfrom the valueof a, usedin the analysisofthe sumrules. Since the perturbativeasymptoticrepresentationof the sum rule is only known to firstorder in a, the methoddoesnot allow oneto expressthevalueof thisparameterin termsof theknownrunningcouplingconstanta,(p.). Onehasto calculatethe secondordercontributionsto find out whichvalueof a, to usein the first orderexpression.Thevaluedependson the quantity oneis looking at. Inthecontextof Laplacetransformedsum rules(exponentialmoments)the valueof a, to beuseddependson the scaleM at which the Laplacetransformis evaluated— thereis no universalprescription(like theprescriptionthat a,(2inb) 0.17 shouldbe used).At a low valueof M the sumrule is almostsaturatedby the groundstate;in this casethe effectivecouplingstrengthis presumablydeterminedby the size ofthe system,or, equivalently,by thelevel splitting [a, = 0.25 to a, = 0.30] ratherthanby its mass(seee.g.refs. [211,202]). Theresult for inb(inb) increaseswith the valueof a~usedin the analysisof the sum rule(a. = 0.17 leadsto inb(inb) = 4.19GeV whereasa, = 0.25 gives inb(inb) = 4.26GeV). With the estimatesfor M(—m2)given by Novikov et al. [201],Voloshin [211],Reinderset al. [204,205], Guberinaet al.[206],Bertlmann[218],Iwao [212] and Shifman [208] we obtain the following values for the runningmassesat scalein:

mc(mc)=1.27±0.05GeV

mb(mb)—4.25±0.10GeV. (16.5)

The quark mass values found within nonreiativistic models for the bound statemotion (for anSiSiSiSiStaaaaaaa55


up-to-datereviewseeref. [213])areconsiderablyhigher thanthesecurrentquark masses.Unfortunatelythe connectionof the potential model parametersto the Lagrangian of the theory is not easytoestablishin the caseof the quarksthat arepresentlyavailablefor experimentation.However, if naturedoesprovideus with avery heavyt-quarkthenthe boundstatesof this particlewill be a sensitiveprobefor measuringthe parametersof the QCD Lagrangian.For very heavy quarks the perturbativenonrelativisticboundstatepictureis self-consistent.The levelsaregiven by theBalrnerforinu/a

M,, = 2Mq — ~ Mq(~as)2. (16.6)

For very heavy quarksthis formulashould be very accurate:both perturbativecorrections(of orderMqa~)andnonperturbativecorrections(of order (B2)M~3a~4,see below) aresmall if the massof thequark is sufficiently large.Thequark massMq that appearsin thisformulamaybecalculatedin termsofthe runningmassasapowerseriesin therunningcouplingconstant,by solvingthe equation

M(M2q) = Mq (16.7)

which determinesthe positionof the pole in the propagatorandthe startof the cut: Mq is the dressedmassof the quark (in contrastto the function M(p2) itself, thedressedquark massis a gaugeinvariantnotion).The conceptof massfor an isolated,dressedquark of courseconflictswith the confinementofcolour. The abovedefinition of the quantity Mq maywork orderby order in perturbationtheory— itfails in the real world wherethe propagatoris not supposedto havea pole.The propagatorS(p)of aheavyquarkonly looks like the free fermionGreen’sfunction as long as p2 stays away from the regionp2 = M~,where nonperturbativemethodshave to be used to get a reliable representationfor thepropagator.Within perturbationtheory the physical mass of the quark howeverappearsto be awell-definedconcept(in particularTarrach[214]hasshownthat Mq is infraredfinite to two loops).

To lowest order the connectionbetweenthe running massand the massto be used in the Balmerformula maybe readoff from (16.2):

Mq = m(m) {1 + ~a,/IT + O(a~)}. (16.8)

It is clear that one needsthe nextorder term in this expansionboth to determinewhatvaluefor a, touseandto havesufficient accuracyto considerbinding effects— whichis what the Balmerformuladoes.To our knowledgethe higher ordercorrectionshavenot yet beencalculated.

Using the numericalvalue(16.5) the massof the dressedb quark becomes

M — f4.6±0.1GeV, a,=0.2b~~48±01GeV a,0.3. (16.9)

The valueof the dressedb quarkmassobtainedin this manneris consistentwith the value obtainedwithin a nonrelativistic treatmentof the bound state motion, provided the leading nonperturbativecontributions are accountedfor. The leading nonperturbativecorrectionswere first calculated byVoloshin [211]who finds

Mb = 4.795±0.025GeV (16.10)


for a, = 0.30±0.03. As shownby oneof us (Leutwyler [215])the leadingnonperturbativecorrection to the

Balmerformula maybe written in the form

= 2Mq_~Mq(~as)2+ (~n64~1 (16.11)

fl q[3as]

wheree,.,~is a rationalnumberof orderone (E10 = 1.468, e20 = 1.585, r2,= 0.998). The formula (16.11)only appliesif the quark massis sufficiently heavysuchthat the nonperturbativeeffectsproportionalto(B

2) do not distort the wave function too strongly. To estimate the size of the nonperturbativecorrectionwe first look at the bb ground state in perturbationtheory.To lowest order the levels aregiven by the uncorrectedBalmer formula.The perturbativecorrectionsareof orderMba~.In QED theanalogouscorrectioncan be absorbedby a propernormalizationof e simultaneouslyfor all levels ofpositronium(one simply hasto identify e with the chargeof an isolatedelectron).This is not the caseinQCD becausethe diagramsin which the exchangedgluon virtually dissociatesinto a gluon pair isinfrared divergentif the gluon momentumvanishes.(There is a similar effect in muoniumwherethevacuumpolarization dueto electronpairs producesa correctionproportionalto M~a5(M~/M,.)2whichexplodesas Me~~*0. For masslesselectronsthe correctionis of ordera3 ratherthan a5.) If onewantstoabsorbthe termsof orderMba,~in the uncorrectedBalmerformulaby a renormalizationof a, thenonehasto choosea different valueof a, for every level of the system.The valueof a, to be used is therunningcouplingconstantat ascaleof the orderof the inverseBohr radiusof the orbit in question[216]

a, = aS(3MbaS/n). (16.12)

For the ground state (n = 1) this gives a, 0.3, Mb = 4.83GeV. The correspondingBohr radius isaBohr= (Mb~a,Y

1= (1 GeV)~’,the kinetic energyamountsto EkI,, = 193MeV, the expectationvalueofthe Coulomb potential is twice as large: Ecb= —386MeV. With the same value of a

5 the non-perturbativecorrection[evaluatedwith (B

2)= 0.12GeV4 as given by Vainshteinet al. [217]]shifts thequark massto Mb = 4.79GeV [compare(16.10); the kinetic energyand the expectationvalueof theCoulombpotentialarealmostunaffected;the nonperturbativecorrectionamountsto E~

2= +62 MeV].If we insteadpick a, = 0.25 in the uncorrectedBalmer formula thenwe get Mb = 4.80GeV, Ecb=

—266MeV and aBOhr = (800MeV)~’.For a8 = 0.25 and (B2) = 0.12GeV4 the leading nonperturbative

correction shifts the quark mass to Mb = 4.73GeV [the leading nonperturbativeterm amounts to~ = +135MeV, i.e. to 50% of the Coulomb potential— the net binding energyvanishes].It is clearthat a frameworkwhich treatsthe nonperturbativeeffectsas smallcorrectionsis not reliablefor the bbgroundstateif theeffectivecouplingstrengthfor the boundstatemotion is belowa, ~ 0.25: the massofthe b quark is then too light and we haveto expecta sizeabledistortion of the wave function. Thepressureexertedby the vacuumfields tendsto squeezethe system andhenceto increasethe inverseBohr radius. Since the ground state realizesa minimum of the energywe neverthelessexpect theformula (16.11) to give a respectableestimatefor the energyof the state,despitethe fact that the wavefunction that underliesit is too broad.[This is confirmedin a recentpaperby Bertlmann[218]whofindsa, = 0.25 for the effectivecouplingstrength;usingthe value(B2) = 0.16GeV4 (obtainedfrom an analysisof the cë system)and working with exponentialmomentshe obtainsMb = 4.71 GeV. If one usesthevaluesa, = 0.25, (B2) = 0.16GeV4 in (16.11)one indeedobtainsMb = 4.70GeV.]

What theformula (16.11)clearlyshowsis thatthe excitedstatesof the bb systemcannotbedescribedintermsof correctionsto apositronium-likeboundstate: the nonperturbativetermexplodeswith the sixth


powerof the principalquantumnumber.It ruins QCD perturbationtheoryunlessthe quark is verymuchheavierthantheb quark.In thecaseof thec~systemthenonperturbativeeffectswin alreadyin thegroundstateandit is not clearwhethertheperturbativenotion of a dressedcquarkmass(M~ 1.4 GeV)is of anyuse.

To comparethe massesof the heavy quarkswith the runningmassesof the light quarksat the scalep. = 1 GeV we haveto convert the above values into the correspondingrunning massesinc, inb atp. = 1 GeV. In the case of in,~ this is a straightforward matter. The renormalization factorin~(1.27GeV): tn~(1GeV)maybe extractedfrom table1. With avalueof A in therange100MeV <A <

200MeV (3 flavours) onefinds

m~(1GeV)=1.35±0.05GeV. (16.13)

In the caseof inb thereis a complication: the changein scaleinvolves crossingthe thresholdfor cëproduction.To ourknowledgetheconnectionbetweentherunningmassin theMS schemefor Nf flavours(appropriatefor scaleslargein comparisonto all quarkmasses)andthe correspondingrunningmassforN5 — 1 flavours(appropriatebelowthethresholdfor theheaviestquark)isnot availablein theliterature[seeOvrutandSchnitzer[219]andBernreutherandWetzel[220]for thecorrespondingconnectionbetweenthevaluesof A]. If oneevaluatestherenormalizationfactorr = in (1): in (4.25)for N5 = 3 accordingto (3.4) onefinds r = 1.26, 1.32, 1.38 for A3 = 100, 150 and200MeV respectively.To calculatethe renormalizationfactor for Nf = 4 one hasto adjust the value of A; accordingto Bernreutherand Wetzel [220] thecorrespondingvaluesof A areA4 = 76,120and165MeV. Usingthesevaluesin (3.4) onefinds r = 1.25, 1.31,1.38respectively.Thisshowsthatthemassrenormalizationfactorsarealmostunaffectedby thepresenceofan additional quark flavour, but do of coursedependon the valueof A, which is still not knownveryaccurately.

The runningb quark massat p. = inb is determinedon the basis of sum rules that only includefirstorder corrections.Sincethe first order mass renormalizationis independentof the numberof quarkflavours the methodis not accurateenoughto requireadistinctionbetweendifferent definitions of therunning mass. Using the value mb(mb) = 4.25±0.10 GeV the running b quark mass at p. = 1 GeVbecomes

15.3±0.1GeV, A3= 100MeVmb(l GeV)= 15.9±0.1GeV, A3 = 200MeV. (16.14)

17. Quark massesfrom SU(4) symmetry

The mass of the charmed quark is of the sameorder as the scale of QCD. It is therefore doubtfulwhetheronemaytreatit as a perturbationandcompareboundstatescontainingcharmedquarkswiththe analogouslight quarkboundstates,say cl’ with ~ or A~with the nucleon.To testthe assumptionthatSU(4) is an approximatesymmetryonemaycomparethecli and~ wavefunctionsat theorigin, whicharemeasuredin the decaysinto eke. If SU(4) is an approximatesymmetry at infinite momentumthenf~,, f’,; experimentallyfq. = (1.69±0.14)fç. The symmetry is obviously rather strongly broken incomparisonwith SU(3) for which the prediction f5, =f,. is well obeyed: f4,, = (1.11±0.07)f~.Theperturbationcausedby the massdifferencem~— 01 which playsthe role of the SU(4)symmetry-breakingparameteris not very smallandoneshouldthereforenot expectSU(4)predictionsto bevery accurate.


The pattern of SU(4) breaking observedin the meson and baryon massescan neverthelessbeunderstoodon the basisof first orderSU(4)massformulae[221—230].The patternis determinedby theratio of SU(4)breakingto SU(3)breakingquark massdifferences

C= (mc— th)/(m, 01). (17.1)

The valueof thisratio is particularyinteresting,becauseit relatesthe light quark massesto the charmedquark mass which is known ratheraccurately.If the quantity C can be extractedfrom the observedmeson and baryon massesthe measuredvalue mc (1 GeV) = 1.35±0.05GeV then determinestherunning light quark massesat the scale p. = 1 GeV. [As discussedin section 13 the analogousratioR = (in, — 01): (ind — m~)determinesthe size of the SU(2)splittings in termsof the SU(3) splittings; wedisregardisospin breakingfor the momentandput m~,= ind.]

The first order SU(4) massformulae (see appendixA) imply the following relationsamong theobservedmasses:

C_~N_~N_DFK_Dp~1~ 172A_N~_NK_IT K-IT K*_P K*_p ( .)

[Exceptfor the relationsinvolving the GoldstonebosonsIT, K, D, F — for which onehasto useM2— thefirst order quark mass expansiondoes not distinguishbetweenmassformulaelinear in M andmassformulaefor M2.] Theisoscalarparticlesundergomixing.The 16-pletof vectormesonsis nearly ideallymixed [see appendix B for a discussionof the OZI rule in the case of SU(3) nonets] and thecorrespondingfirst order massformulaefor ~‘pand~/simply the furtherrelation

(17.3)

The 16-pletof pseudoscalarsis not ideally mixed— we do not discussthe SU(4)massrelationsfor i~,‘q’

and?lc here.Considerfirst the linear massrelations.The observedmassesAc(2273),£~(2430),D*(2007), F*(2140)

and t/i(3097) imply CA,= 7.6, C~= 5.9, CD~= 10.6, CF. = 10.8 and C,,., 9.5 respectively.Thescatteringof thesevaluesis consistentwith what oneshouldexpect:the ratio of thelargestto the smallestvalueisCF.: Ci.,, = 1.8, comparableto the asymmetryin the vectorcouplingconstantsmentionedabove.Usingthe average

C = 8.9i~ (17.4)

of the abovefive valuesandusingthe ratio m,:01 = 25 the value inc (1 GeV)= 1.35 ±0.05GeV leadsto

thefollowing estimatesfor the light quark masses:

01 (1 GeV)= 6.3ii~MeV, m, (1 GeV) = 160~MeV. (17.5)

As mentionedabove it doesnot makesenseto uselinear massrelations for the light pseudoscalars.Using quadratic mass relations rather than linear ones throughout we get CA,, = 11.8, C~= 9.3,CD = 15.3, CF = 17.2, CD. = 17.7, CF. = 19.6, C,,, = 20.5. As was to be expected the scattering issubstantiallylargerhere(relative splittings in M2 aretwice as largeas relativesplittings in M). What isworseis that the valueof C,,, obtainedfrom the quadraticmassrelation differs from the oneobtained


from the linear massformula by almost a factor of two. It is howevernot difficult to understandtheorigin of thisdiscrepancy:the bulk of M,,, comesfrom therestenergyof the c-quarks— if amassformulaneglectsthis contribution it is bound to fail. The first order SU(4)massformula for the squareof themassof the 16-plet of vectormesonsmaybe written as (OZI rule, ideal mixing):

M~=A+B(m1+m2) (17.6)

where in1 and m2 are the massesof the two quarkscontainedin the state in question.The formulaobviouslyneglectsthe term (in1 + in2)

2 that correspondsto the quark rest energy.In the caseof lightquark bound statesthe terms of order (inquark)2 in the quark mass expansionof M2 are smallcorrections,for heavyquarkstheycontainthe bulk of the boundstatemass.[For very heavyquarksM2is given by (in

1 + in2)

2 up to effectsof order v2/c2— ai.]The linearSU(4)massformula

= a + b(ini + in

2) (17.7)

does not have this obvious shortcoming:if b is close to 1 then the linear mass formula may beapproximately valid even if one or both of the quarks are heavy. It is therefore not a prioriunreasonableto usefirst orderformulaelinear in the massfor particlescontainingheavyquarks,but it isunreasonableto usequadraticfirst orderformulae. In theprecedingsectionswe haveshown that in thecaseof light quarkboundstatescrudeestimates(additivity rule) for the termsof order (mquark)

2tendtotransformthe quadraticmassformulaeinto the correspondinglinear ones.Onemay of courseapply thesameanalysisalsoto the SU(4)relations.Sinceoneis howevernot looking at small correctionsheretheresultsarerathersensitiveto the mannerin which oneanalyzesthe termsof order (inquark)2 andit is notclear that one learnsa greatdeal from this analysis. We find it moreinstructive to briefly discussanextremelysimpleansatzthat allows one to understandthe main featuresof the observedpatternofsymmetrybreakingevenif the quark massesare not small [232].For the Goldstonebosonswe take aquadraticSU(4) massformulaof the type

M~=A + E(m1+ ,n2)+ C(m1+ rn2)

2. (17.8)

[We do not considerthe isoscalarstates~, ri’, m, which arecomplicatedby mixing.] Sincethe massofthe multiplet vanishesin the chiral limit, the constantA is zero. In the oppositeextremeof very heavyquarksthe massof the boundstatemust tend to the sum of the quark masses.For the massformula(17.8) to havethis propertythe constantC mustbe equalto 1:

M~=.~(m1+m2)+ (mi+ rn2)

2. (17.9)

For the vector mesonswe take the first order SU(4) formula (17.7) linear in M and again require thatthis formulahasthe right behaviourfor very heavyquarks:

M~=Mo+m1+m2. (17.10)

[We couldjust as well also usea formula of the type (17.8)for the vectormesons;this would introducean additional free parameterthat could be tuned to get even better agreementwith the observedspectrum.Thelinear formula (17.10)correspondsto the choice A = M~, B = 2M0, C = 1 in (17.8).]We


then notethat thedifferencesM~— M~.= 0.58GeV2, M~.— M~= 0.55GeV2, M~.— M~= 0.55 GeV2areessentiallyflavour independent.The massformulae(17.9) and (17.10)havethis property, providedB = 2M

0:

= 2Mo(m1+ m2)+ (in1 + in2)

2. (17.11)

Apart from the quark massesin,,, ind, in,, m~the mass formulaefor the pseudoscalarand vectormultipletsthus involve a single unknown,the massM

0 of thevectormesonsin the chiral limit, given byM~= M~— M~..We may thereforedeterminethe quark masses01, ins, in,, (recall that we disregardthemassdifferencem~— lfld) from the massformulaefor IT, K andD with the result

01=6.2MeV, m,= 140MeV, m,,= 1250MeV, m,/th=22.7, C9.3. (17.12)

The linear massformulafor the vectormesonsthenof coursereproducesthe massvaluesof K* andD*ratheraccurately— this is built in. What is not trivial is that the simple linear formula (17.10) alsogivesacceptablevaluesfor the massesof ço(1020), t/i(309

7) and F*(2140): Ms,, = 1044MeV, lvi,,, = 3256MeV,MF* = 2150MeV.

The model is too crudeto allow us to specifytheprecisenormalizationof thequark massparametersin termsof the quark massesthat appearin the Lagrangian.In view of this the fact that the absolutevalues of the massesoneobtainsfrom the model are consistentwith the estimatesfor the runningmassesat a scaleof order 1 GeV maynot appearto be very significant. Note that in the linear massformula (17.10)one might just as well think of somesort of constituentmassesthat remain finite in thechiral limit — what countsthereis only the propertythat the quark massdifferenceshavetheright value.In the massformula (17.11) for the Goldstonebosonsit is howeverevident that the ansatzdoesnotinvolve constituentmasses:the massformulacan only be right if the quark massesthat appeartheremeasurethe breakingof chiral symmetry.Sincethe algebraicstructurein the chiral limit is correctwebelievethat the model does allow one to estimatequark mass ratios in a meaningful manner.Theestimatefor in,: 01 is indeed essentiallythe one given in section 8. The value C = 9.3 basedon themassesof p, IT, K, D confirms the estimateC = 8.9i~derivedabovefrom the massesof A,,, D* and cl’.If the valueC = 9.3 is correct,thenthe absolutevaluesof the running light quark massesat p. = 1 GeVfollow from themeasuredquantitym~(1 GeV)= 1.35GeV andthevalue in,: 01 = 25 given in section13:

01 (1 GeV)= 6.0MeV; in, (1 GeV)= 150MeV. (17.13)

Finally, the ratio R = 43.5 determinesthe massesin,, and ind individually:

in,, (1 GeV)= 4.4MeV, ind (1 GeV)= 7.7MeV. (17.14)

18. Quark massesfrom grand unified theories

In the precedingsectionswe haveexclusivelyconsideredlow energyphenomenaand haveexploitedthe fact that at low energiesQCD + QED representsan accurateeffective Lagrangian. In thatLagrangianthe fermionmassesareexternalparameters.To understandthe sizeof theseparametersonehasto go beyondthe effective low energytheory andstudythe degreesoffreedomthat are frozenat lowenergies.

StSt

StStStStSt


Oneset of degreesof freedomthat certainlyplaysa centralrole in the massproblemis known: theelectroweakgaugefields.The electroweakinteractioncan be formulatedin a consistentmanneronly ifthe Lagrangianof the theory doesnot contain fermionmassterms.In the standardframeworkbasedonSU(3)x SU(2)x U(1) the presenceof mass termsin the low energyeffective Lagrangian is associatedwith a spontaneousbreakdownof the SU(2)gaugesymmetrywhich gives massnot only to the fermionsbut at the sametime alsoto the W andZ gaugefields. In fact, the electroweakSU(2)groupcannotbe asymmetryof the groundstateif the dynamicsof the stronginteractionproducesa fermion condensate:the vacuum expectationvalue (OIüu!0) = 2(0!fiRuLIO) necessarilybreaksSU(2), becausethe left- andright-handedfields transformin a differentmannerunderthis group. It doeshowevernot seemto bepossibleto understandthe observedmassscaleson the basisof the SU(2)asymmetriesdue to the quarkcondensatealone.Other degreesof freedomthan thosecontainedin the observedfermions and theSU(3)X SU(2)X U(1) gaugefields seemto be involvedin the massproblem.

The Higgs modeldemonstratesthat thereexist renormalizableLagrangianswhich do reduceto thepropereffectivetheoryat low energies.It suffices to introducea doublet of scalarfields with a suitableself interaction to get spontaneousbreakdownof SU(2). The observedfermion mass matrix can bearrangedby giving the Yukawacouplingsof the scalarfields with the fermions suitablestrength.In thisform the Higgs modelhoweveronly demonstratesthat thereexists atheoreticallyconsistentframeworkwhich doesreproducethe low energyphenomena— it doesnot shedany light on the fact that ine is sodifferent from in~or on the origin of the flavour asymmetriescausedby in,, < ind < in, < in,, < inb < in~.

These flavour asymmetriesare introduced into the Lagrangianby hand in terms of widely differentYukawa couplingconstants.

Thereareother theoretical indicationsfor the needofa larger framework:(i) The standardmodel doesnot explain why the proton and the electron have exactly opposite

electriccharge(quarksandleptonstransformin a differentmannerunderthe U(1) group;the standardmodel doesnot requirethe eigenvaluesof the U(1) generatorsto be relatedin sucha mannerthat saythe d quarkcarriesonethird of the electricchargeof the electron).

(ii) In the standardmodel it is a mystery why thereremainedan appreciablenumberof protons,neutronsandelectronsafterthe big bang was over; sincethe modelseparatelyconservesquark numberand lepton number it calls for a tiny excess of quarks over antiquarks in the very beginning(alternatively,the universemusthavemanagedto keepislandsof quarksandantiquarksseparatedwellenoughfor them to escapeannihilation— thispossibility looksevenmoremysterious).

(iii) The valuesof the threecouplingconstantsin SU(3) x SU(2)x U(1) arearbitrary— theoretically,the modelwould also makesenseif theseconstantsweredifferent from what is observed.

Grandunifiedtheories[234,235] proposearemarkablysimple andbeautiful solutionto thesepuzzles.In the minimal model [235]the groupSU(3)x SU(2)x U(1) is a low energyrelic of the groupSU(5) thatunifies all known interactionswith the exceptionof gravity in a gaugefield theory containinga singlecoupling constant.Since the generatorof the U(1) subgroupis amongthe generatorsof the simplegroup SU(5) its eigenvaluesare integer multiples of the eigenvaluesthat occur in the fundamentalrepresentation.This explainswhy the electric chargeis quantized.The irreducible representations~and ~fflunify quarksand leptons;the interactionsmediatedby the gaugefields of SU(S) transformquarksinto leptonsand thereforeallow antiprotonsto convertinto electrons.The conversionmay besufficiently inhibited at low energiesto haveescapeddetection,but yet be sufficiently fast in the earlyhot phaseto explainwhy asmall CP violating componentin the massmatrix will preventthe baryonsfrom disappearingaltogetherafter the universehadbecometoo coldfor baryonpair formation.For themodel to be realisticit is essentialthat the SU(5) symmetrybreaksdown at a massscalefor which thevalue of the effective couplingconstantg2/4IT is close to ~. This value appearsto be too small for a

134 1 Gusset and H Leutwyler, Quark masses

spontaneousbreakdownto occurwithin the theory itself. Onemayhelp it alittle by introducinga scalarfield ~ thattransformsaccordingto the 24 dimensionaladjointrepresentationof SU(5)andthat hastheproperself interaction to makeit energeticallyfavourablefor the groundstateto be asymmetricunderSU(5), (01410) � 0. This expectationvaluegives massto the gaugefields X, Y that transformquarksintoleptons (the fermions remainmasslessbecausethe left-handedfields transformdifferently from theright-handedfields underthecenterof SU(5)— theHiggs fields areinvariantunderthe centerandhencecannotproduceany left—right transitions). If at the scale M~ the effective coupling constantg2/4ir iscloseto ~ then onecan understandwhy a

5, a and0w havethe valuesobservedat low energy:for the

stronginteractionto bind light quarks,to form a fermion condensateandall that the effectivecouplingconstanta, must be of orderone. Since this valuehasto grow out of ~ the size of the boundstatesproducedby the stronginteraction is larger than the Comptonwave length of X and Y by about 14orders of magnitude(refs. [236,237];see Antoniadis, KounnasandRoiesnel [238] for more precisestatementsandreferencesto the recentliterature):

M~ 1.5 x 1015A~. (18.1)

The fine structureconstantdecreasesduring this huge interval from ~ . = th to ~ and sin2 O~decreasesfrom the SU(S)Clebsch—Gordonvalue~ to 0.22.

What remainsa mysteryin thisscenariois why the symmetrybreakdownof SU(2)necessaryif SU(3)is to produceafermion condensateoccursat a massscaleM~, M~ 100GeV, 12 ordersof magnitudesmaller than M~, but two ordersof magnitudelarger than the scale of the stronginteraction boundstates(Gildener[239];seeEllis [240]for an excellentreviewof the problem).

Georgi andGlashow[235]proposedthat the secondbreakdownis associatedwith a scalarHiggs fieldtransformingaccordingto the representation~ that happensto get a relatively small vacuumexpec-tationvaluewhich breaksSU(2)andleavesus with the effectivelow energytheory SU(3)X U(1). TheWand Z gaugefields get the massesneededto reproducethe Fermi constantif the vacuumexpectationvalue hasthepropersize andthe fermionsobtain massesin proportionto the Yukawa couplings.Thesymmetrygroup assertsthat the Yukawacouplingconstantsaredeterminedby two free parametersforeachoneof the threegenerations(moreprecisely,two for eachpair of generations).The massesof theu, c andt quarksandthemassesine, m~and inr maybeusedto tunetheseconstants.The symmetrythenpredicts

indine, in,1fl~, inbin~. (18.2)

[Quite apartfrom the questionof how to get theproperscalefor M~ andM~ anyHiggs modelwhichtreatsthe threefermiongenerationsas independentsuffersfrom the defectmentionedabove:someofthe flavour asymmetriesare still introducedby hand in the form of different values for the Yukawacouplingconstantsassociatedwith the threegenerations.]The massesare renormalizationdependent;sincethe leptonsare not surroundedby gluonstheir masseschangemuchlessasthe scaleis changed.Ifrelationslike (18.2)hold at very short distancesat which all degreesof freedomof SU(5) are in action(p. ~ M~) thenthe b quarkmust be considerablyheavierthan the T-lepton at low energies.In orderofmagnitudethe observedratio

S

inb(mb)/in~ = 2.38±0.06 (18.3)

indeedagreeswith the calculatedrenormalizationeffect in leading log approximation[241].A more

55555555St5555Si

I Gasser and H Leuiwyler, Quark masses 135 I

5555

precisecalculationof the renormalizationeffect appearsto give a value for the ratio inb.in,. that issomewhatlargerthanwhat is observed[242,243].

To analyzethe relationfor in, we considerthe ratio mb. in, whichin the minimalsubtractionschemeis renormalizationindependent.With the value inb (1 GeV)= 5.6±0.4GeV given in section 16 theSU(5) relation

in, = ~ inb (18.4)

implies

in, (1 GeV)= 330±25 MeV (18.5)

avaluethat is substantiallyhigher thanthe estimatesgiven in the precedingsections.Note that with theratio in,: 01 = 25.0±2.5 the abovevaluecorrespondsto 01 (1 GeV)= 13.2±1.8MeV.

Finally, the relation

m,Jmd = in/line = 206.8 (18.6)

maybe compareddirectly with the quark massratiosobtainedin sections13 and14 whichimply

in,/ind 19.6±1.6 (18.7)

in obviousdisagreementwith (18.6).Sincethe massesind andm~arevery small it is conceivablethat therelation(18.6)is affectedby somesmall residualinteractione.g. dueto aheavyfermionwith a massthatmay be as large as the Planck mass [244].Georgi and Jarlskog[245] insteadproposeto replacetherelations(18.2)by

ind3ine, in,~

3in,, inb in,. (18.8)

andshowthat theserelationsmaybearrangedtogetherwith the familiar formula for the Cabibboangle[246,247]

tg2 0,, = ind/in, (18.9)

if oneconsidersa richer spectrumof Higgs particlesandchoosesthe Yukawacouplingsappropriately.[Note, however, that it is difficult to tune the Cabibboangle “naturally”; see Gatto, Morchio andStrocchi [248]; Sartori [249] and the referencesquoted in these papers.] We do not discussthetheoreticalstatusof the aboverelations[250—253],but simply comparethem with the quark massvaluesobtainedin the precedingsections.The prediction (18.5) is lowered by a factor of 3 and now givesin, (1 GeV)= 110±8 MeV, which is somewhatlow. The ratio in,: 1~d is modified by a factor 9 toin,: ind = 23, which is somewhathigh, but is not inconsistentwith the measuredvalue(18.7).

It is clear that the presentunderstandingof the massproblemin the framework of grand unifiedtheoriesdoesnot give reliablepredictionsabout the fermion masses.Many basicproblemshaveyet tobe solved: (i) It is not clear why the curvatureof space-timeis determinedby the physical energy-momentumtensor,which measuresthe differenceof the actual local energydensity and the energy

136 J. Gasser and H. Leutwv/er, Quark masses

densityof the asymmetric,physicalgroundstate— in the absenceof a principle that guaranteesthis thetheory is likely to predicta cosmologicalconstantof order GM4X. Unlessgravity is modified well beforethe Plancklength is reachedit would generatecurvatureof order (10-20cm)2 or more andruin eventhe mostbeautiful scenario.(ii) A theory that embedsthe effective low energyLagrangianSU(3)x U(1)in a simple gaugegroup suffering spontaneousbreakdownleadsto heavy monopoles[254,255]. Thephysical significanceof theseexotic objects is not understood.(iii) It is not clear that all the fieldsappearingin the effective low energy Lagrangian are in fact still elementaryat distancesof orderM~t_theymay turn out to be compositealready at a much larger scale. In particular,the fieldsresponsiblefor the spontaneousbreakdownof SU(2) may havean interestinglife of their own with acharacteristicscaleperhapsof order I TeV (seee.g. refs. [256—261]).(iv) The occurrenceof two scalesM~, M~ is difficult to understandin terms of a hand made Higgs potential.Whethera consistentrealistictheory emergesif oneassociatessomeof the forcesresponsiblefor symmetry breakdownwithradiativecorrections[262,264] remainsto be seen.(v) Radiativecorrectionsmay alsobeat the origin ofsomeof the fermionmasses[263].Thesmallnessof ratioslike ine: m~or m~:m~suggeststhat only someof the fermionsget their massdirectly from the samesourceas W andZ. At any rate a solutionof thegenerationproblemis not in sight.

Theideathat the asymmetriesin the effectivelow energyLagrangianarenot relicsof an asymmetryin the underlyingtheory,but are there as a consequenceof the fact that the stateof lowest energyhappensnot to be symmetricis too beautiful to be wrong— the availablepartial realizationsof this ideaprovide us with instructive models showing how this could happen; if thesemodels are howeverequippedwith all the parametersnecessaryto reproducethe properlow energytheory theybecometoougly to be right.

19. Summaryandconclusions

1. At low energies(p. ‘~ 100 GeV)all degreesof freedomexceptthe quarks,the gluons,the leptons,the photonand gravity arefrozen.Low energyphenomenologyis consistentwith the standardpictureaccordingto which QCD+ QED is a very accurateeffective low energy theory (unlessone considersvery largesystemsfor which gravity cannotbe neglected).

2. The effectivelow energytheory involves two couplingconstantsg, e andthe massmatricesof thequarksandleptonsas externalparameters.The theory is renormalizableand maybe characterizedbythe on-shellvaluesof e, mlept andby therunningcouplingconstantg andtherunningquark massmatrixat scalep..

3. We havereviewedthe information aboutthe eigenvaluesm,,, m,,~,in,, m,,, mb of the quark massmatrix. We use the MS schemeand normalize at the running scale p. = 1 GeV. [Sincethe renor-malizationgroupinvariant scaleA is small, a changein p. producesonly a small changein the valueofm(p.): for A <200MeV the value of ni(p.) differs from in (1 GeV) by lessthan 10% if 0.80GeV< p. <

1.3GeV, comparetable 1.] In contrast to the value of the running mass at a given scale therenormalizationgroupinvariant mass01 is sensitiveto the value of A which is not known to sufficientaccuracy.[If we wantedto quoterenormalizationgroupinvariant masseswe would alwayshaveto giveseveralnumbersfor a representativeset of values of A.]

4. To first orderin e2 the effectiveLagrangiandescribingthe propertiesof hadronicboundstatesisgiven by

~ (19.1)


The interaction due to photonexchangeis ultraviolet divergent and must be renormalized.As it isfamiliar from QED the counter terms i~E,Ainq, i~g are not unique— the total Hamiltonian ofQCD + QED is uniquelydefinedby the constantse, m1~~5,g(p.), inquark(p.), the splitting into pureQCDandan effectiveelectromagneticinteractionis not. Numerically,the inherentambiguity in the notion ofa quark mass in pure QCD is howevernot significant: A changeof the renonnalizationpoint thatdefinesthe renormalizedelectromagneticinteraction by a factor two affectsthe quark massonly byabout l%~for charge~ andby ~%ofor charge~— this is far beyondthe accuracyto which quark massescan be determinedatpresent.

5. In the limiting caseof a theory involving only masslessquarks the counterterm ~inq is absent.Apart from the renormalization~E of the vacuumenergythe divergentpieceof the electromagneticinteraction is absorbedin a suitable shift of the strong coupling constant, i.e. in a changeof therenormalizationgroupinvariant scaleA. A changein the scaledefining the renormalizedelectromag-netic self-energyamountsto a universalmultiplicativerenormalizationof all physicalmassesin pureQCD (seesection11). A changein the scale by a factor two changesthe massof the proton in pureQCD by 0.06MeV. If m~and ind vanish all isospin breaking inass differencesare finite and may beworkedout on thebasisof the unrenormalizedCottinghamformula.

6. In the realworld with m~� ind the Cottinghamformulafor isospinbreakingmassdifferencessuchas M~— M~ must be renormalized.The fact that the divergenceonly hasa tiny coefficientproportionalto am,,,aind is consistent with the analysisof the Cottinghamformula: the contributionsfrom the deepinelasticregion that areresponsiblefor the divergenceareindeedinvisibly small.

7. We have reanalyzedthe electromagneticself-energiesof the qqq and q~ground states. Theresultingvaluesfor the isospinbreakingmassdifferencesin pureQCD aregiven in table3. Thereis animportantcheck: in pure QCD the massdifferences~ + .~— 22°and IT~— IT

0 vanish up to termsoforder (in,, — md)2 [~O — A and IT0 — i~mixing and higher order terms in the quark mass expansionproducea contributionestimatedat (.~+ £ — 2.~°)ocD= ±0.02MeV, (ITt — IT°)OCD= ±0.1MeV]. Thecalculatedelectromagneticcontributions to thesemass differencesindeed agreewith the observedvalueswithin the errorsof the calculation.

8. In pure QCD the massesof the boundstatesare determinedby A and by the quark masses:M,,(A, in,,, ind, in,, inc, . . .). Since the massesof the light quarks u, d, s turn out to be small incomparisonto their typical kinetic energyit makessenseto expandthe massof the boundstatesinpowersof in,,, ind, m,, keeping in,~,inb,... fixed at their physical values.For the massof the proton,e.g., the quarkmassexpansionis of the form

M~= a + binquark ~ (19.2)

(the first order expansionfor M~is of the sameform—first order massformulaedo not distinguishbetweenM and M2). In the case of the pseudoscalaroctet the mass vanishesin the chiral limitin,, = ind = in, = 0. The expansionfor M,,. takesthe form

M~ = Bin quark (19.3)

(a quark mass expansionholds only for M~ the quantity M.,r has a squareroot singularity atinquark = 0).

First order massformulaesuch as (19.2) and (19.3) indeedgive a satisfactoryaccountof the SU(3)asymmetriesin the boundstatemassesdueto in,, < ind< in,. In particular,the first ordermassformulaeimply the Gell-Mann—Okuborelationswhich arevery well satisfied.Furthermore,the ratio


R = (in, — 01)/(md— in,,); 01 = ~(m~+ ind) (19.4)

determinesthe size of the SU(2) splittings producedby in,. � ind in comparison with the SU(3)splittings generatedby in, � 01: if R is known then the mass differencesp — n, .~— ~ o —

~ and K~— K°can be calculated from the observedSU(3) splittings ,~‘— N, .~ — A and K2— IT2.

Thevalueof R that fits K~— K°roughly agreeswith .~— .~ and.E°— ~. This valuehoweverleadstoap — n massdifferencethat is considerablylargerthan whatis observed.Quite apartfrom the needtoresolvethis discrepancyone hasto studythe higher order termsin the quark massexpansionto see towhat extentfirst ordermassformulaecan be trusted.

9. As pointed out by Li and Pagelsthe quark massexpansionis not a simple Taylor series— thetermsneglectedin (19.2) and (19.3) are not of order (inquark)2. In the chiral limit aroundwhich one isexpandingthe theory containsmasslessphysical particles (Goldstonebosons)which generateinfraredsingularities.The coefficients in the formal Taylor expansionare infrared divergent.The properquarkmassexpansionfor the proton masse.g. hasthe structure

lvIi. = a + binquark + C(inquark)312 + . (19.5)

The sizeof the nonanalyticterm is determinedby current algebrain termsof f,r, g.~.,M,,., MK andM~.Numerically, the nonanalyticterm is a disaster:it is as big as the first order term and thereis everyreasonto expectthat the higherorder termsin thequark massexpansionareasbig asthe termretainedin (19.5). If one usesthe quark massexpansionin this rawform (chiralperturbationtheory) onedoesnot get meaningfulresults.To obtain a reliableapproximationschemechiral perturbationtheory mustbeimprovedby reorderingthe expansionandsummingup leadinginfrared singularitiesthat occurto allordersin the quark mass(seeappendixC). A simplecoherentmethodthat achievesthis is basedon thenotion of an effective chiral Lagrangian.As long as the effective Lagrangiangives the particles thepropermassesandsatisfiesthe soft-piontheoremsit alsoproducesthe correctleadingnonanalytictermsin the quarkmassexpansion.To purify the massof theprotonof its leadingnonanalytictermsit sufficesto subtractthe self-energydiagramcorrespondingto a suitableeffectiveLagrangian.If the nonanalyticterms are handledin this mannerone finds that the correctionsto the first order massformulaeareindeedsmall.They areneverthelesssignificant as they accountfor the flavour asymmetriesin the massspectrumcausedby the fact that different membersof a multiplet get an unequalshareof mesoncloudenergy. In fact, the correctionsrequiredby improved chiral perturbationtheory not only lead to aperfectly coherentapproximationscheme,but explainwhy earlierdeterminationsof the ratio R did notgive a consistentpicture.

10. Apart from the nonanalyticcontributionsthe quarkmassexpansionalsocontainsregularhigherorder terms, presentevenif the massesof the boundstateswereanalyticin in,,, ind, in,. We givesimpleestimatesof thesecontributionswhich show up e.g. in the differencebetweenthe massformulaefor Mandfor M2.

11. Armed with theseestimatesof the higher order terms in the quark massexpansionwe thenanalyzethe two quark massratios inu m,~:in, on the basisof the observedmesonandbaryonmasses.We determinethe ratio R on the basisof five independentmanifestationsof isospinbreaking(K~— K°,p— n, .~— ~, ~!°— E and p — w mixing, see table 4). The resultscluster aroundR = 43.5 which iswithin the estimateduncertaintyof eachone of the five values.Treatingthesevalues as independentmeasurementswe obtain

R = 43.5±2.2 (19.6)

SiSiStStStStStStSt

1 Gasser and H. Leutwyler, Quark masses 139

(the individual error barsrangefrom ±4to ±10). Note that the nonanalyticterms in the quark massexpansionproducea substantialcorrection to the R-valueobtainedfrom p—n; the value extractedfrom the bare first order mass formula is inconsistentwith (19.6). The information on the ratio Rderivedfrom i/i’ decayandfrom theD~— D°massdifferenceis not inconsistentwith the value43.5, butis too uncertainto give comparableaccuracy.We do not attemptto extracta valueof R from the decay

—*3IT, becausethe theoreticalpredictionfor the rate of the decayi~—~2y (which normalizesthe other

experimentalrates)disagreeswith the availabledata.Thevalueof the ratio in,: 01 maybe extractedfrom the observedmassesof IT, K and~. On the basis

of an analysisof thecorrespondinghigher ordertermsin the quark massexpansionwe obtainthe value

in,/01 = 25.0 ±2.5. (19.7)

The aboveresultsfor R and in,: 01 imply the following valuesfor relatedquark massratios:

ind/inu = 1.76±0.13, m,,/ñi = 0.72±0.03

in,/ind 19.6±1.6, ind/01 = 1.28±0.03

m,/in,,= 34.5±5.1, (m~— ind)/(in,, + ind) = —0.28±0.03. (19.8)

[Note that if the u quark mass would vanish the two ratios in,: 01 and R would be related byin,: 01 = 1 + 2R, in total disagreementwith the abovevalueswhich give 25.0±2.5 and88.0±4.4 for theleft- andright-handsides of this relation.]

12. The baryonmassesalso provideus with an estimateof the u-term in pion—nucleonscattering.Itturnsout that the higher order termsin the quark massexpansionincreasethe valueof the u-termbyabout 10 MeV. The result is

cr0=35±5MeV (19.9)

wherey measuresthe protonexpectationvalueof the operatorss:

2(p~ss~p) (19.10)

(pluu + ddjp)

Thevalueof y is not known; it is correlatedwith the valueM0 of the nucleonmassin the chiral limit[y = 0 correspondsto M0= 870MeV, for y= 0.2 one finds M0 = 670MeV]. We considerthe estimatey<0.2which leadsto u <50MeV as very conservative.The observedvalueu(2M~,.)= 65 MeV is notconsistentwith our analysis(for detailssee appendixD, wherewe alsodiscussisospin breakingin theu-terms).

13. The absolutevalue ofthe light quark massesis not known very accurately.We haveperformedanew evaluation of the QCD sum rules for the axial divergenceand show that the sum rules areconsistentwith the following runningmassesat scalep. = 1 GeV:

01 (1 GeV)= 7±2MeV, m,(1GeV)= 180±50MeV (19.11)

[the quantity 01(1 GeV) stands for ~{m~(1GeV)+ md(1GeV)}]. Smaller values of 01, in, are in-consistentwith the boundsgiven by Narisonand de Rafael, largervaluescall for an exceedinglylarge

140 J. Gasser and H. Leutwv/er, Quark masses

spectralfunction in the intermediateenergyregion between1 GeV and2GeV whichwould representaremarkablephenomenonin itself.

The evaluationof otherQCD sumrules that provideinformationaboutlight quark massesis subjectto largeruncertainties,becausethereis less information about the behaviourof the spectralfunctionthanin the caseof the axial divergence.To within the ratherlargeerror bars the resultingquark massestimatesareconsistentwith the abovevalues.

14. With the valuesfor the two ratios in,.: m~.in, given abovethe estimate01 (1 GeV)= (7 ±2) MeVamountsto the following resultsfor m~,ind and in5:

m,.(1 GeV)=5.I±1.5MeV

ind (1 GeV)= 8.9±2.6MeV

in, (1 GeV)= 175±55 MeV. (19.12)

The correspondingrenormalizationgroup invariant massesmay bereadoff from table1 (the valueof A

refersto the MS schemewith 3 flavours):

A=100MeV A=200MeVni,,=7.6±2.2MeV th~=6.3±1.9MeV

= 13.3±3.9MeV thd = 11.0±3.2MeV01, = 260±80MeV th,= 215±65MeV. (19.13)

15. Accordingto Gell-Mann, Oakesand Rennerthe pion massis determinedby the productof thequark mass01 andthe orderparameter(0Iüu~0)= (0~ddI0):

~ = —4th(0IuuI0)+~.~. (19.14)

The measuredvaluesof M,,. andf,,. thereforefix the orderparameteroncethe quark mass01 is known.With the value01 (1 GeV) 7±2MeVwe obtain

(OIfluIO)= —(1.1 ±0.3)x102GeV3

= —(225 ±25 MeV)3. (19.15)

16. The massesof the c andb quarksareknownmuchmorepreciselythanthe massesof u, d ands.Thevaluesobtainedfrom the QCD sum rules involving the electromagneticcurrentmaybewritten inthe form

m,,(m,,)= 1.27±0.05 GeV

mb(inb)=4.25±0.1OGeV. (19.16)

[The function m,,(p.) is the running c quark mass,the quantity m,,(m,,) is the solution to the equationm,,(p.)= p..] The correspondingvaluesat scalep. = 1 GeV are

m,,(1GeV)=1.35±0.05GeV

5.3±0.1GeV; A = 100MeVinb (1 GeV)= ~ ±0.1GeV; A = 200MeV. (19.17)

StStStStStSt


17. For very heavyquarksthe perturbativenotion of a dressedquark massMq (the massthat appearsin the Balmer formula) is ameaningfulconcept.The information aboutthe valueof Mb is reviewedinsection16— thenonperturbativeeffectsin boundstatescontainingc quarksarepresumablytoo largeforthe perturbativenotion of a dressedc quark massto beuseful.

18. In contrastto in,,, ind, in, the mass of the c-quark is not small in comparisonwith thecharacteristicscale of boundstates.One thereforehasno right to expect the SU(4) mass relations(which are obtainedby treating in,,, — 01 as a perturbation)to be reliable. An analysisof thesemassrelationsneverthelessleadsto a remarkablycoherentpicturewhich allows oneto extractan estimateofthe ratio

C= (in,,— th)/(in,— 01) (19.18)

that measuresthe sizeof SU(4)splittings in comparisonto SU(3)splittings:

C=9±2. (19.19)

[It is difficult to assessthe systematicerrorof the analysis— wethink that the error barsgiven in (19.19)representarealisticestimatefor the rangeof C valuesthat areconsistentwith brokenSU(4).] With themeasuredvalueof the running c-quarkmassat 1 GeV and with the ratios in,,: m,~:in, given abovetheestimate(19.19)implies the following valuesfor the running light quark massesat 1 GeV:

01 (1 GeV)= 6.2±1.9MeV, md(1GeV)=7.9±2.4MeV,in, (1 GeV)= 4.5±1.4MeV, in, (1 GeV)= 155±50MeV. (19.20)

Thesevaluesareperfectly consistentwith thoseobtainedfrom QCD sum rules.19. The recentdevelopmentsin the numericalevaluationof QCD correlationfunctionson a lattice

(for an excellent review see Hasenfratz [265,266])have made it possible to calculate observablequantitiesin termsof the parametersof the Lagrangian.Thenumericalresults[267—271]areimpressive.The evidencegiven for spontaneousbreakdownof chiral symmetryis ratherconvincing.The numericalvaluesfor the orderparameter(0I~qI0)and for the quark masseshoweverhaveto be takenwith somecaution,becausethe latticeregularizationviolateschiral symmetry ab initio; a moredetailedanalysisisrequiredto clearlyseparatespontaneouschiral symmetrybreakingfrom theresidualsymmetrybreakingeffects dueto the regularization.It is not a trivial matter to connectthe latticequark massparameterswith the running quark massesof the continuumtheory. (The quark massvalues given by differentgroupsdiffer by a factor two.) Mass ratios are scale independentand may be less sensitive to theuncertaintiesof the method.If the ratiosof the latticequark massparametersgiven by Hasenfratzet al.(01 : in,: in,, 5.6: 144: 1213)can be taken as a measurementof the ratios of the massesthat occur inthe Lagrangianof QCD then the known value (19.17) of the running c-quarkmass at 1 GeV implies01 (1 GeV)= 6.2MeV, in, (1 GeV)= 160MeV, in remarkableagreementwith the values given above(The two ratiosof latticequark massesamount to in,: 01 26, C 8.7.) At anyrate the numbersgivenfor the light quark massesconfirm the standardpicture underlyingthis review andit maywell turn outthat in the future lattice calculationswill provide us with the most accuratemethod to measuretheparametersof the QCD Lagrangian.

20. Many interestingattemptsto find a nonperturbativeapproximationschemefor QCD that doesnot rely on computer-basednumericalmethodshavebeenmade.Someof thesedynamicalschemesareindeedconsistentwith a spontaneouslybroken chiral symmetry and a non-vanishingvacuumexpec-tation value of 4q whose size can be estimated(seee.g. refs. [272—2771).The resulting quark mass


estimatesagreewith the valuesgiven aboveto within a factor of two. More work is neededto establishthat masslessQCD indeedimplies a nonvanishingvalueof (0~~qt0).In fact, the literaturealsocontainsaconsiderablenumberof papersthat arebasedon the (explicit or implicit) assumptionthat the vacuumexpectationvalue of ~q vanishesin the chiral limit (see appendixE). Even if this alternative istheoreticallynot very attractivewe emphasizethat it is not ruled out on phenomenologicalgrounds.Aclean calculationof the orderparameter(OIqqIO) in masslessQCD would decidewhetheror not QCDindeedrealizesa spontaneouslybrokenchiral symmetry in the form proposedby Gell-Mann, Oakes,Renner,GlashowandWeinberg,andwould at the sametime provideuswith an approximatevalueforthe meanmassof the u andd quarksif the answeris affirmative.

21. Like the lepton massesthe quark massesare scatteredover more than three orders ofmagnitude. Leaving the neutrinosaside the first generation(e, u, d) populates the interval from0.5MeV to 10 MeV, the second(p., c, s) coversthe rangefrom 100MeV to 1.4 GeV, the third (r, t, b)extendsfrom 1.7 GeV to the massof the t quark whosevalueis not knownyet. The leptonsarelighterthanthe quarksof the samegeneration.Otherwisethe distributionof the masseswithin the generationsdoeshowevernot indicateany conspicuoussystematicfeatures[normalizingthe runningquark massesat 1 GeV the ratios are (me:m,,:md)= (1:9:16); (in,., :m,,:m,)= (1:13:1.5); (m,.:mt:mb)=(1:?:3.1)].Theorder of magnitudeof the ratio ins: m,. maybe understoodon the basisof SU(5)grand unification.Theotherratioscanbe accommodated,but thereareno convincingargumentsto show why theycouldnot just as well haveany otherrandomvalues— the origin of the observedpatternof flavour symmetrybreakingremainsmysterious.

Acknowledgements

It is a pleasureto acknowledgeinformative discussionswith M. Böhm, I. Files, H. Fritzsch, F.Jegerlehner,S. Mallik, E. de Rafael,M. Roos,J. Stern,T.N. Truong,D. Wyler andwith our colleaguesatthe institute, in particularwith P. Minkowski. Wehavemadea seriouseffort to give referencesto theliterature, but we are awareof the fact that our list is unlikely to be complete, particularly sinceit islong. We apologizefor the omissions.(Thematerialcontainedin sections15, 17 and 18 only concernsthoseaspectsof QCD sum rules, SU(4) symmetry and grand unified theoriesthat are of immediaterelevanceto this review; thereferenceswereselectedaccordingly.)

Appendix A. Other multiplets: 3, 3*, 6 and 10

A.]. SU(3) symtnelly

Mesonsconsistingof oneheavyquark and onelight quarkandbaryonscontainingtwo heavyquarkstransformaccordingto the representations3 or 3*• For baryonswith oneheavy and two light quarksboth representations3* and 6 are possible.The matrix elementsof the operator4q betweenstatestransformingaccordingto the representations3, 3*, 6 or 10 aredeterminedby two constants.

In the following we exhibit the mass formulaefor the multiplets ~, ~, ~ and~ in a somewhatcondensedform. We shall assume that the unperturbed multiplets (in,, = in,,, = in, = 0) are well

575575515515515555515‘S

I Gasser and H. Leutwyler, Quark masses 14357~

separatedfrom one another,such that the mixing phenomenondiscussedin appendixB is absent.Wedenote the matrix elementsof the quark bilinearsin the statewith the largest value of isospin T,T3=+TbyB”,B’~,B’: 1

= (Tmax, Ti’3 = TmaxIÜUIT3 = Tmax, Tmax) etc. 1

Thenwe find the following first order quark massexpansions: I

Triplet

Baryons Mesons

Iuhh,dhh,shhl Iuh,dh,sh~

~+± ± fl±J ~ D, F~J

= A3+ ~(m,,+ ind+ m,)(B~+ 2B~)+ 2(B~— B~)[~(m,,— ind)T3+ (01 — m,)(T—~)]

wherewe used

— DdLI 3 — LI 3

Antitriplet

Baryons Mesons

f ush,dsh,udh ‘~ f hd,hU, h~lE”~ A0 1D~,D°,F~

M

2= A3.+~(,n,,+ind+ ~ md)T3 (01—

with

— DU

L)3* —

Sextet

fuuh, udh, ddh,ush,dsh, ssh

‘~ i÷~°s~s°a

M2 = A

6 + ~(m,.+ ind + m,)(B~+ 2B~)+(B~— B~)[~(m,,— md)T3 + (01 — m,)(T—

with

13’ —


Decuplet

f uuu, uud,udd,ddd, uus,uds,dds,uss,dss,sssLi±± ~+ ~o ~ ~*± ~*o ~*— ~~*o E~ Q—

M2 = A

10 + ~y(inu + ind + m,)(B~0+ 2B~)+ ~(B~0— B~)[~(m,,— md)T3+ (01 — m,)(T— 1)]

with

B~0=B~. (A.1)

For the multiplets3, 3~,~ andj~QSU(3)-symmetryamountsto quarkcounting(equalspacingrule): for

thesemultiplets the SU(3)massformulamaybe written as

M2= a+ b(N,,mU+Ndmd+N,m,)

whereN,. is the numberof up quarksin the statein question.

A.2. SU(4) symmetry

In the limit in,, = ind = m,= in,, = 0 the ground statebaryonsconstitutea 20-dimensionalrepresen-tation of SU(4)andthe mesonmultiplets transformaccordingto 1~J~31.lithe quark massexpansionofthe protonmassin powersof in,,, ind, in, and in,, is known,onecan calculatethe expansioncoefficientsfor all membersof the 2!2 = 8 ~ 6~ 3~ ~ Thecoefficientswhich occurin thequark massexpansionof

~ and3* with respectto in,,, ind and m, may then begiven in termsof the quarkmassexpansionofthe proton.

In fact let

= A + Eum + ~dmd+ Bm,+ ~cin

Eu(pIuuIp) etc.

Then onehasthe following relations(valid to leading orderin an expansionin in,,):

A

3 = A + m,,E~ A3~= A + ~m,,(2E’— Ed + 2Ev); A6 = A + in,,Ed

B~=1i”; B~.=B;~=~(E’+4fr’+E”); B~=A~

B~.=E’; B~=B’6=E’.

Equivalently,one may expressthe massesof the statesin 3, 3*, ~ and~ in termsof the massof theT = 0 statein the octet,M~, and the F- andD-couplingsof the octet

~

D=~(th— m,)(Eu+E5_2E)=~(M~_M~,).

555‘S‘S‘S‘S‘S‘S‘S

J. Gasser and H. Leuiwy/er, Quark masses 145

‘S

We summarizethe resultingexpansionbelow [2781:

Triplet

~ms—rn rn~—in

Antitriplet

= M~+ (i - I ‘~)(F + ~D)+ (F - ~D)(rnd ‘~ T~- 2T)

Sextet

M~= M~ _(i+ in~ ‘~)F+~(1+3 in~ ~~)D+F{in~ mU T3+2T1m,—m 3 ,n,—m in,—m j

Octet

(A.2)

Thequark massdifferences01 — in, and in,, — ind inducemixing in zerothorder betweenstatescarryingthe sameelectric charge,strangenessand charm quantumnumbers.(i) The mass difference 01 —

mixes the isospindoublets(usc, dsc) that occur in the representations~3”and~. The mixing angleis oforder (in, — 01): (in,, — 01). (ii) The isospinviolating massdifferencein,. — ind generatesmixing of the twooctet states.~‘°, A. The mixing angle is of order (ind— m~):(in,—01) [seesection 9]. (iii) The massdifference in,, — ind alsomixes the isosingletudcin the representation~3~”with the isovectorstateudcinthe representation~. Themixing angle is of order (rnd— m~):(m,,—01). [me—ind also slightly modifiesthe mixing amongthe isospindoubletsmentionedabove.]

For these8 statesthe massformula(A.2) thereforeonly holdsup to termsof order (in, — 01)2: (in,, —

01), (ind— m,,)2:(m,—01), (ind— mu)2:(m,,—01) and (ind~m,,)(m,—01):(m,,—01) respectively.[For a

moredetailedanalysisof the mixing problemin the casein,, = ind see Borchardt,Mathurand OkuboJ279].]

The mesonsoccur in 16-plets: on accountof the OZI rule the representationsi~and~iarenearlydegenerate.We refer the readerto the literature[279,280] for a detailedaccountof the generalmassformulae. For the unmixed statesthe first order SU(4) mass formulae are equivalentto the quarkcounting rule:

a + b(N~m~+ Ndind + N,m,+ N,,m,,).

This formulais alsovalid for the mixed states,provided the mixing is ideal (OZI rule). As discussedinappendixB (in the context of nonetdegeneracyin SU(3)) the mixing is nearly ideal for all multipletsexceptfor the pseudoscalarsfor which ~ is nearlypure~c,the ~ is closeto the SU(3)octetstateandthe ~‘ is close to the SU(3)singlet state(uu+ dd + ~s)/\/3 [with asizeablecontaminationfrom FF].

146 1 Gasser and H. Leutwy/er. Quark masses

Appendix B. Mesonnonets

As mentionedin section 7 the OZI rule implies that the singlet and octet mesonscontaininglightquarksare nearly degenerate.For definitenesswe use the notation appropriatefor the vectormesonnonet.Since the width of someof thesemesonsis not small in comparisonwith the mass splittings(F~= 158MeV, MK. — M~= 116MeV) we haveto considerboth the real andthe imaginarypart of theposition E = M — ~iFof the pole in the propagator.Both M and F arefunctionsof A andthe quarkmasses.The width is very sensitiveto the positionsof the thresholdfor decay(phase-space)— it is ahighly nonanalyticfunction of the quarkmasses.If we would e.g.put the massesof the u andd quarksequalto the physicalvalueof m,(in~= ind = in, = in

0) thenthe entirenonetwould be stable;both octetand singlet would be close to 1020MeV whereasthe pseudoscalarmesonswould occur at the mass(in,/th)~

2M,.= 700MeV, too large for the vector mesonsto decay. As the massesof u andd aredecreasedat fixed in. the multiplet splits, the statescontainingpredominantlyu andd quarksbecominglighter. When the meanmass01 = ~(inu + ind) hasreacheda valueof the orderof ~m,thedecaychannelfor thep is openedandthe polestartswalking off the real axis. At the physicalvalueof 01 the realpartreachesthevalue776MeV andthe imaginary part is —79 MeV. Thepoles of the otherstatesstaycloserto the real axis: the imaginary parts of M,,, MK* and M

55 are —5MeV, —25 MeV and —2 MeVrespectively. First order perturbation theory only shifts the real parts, by an amount given by theexpectationvalue of the perturbationc~j(m— mo)q in the unperturbedstates.It is difficult to control thehigher order terms;the main effect of thesehigher order termsis to shift the polesin the direction ofthe negativeimaginary axis.

To be on safegroundswe first analyzethe massformulaefor unphysicalvaluesof in,,, ind that arelargeenough for the vector mesonsto be stable(say 01 —~km,) andshowthat in this region of quarkmassesthe ~ state is almost ideally mixed. Sincethe decay channelsrespectthe OZI rule we thenconcludethat the ~ remainsideally mixed when the massesof in,, and

1~dareloweredto their physicalvalues.Finally we will analyzethe mixing of p°and w for which it doesnot makesenseto ignore theimaginarypart of the masses,but for which only the small isospin breakingpiece of the quark massperturbationcounts,such that a first ordercalculationwith respectto that perturbationis appropriate.

We first put in,, = ind = 01 andsupposethat 01 is large enoughfor the nonet to be stable.The firstordermassformulaefor the unmixedstatesare

M~=A8+201Bu+m,Bd

M~.=A8+th(B~+B”)+in,B”. (B.1)

(The octet matrix elementsof the operatorsüu, dd and ~sinvolve only two unknownsbecausetheF-coupling vanishes;in the notation usedin section9 for the baryon octet, the equality M,,~= M~-

requiresB’ = B”.) We denotethe masseigenstatesin the limit in,, = m,~by ~,1,5), K*), ~). The statesk~)andI~)arelinear combinationsof the singlet and of the octetmemberwith T = 0:

I~)=cos~I1)+sin~I8,T=0)

~)=sin~ I1)—cos~8, T=0) (B.2)

wherewe haveintroducedthemixing anglein sucha mannerthat for tg ~ = 1/V2 the stateki~)becomes(uu+ dd)/V2whereas~)becomespure~s.

SiSiSiSiSiSiSi


To work out the massesof thesestateswe needthe matrix elementsof the perturbation

(8, T= 0~4mqJ8,T= 0)= ~(01+ 2in,)B” + ~(401— in,)B”

(8, T = 0~4mqI1)= ~\/2(01— m,)B18

(1Jc~mqI1)= ~(2th+ in,)B1. (B.3)

In termsof thesematrix elementsthe massesare

M~=a_Vb2+c2

M~=a+Vb2+c2

a ‘M~+~(th+2m,)B”+~(401— ms)B’~’c

b = ~V2(m,—th)B18

c = ~(A8— A1)+ ~(01+ 2m,)B” + ~(4th— in,)B” — ~(201+ m,)B1 (B.4)

andthe mixing angle is given by

M2 ~ 2 +M2

sin2 ~= ~ —M~,) ~ (B.5)

TheOZI rule requiresA8 — A1aswell asthematrixelements(uu 4mq!dd),(uu~4mq~s)and(ddl~mq~.~s)to

besmall. In termsof the invariant matrix elementsthis implies

A8=~A1

B18 (B~— B”) [OZI] (B.6)

B1=(B”+~B”).

It is easyto checkthat if theserelationshold, then

M~= M~,

M~— M~= M~.— M~= (in, — th)(B” — B”) [OZI] (B.7)

tg~’=1I’F2; ~=35.3°.

The deviationsfrom ideal mixing are controlledby the ratio of the OZI violating amplitudessuch asA8— A1 in comparisonto the mass splitting producedby the quark mass term, which is of order(in, — th)(B” — B”). The smallnessof the ratio (M~— M~): (MK* — Al,,) 5% shows that the OZIbreakingamplitudesare very small. To the accuracyof first order massformulaethe mixing angle isideal, the I~)is pure cs, the I~~ii)is pure (üu + dd)I\/2.

As discussedaboveit is not easyto predictexactlyhowthe poleswill movewhen the decaychannelsopen. The real parts of the physical massesof ~,K* and p satisfy the equal spacing rule (B.7)remarkablywell for the linearmassformula.The differencesshowingup in the quadraticmassformula


arewell accountedfor by the simple recipefor estimatingtermsof order (inquark)2 given in section10.

The asymmetriescausedby the decay channelsdo thereforenot seemto havesizeableeffects on thereal parts(this conclusionis confirmedby the successof the equalspacingrule in the baryondecuplet).

We now considerthe perturbationcausedby the isospinviolating pieceof the quarkmassterm.Theunperturbedstatesarethe isospinmultiplets o3), 1,5), K*), I~).The unperturbedHamiltonianincludesthe physical quark mass term except for the piece ~(m,, — ind)(UU — dd). To first order in mu — m~thesplittingsaregiven by the expectationvaluein the unperturbedstates:

Al2 ~.q2 — ~ ~

IIK**JVJK*o_ —01 I~JViK*IVip

M~+— M~o= 0. (B.8)

As was discussedin section9 the OZI selectionrule inhibits the stateI~)to mix with I~)and ,5°).Toanalyzethe mixing of 1CZ) and I,~°)producedby the perturbationit is important to accountfor the factthat the imaginary parts of their massesare very different. To first order in in,., — m~the effectiveHamiltonian that reproducesthe propertime evolutionof the states(seeappendixF) hasthe form

H_IMP21F0 Mr,., B9~ M,,.~ ~ ( .)

with

= ~-~-(m,,—md)(/5°Ifiu— ~dI&). (B.10)

Theeigenstatesare

0 -0 M,.,~ -

I~)= Iii

(B.11)

There is no first order shift in the masses.Note that the stateslp°)and 1w) are not orthogonalonaccountof I~,.� F,.,. The ratefor w to decayinto IT~IT is given by

MI2

IT = (M~— M~)2+~(F,,.— F~)2FPo~*IT-. (B.12)

Thetransitionmatrix elementM,.,,,, maybe expressedin termsof the invariantsB”, B”, B18:

M~= M,.,,, = m,~_fh~~{sin~ (BU — B”)+ cos~ V2 B18}. (B.13)

Using the OZI relations(B.6) thisbecomes I

SiSiSiSiSiSiSiSiSiSt

1 Gusset and H. Leutwyler, Quark masses 149

M~=-”~(MK.-A~). (B.14)

The experimentalinformationon M~andtheresultingvaluefor the ratioR = (in, — 01): (inn— m~)arediscussedin section14.

Appendix C. Improvedchiral perturbationtheory (ICPT)

We split the Hamiltonian of QCD into the chirally symmetricpart H0 andthe quark massterm H1anddiscussthe perturbationexpansionin powersof H1. The formal perturbationseriesfor the massofa physical statereads

M~= M~+ (p~4inqIp)- ~ifd~x(pJT~(x)mq(x)q(0) mq(0)Jp)~ (C.1)

where M0 denotesthe mass of the particle in the chiral limit and I~)standsfor the unperturbedeigenstateof H0. Since the energiesare measuredwith respectto the physical ground state thetime-orderedproductappearingin thesecondorderterm only includestheconnectedpartof the matrixelement.The contributionof the statel~)to the sumover all intermediatestatesimplicit in this matrixelementhasto behandledwith care— in thecorrespondingsecondorderformula for theshift of energylevels in quantummechanicsthe sum only extendsover the intermediatestatesorthogonalto p). Toeliminatethe correspondingcontributionin the covarianttime-orderedproduct, one maylook at theamplitude

T(p, q)= i f d~x~55x (~IT~(x)mq(x)4(0) mq(0)~p),,,,~~. (C.2)

wherean averageoverthe spin directionsof the stateI~)is understood.We areinterestedin the limit ofT(p, q) as q—~ 0. The contributionof the intermediatestateI~)shows up in the form of pole terms‘—[M~—(p±q)

2}~which explode near q = 0. The proper second order contribution in the massfonnula (C.1) is obtainedby first taking the limit q2—* 0 andthentaking the limit pq —~0 [282].

Alternatively,onemay considerthe one-particle-irreduciblepieceof T(p,q). This notion is howeveronly definedwithin a frameworkthat identifiesthe stateI~)as the asymptoticstateof an interpolatingfield with specifiedpropagationproperties.We want to avoidthe use of sucha formalismhere,becauseit obscuresthe fact that we areonly interestedin physicalmatrix elementsof the operator4inq. Insteadwe maydefinethe amplitudeT’(p, q) by

T’(p, q) = T(p, q) — {[M~ — (p — q)2]~+ [M~ — (p + q)2]1} G(q2) (C.3)

whereG(q2) is theresidueof the polein T(pq,q2) atpq = ~q2.The prescriptiongiven aboveamountstothe statementthat the secondorder term in the series(C.1) is the value of —~T’(p,q) at q = 0. Forscalarparticlesthestandardone-particle-irreduciblepiececoincideswith T’; for particlesof spin ~this isnot the case,becausethe propagatorleadsto a polynomial in pq — the one-particle-reduciblematrixelementdoescontributeto the limit specifiedaboveandhencecontributesto the massshift [282].This

150 1 Gasser and H. Leutwyler. Quark ,nasses

differencein the one-particle-reduciblematrix elementsreflectsthe fact that the Lagrangianfor scalarfields is quadraticin the mass,while the Lagrangianfor spin ~fields is linear in the mass.

After thesekinematicalpreliminarieswenow turn to the dynamicalproblemsgeneratedby the factthat the unperturbedsystemcontainsN,~— 1 masslessGoldstonebosons(N1 is the numberof flavourswhosemasstermsareincludedin H1). We first considertheperturbationseriesfor boundstatesthat aremassivein the chiral limit, M0 � 0. Since thesestatesoccur in degeneratemultiplets of SU(Nf) it isimportant that the unperturbedbasis vectors are chosen such that the perturbationis diagonal(A—1~°-mixing).The expectationvalueof the perturbationthen determinesthe massshift in first orderof inquark. The second order term in (C.1) is infrared divergent. The divergencearises from theintegrationover x: the integrandexists,but doesnot decreasesufficiently fastas x—~ ~. The divergencehasto do with the fact that a chiral proton is surroundedby a cloud of masslessvirtual pions. (Fordefinitenesswe discussN1 = 2 andconsiderthe expansionof the proton massin powersof m~,in,,, atfixed m,. In this caseonly the pions producean infrared divergence.)The probability for finding avirtual pion in the vicinity of the proton only falls off with a powerof the distance.Becausethe matrixelement(ITI4mqIIT) is different from zero[to lowestorderthis matrix elementis the (mass)

2of the pion]the quantity (p~T4mq4mqIp)containscontributionsthat only decreaselike a power of x. The crucialpoint hereis that thesecontributionsarisefrom configurationsin which the pion is far away from theproton. The coefficient of the leading infrared divergenceis determinedby physicalpropertiesof thechiral theory: by the ~r—pscatteringamplitudefor pions of zero momentumand by the one particlematrix elementsof theperturbation.The low energytheoremsof currentalgebrafurthermoreshow thatthe scatteringamplitude is determinedby the one particle matrix elementg.~of the axial current(Goldberger—Treimanrelation).Any model Hamiltonianof the typeH

0 + H1 for which(i) H0 hasthe sameoneparticle spectrumas the chiral Hamiltonianof QCD,(ii) H0 conservesan axial vectorcurrent with the correctmatrix elementsbetweenthe vacuumand

the pseudoscalaroneparticle statesand the correct value gA for the matrix elementbetweenoneparticlestatesat zeromomentumtransfer,

(iii) H, hasthe sameoneparticlematrix elementsas the quark massterm in QCD,producesthe same leadinginfrared divergencesin the perturbationseries.[Note that only the linearinfrared divergencein the secondorder term is determinedby g.~. If gA vanishesfor the particle inquestion, the second order term in the quark mass expansiondoes not contain a linear infrareddivergence,but may containa logarithmicone.The conditionsgiven abovehaveto be sharpenedif theyareto guaranteethat the modelHamiltonianreproducesthe logarithmicsingularity correctly.]Effectivechiral Lagrangianswhich describethe pseudoscalarmesonsandthe nucleonsas elementaryfields havethe above properties,provided one restrictsoneself to tree diagrams.Fig. 3a shows the diagramresponsiblefor the leadinginfrareddivergencein M~in the languageof such a chiral Lagrangian.Notethat we are not usingthis Lagrangianbeyondthe tree graph approximationhere:we arenot carryingout the loop integral, but only look at the behaviourof the integrandat small values of the pionmomentumk — thereis no significanceto this loop integral for largevalues of k. Thebehaviourof theintegrandneark = 0 producesa linear infrareddivergence— the secondorder massformula (C.1) doesthereforenot makesense. Si

What onehasto do to carry the quark massexpansionbeyondfirst order is to reorderthe seriesbysummingup dangerousdiagrams[281].In fact the diagramsdepictedin fig. 3b producefactorsof thetype [(irI4inq~IT)/k

2]”which aremoreandmoreinfraredsingularas the ordern increases.The diagramsinvolve the samechiral one particle matrix elementsand it is easyto sum them up: one merelyhastoreplacethe masslesspion propagatorin the chiral effective Lagrangianby a massivepropagatorwith

5’‘S

Si55555555Si

1 Gasser and H. Leutwyler, Quark masses 151 1Si

I I I I

a) b) c) S

d) e)

Fig. 3. Summingup infraredsingularities.Single lines representbaryons(—) andmesons(———) in thechiral limit, thecrossesdenotemassinsertionsgeneratedby the chiral symmetry breakingterm 4mq. Open circles indicate derivativecoupling proportional to g~/f,,.Double lines denotepropagatorswith physicalmasses,andthefull circlesrepresentthecorrespondingvertexfor physicalparticles,proportionalto theaxialvectorform

factor.

M~.= (ITI4mqjIT) (fig. 3c). In addition to thesum of the contributionswith two or moremassinsertionsthis diagramalsocontainsacontributionwith no massinsertion anda contributionwith oneinsertion.We could subtractthesepartswhich arenot infrared divergent.This is not necessary,however,for thefollowing reason:the diagramwith no massinsertion only changesthe value of Mo; the diagramwithone mass insertion is linear in the quark masseswith coefficientsthat respectthe SU(N1) symmetrypropertiesof the chiral limit. This contributionmay thereforebe absorbedin the first order term — itmerelychangesthe valuesof the coefficientsB~,B”, B’ withoutaffecting their symmetryproperties.Aslong as we only extractquark massratiosfrom the spectrumof the physicalparticleson the basis ofthesesymmetrypropertiesandarenot interestedin the valueof the nucleonmassin the chiral limit orin the coefficientsB”, B”, B’ we do not haveto subtract thesecontributions. (To extract the u-termsonedoeshaveto takethesemodificationsinto account;seeGasser[282].)

The sumof the leadinginfraredsingularitiesregularizesthe divergencein the secondorder term: thediagram 3c, evaluated with a massive pion propagator is convergent. The divergence—fd~kk,., k~(k

2)3(pk)’ is convertedinto a factor 1/MIT. The entirecontributionfrom graph3c hasanexpansionof the form a + bM~+ cM~.~ . The third term in this expansionis proportional to(in,, + ind). The quark massexpansionfails, becausethis function is not analyticat in,, = ind = 0.

To reproducethe leadingnonanalyticterm we mayin factusethe physicalpion massratherthanthefirst orderexpressionM~= (ir~4mq~ir),becausethe differencebetweenthis expressionandthe physicalpion (mass)2is of orderM~In M~and henceonly showsup in the termsof order in~In in,,,.

The resultis very simple: to get rid of the leading nonanalytictermsin the expansionof the proton

mass M0(A, in,,, in~,in,,...) in powers of in,, and in~ onesubtractsthe correspondinglowest Qrder

diagram, fig. 3c, using the physical mass in the pion propagator.The nucleon propagatorand thepion—nucleonvertex are takenin the symmetrylimit. The behaviourof the vertex at nonzerovaluesofk is irrelevant as longas it is symmetricunderSU(Nf) and reducesto g~/f,. for k —* 0. The integraloverk may be cut off at some kmax or, alternatively,one may suppresslarge valuesof k by using a form


factorat the vertices:

j~T~r~=M~—~—-2~J(Mo,M0, MIT)

J(M1,M2, in) = ~ J rn-~~G2(k2) u(p)Xy

5 ~ ~Xy5 u(p)

p2=M~ (C.4)

whereG(k2) is the axial vectorform factor with G(0)= 1. In contrastto the proton massitself, thequantityM~is analytic in thequark massesin,,, in,,, up to termsof order in~In in,,,.

The next-to leadinginfraredsingularities(ln kmin ratherthan 1/kmjn) are lesseasyto dealwith. Onecontributionarisesfrom the diagramin fig. 3d: onemassinsertion is attachedto the nucleonline, theotherto the pion line. Thediagramscontainingany numberof nucleonmassinsertions(internal as wellas externallines) areagainsummedup if one replacesthe chiral nucleonpropagator(massM

0) by thecorrespondingpropagatorwith the physical nucleon mass. A prescription that cures the infraredsingularitiesalso on the first nonleadinglevel must involve propagatorswith physical massesboth forthe pion and for the nucleon lines (fig. 3e). It is however not correct to simply usethe symmetriccouplingsgiven by the effectivechiral Lagrangianand to only considerthe chiral asymmetriesdue tomasstermsin this effective Lagrangian(symmetricvertices,propagatorswith physicalmasses).On thenext-to-leadinglevel thereis alsoa contributiondueto the u-termin IT N-scattering.In the quark massexpansionof the nucleon massthe u-term showsup in a nonanalyticpieceproportionalto (p101(uu+

dd)Ip)M~ln M~.In the languageof an effective Lagrangianthe u-term amountsto a coupling of thetype ~ with a coupling constantthat vanishesin the chiral limit. If one wants to include allnonanalytictermsof order in ~ln mq onehasto strengthentheconditions(i), (ii) and(iii) on the effectiveLagrangianand insurethat it reproducesthe low energyscatteringamplitudesnot only in the chirallimit, but up to and including the terms of order in. [Note that the nonanalyticterm proportional toM~.ln M~in the quark massexpansion(C.6)of M~is dueto the analogousvertexcorrection.]

We did not takethesenext-to-leadingnonanalytictermsin the quark massexpansionof the baryonmassesinto account.The quantityM~definedby

= — ~2 75 ~ C~,,.J(M~,Mm,M,) (C.5)IT ~

involves only physical masses.[The constantsC’,.,,, are the Clebsch—Gordancoefficients involving theratio a = d: (f+ d) of the axial vector coupling constants; the integral J is defined in (C.4).] In theabovewe haveshownthat in contrastto M,,(A, in,,, in,,,, ins,...) the quantity M,,(A, in,.,, m,,,, in,,...) isanalyticin the quark massesin,,,

1~d,m, up to_termsof order m~ln in,,,.

Numerical valuesfor the purified massesM,. aregiven in table6. In column I we list the massesofthe particlesin pure QCD, obtainedby subtractingthe electromagneticBorn termsfrom the physicalmasses(we do not indicatethe errorbars associatedwith thesenumbers,but work with the meanvaluesthroughout).In the top half of the tablewe give the mean_massesof the isospinmultiplets, i.e. ignorethe difference in,, — in,,,. Column 2 containsthe valuesfor M,, accordingto equation(C.5) with a = 0.62(SU(3) symmetryfor the axial vectorcouplingsgA/fIT). We haveused the dipole formula for the axialvectorform factor: G(t) = (1 — t/m2)2 with in2 = 0.71 GeV2. The main effect is an overall shift of the

SiSiSiSiSiSiSi

SiSiStStStSSS

1 Gasser and H. Leutwy/er, Quark masses 153SS

Table 6Massesof baryonsstrippedof their mesonclouds.Column1 Containsthe physical masses corrected for electromagnetic self-energy.Columns2 and 3 give thepurified massM, for SU(3)symmetricaxialveCtor couplings andSU(3) symmetricmeson—baryoncoupling con-stantsrespectively.The quantity~ measuresthedeviationsfrom the

GeH-Mann—Okuboformula.

QCD g~If,, renormalized 0 = 0 ~1 2 2a 2b 3

N 939 1082 939 1082 1080A 1116 1262 1119 1262 12301 1193 1331 1188 1331 1289

1318 1460 1317 1460 1406

—0.070 —0.084 —0.084 —0.084 —0.018

p — n —2.05 —2.52 —2.52 —2.74 —2.49.Z~—1 —7.81 —8.49 —8.49 —9.11 —7.97

—5.56 —5.52 —5.52 —5.80 —5.09

massesby about 140MeV. Onemay renormalizethe purified massesby subtractingan SU(3) singlet,e.g. in such a mannerthat the nucleon stays whereit is (column 2a): the renormalizedmassesdifferfrom the physical massesby at most 5 MeV. Alternatively, one may look at the Gell-Mann—Okubocombination

2N+2E-3A -.~

-

The nonanalyticcorrectionsmodify this combinationonly very_little. In column 3 we show the resultsobtainedif the meson—baryoncouplingconstantsgMnB = (gAIV2fIT)(M~ + MB) ratherthan theconstantsg~/f,,-aretaken SU(3)symmetric.Sincethe correspondingleadingnonanalytictermsare the samewehaveno reasonto prefer one prescriptionto the other. According to the coupling constantanalysisprovidedby Nagelset al. [283]theratio d : (f+ d) hasthe valuea = 0.76 in thiscase.As can be seenbycomparingcolumns 2 and 3 the individual shifts are substantiallydifferent for the two prescriptions.Whatmatters,however,arenot the overall shifts, but only the partsthat arenot octetor singlet.Theseparts differ only very little as can be seene.g. from the Gell-Mann--Okuboformula which is againalmostunaffected.

The lower half of the tablequotesthecorrespondingmassdifferenceswithin the isospin multiplets.The massdifference in,, — in~induces mixing between IT

0 and~ aswell as between~X°andA. TheSU(3)symmetriccoupling con~tantsdeterminethe probability amplitudesfor emissionandabsorptionof themesonicoctetstates:the mesonmassmatrix that determinesthe propagationof thesestatesis howevernot diagonalin the octet basis.In order to be able to work with diagonalmesonpropagatorsonehastotransformthe verticesaccordingly.To seethe effect of IT0 — ~ mixing on the massdifferencesonemaycomparecolumns2 and 2b: in column 2 we give the numbersthat one obtainsif the baryon—mesoncouplingconstantsaresubjectto the mixing transformation(with themixing angle0 = 0.57°that followsfrom the mesonspectrum)andin column 2b wegive the correspondingvaluesif the mixing angle is setequalto zero. As discussedin section 10 the nonanalytictermsincreasethe mass differencebetween


proton and neutron in comparisonto the splittings due to m,— 01. The effect is not sensitive to themannerin which onedeterminesthe baryon-mesoncoupling constants(comparecolumns2 and3).

The leadingnonanalytictermsareindependentof the spin of theparticle in question.Whatcountsistheanalogof g,~,thematrix elementof the axial chargeat infinite momentum.For the Goldstonebosons,the constantg.~.vanishes.The leadinginfrared divergencein the secondorder term of the quark massexpansionis thereforeonly logarithmic [284].To reproducethe leadingnonanalytictermsin the quarkmass expansionof the Goldstoneboson masseswith an effective Lagrangian,this Lagrangianmustincludethe relevanta--terms.Thesum of the leading infrareddivergencesis againgiven by the lowestorderself-energydiagramswith physicalmassesin the propagatorsandoneendsup with an expressionanalogousto (C.5). Thecorrespondingleadingnonanalyticcontributionsareproportionalto M~ln M~,,M~In M~ and M~In M~. In fact, the approximation obtained by expanding the result ofICPT andretainingonly the leadingnonanalytictermsis indeedreliablehere[282].This is by no meansthe casefor the baryon massformulae,which becomecompletely unphysicalif one only retains theleadingnonanalytictermsproportionalto M~,M~andM~(seesection10). For the Goldstonebosonsthe net result is that the infrared logarithm is replacedby a logarithm of the mesonmassesand theformulaeanalogousto (C.5)maybe written as

M~.= M~r{1— 3LIT + Lq}

M~=M~{12Lq}

= M~{1— 6LK + 4L~}+M~{3LIT— 2LK — L~}

fvJ~o-M~+= (M~o-M~+){1- 2L~+ M~-M~(LIT- L~)}

L,, = (48IT2f~,)~M2,. ln(M~//L2). (C.6)

In theseapproximateexpressionsthe form factoronly entersthrough the valueof ~t in the argumentofthe logarithm. For a value of ~ in the range 0.5GeV< ~ <1 GeV the mass shifts due to thesecorrectionsamountto lessthan 5%.

We have mentionedin section 8 that the mass difference MIT~— MIT’ vanishesto first order inin,, — ind. The same is true of the correspondingchiral corrections.Retaining terms of order (in,, —

in~)2/(m,— 01) and taking IT°—fl-mixinginto accountthe expressionfor the massdifferencebecomes

M~+_M~o=~ind‘~Bf1— 18LIT + 18LK+ 8L~4 m,—m I.

+ 6M~~2(LIT - LK) + M~~M2IT(L~- LIT)}. (C.7)

Appendix D. The pion—nucleon u-term

The pion—nucleonu-termin IT~p—* ~p scatteringis definedby

u=~j-(pIuu+ddIp) (D.1)

SSt

J. Gasser and H. Leutwy/er, Quark masses 155

where p) denotesthe physical one-protonstate,andM~is the proton mass.In a first stageof ourdiscussionwe shall disregardisospin violating effects due to in,, � in,,, (isospin violation due to photonexchangemust be takeninto accountby correctingthe scatteringdatabefore extractinga valuefor a-.Theseeffectswill not beconsideredhere).At the endof this sectionweshall comebackto the questionof isospinasymmetriesin u-termsin connectionwith otherchannelsof ITN-scattering.

The interestin the quantity a- derivesfrom its connectionwith u-N-scattering[285—2961:Definetheform factor

u(p’)u(p)o-(t)=th(p’Juu+ddlp); t=(p’—p)2.

As shown by Brown, Pardeeand Peccei [297]the on-shell pion—nucleonscatteringamplitude at theunphysical Cheng—Dashenpoint s = M~, t = 2M~.differs from the form factor cr(t = 2M~,)onlythrough termsthat areformally of orderM~.FurthermorePagelsandPardee[298]haveshownthat therelation betweena-(2M~)and the scatteringamplitude at the Cheng—Dashenpoint doesnot containnonanalytictermsonthelevelof M~,butdoescontaincorrectionsof orderM~.In M~.Onthebasisof theiranalysisweconcludethat the on-shellamplitudeat the Cheng—Dashenpointshoulddiffer only very littlefrom u(2M~).To connectthe dataon IT—N scatteringto the forwardmatrix element,i.e. to the properu-term,a- = cr(0) definedin (D.1)onethenonly needsto studythet-dependenceof a-(t).Froman analysisthat is identicalto the onegiven by PagelsandPardee[298]we obtain

u(2M~)= u(0)+ + O(M~.ln M~) (D.2)

with fIT = 132MeV. The correction amountsto 7 MeV (the analogousformula given by PagelsandPardeediffers from eq. (D.2) by a factorof 2). Improving the calculationwith the methodsdescribedinappendixCwefind thatthecorrectionis reducedbyafactorof two; thefinal value,includingcontributionsfrom the kaon and etacloudsis

u(2M~)—o-(O) = 3.5±1.2MeV+ O(M~TIn M~). (D.3)

The remainingcorrectionsof order m~,,arklfl(inquark) havenot beengiven in the literature.We expectthem alsoto bevery small.

Recentevaluations(refs. [293,299]; for a compilation of recentdata, see Nagelset al. [291b])ofscatteringdatagive valuesaroundu(2M~)= 65 MeV with remarkablysmallerror bars.In view of theabovediscussionthis valueimplies

u~60MeV. (D.4)

A popularestimatefor the size of the u-termis obtainedif oneassumesthat the matrix elementsofthe operatorsüu, dd and~ areapproximatelyequalto the matrix elementsof the correspondingchargedensitiesu~u,d~dands~sat rest(additivity rule). In this picture the quark massdifference in, — 01 isgiven by ~(M~— MN) 190MeV whereasthe u-term is given by 301. Using the ratio in,: = 25 onethusgets 01 8 MeV, u 24MeV.

A more precise determinationfollows from the mass formulaefor the baryon octet, discussedinsections9 and 10. To get a value for a- one howeverneedsone additional piece of information


concerningthe SU(3)singlet componentof the quark massoperator.It suffices to knowthe valueof thenucleonmassin the chiral limit or the size of the nucleon matrix elementof the operatorcs. We firstleavethe valueof (plcslp) open andparametrizeit in termsof

= 2(pl~s[p)~

y (pi~u+~dipY

The u-term may then be worked out either by using the observedSU(3)-breakingmass differencesbetweenN, A, ~ andE or the observedisospinbreakingmassdifferencesp — n, — ~,~° — ,E’. Thelinear first ordermassformulaegive

a- = (~_ 01) (1— ~) (M~+ M~— 2MN)

inu+ind , (D.5)\~

t~in~— in,,) ~1 }~)

With y=O, m,:01 =25, (in,—01):(m~—m,,)=43.5one finds a-=26MeV andu=25MeV from thesetwo determinationsrespectively (the contribution of the electromagneticinteraction to the isospinsplittingshasbeencorrectedfor accordingto table3).

Theorderof magnitudeof the correctionsdueto higherordertermsin thequark massexpansioncanbe seenby using first order mass formulaefor M2 ratherthan for M: this gives a- = 31 MeV anda- = 32 MeV respectively.One of us (Gasser[282])has analyzedthe higher order terms in the quarkmassexpansionon the basisof themethoddescribedin section10 andhasshownthat theasymmetriesofthe mesoncloudsincreasethe valueof the u-termby about 10 MeV. Taking the higher order termsinthe quark massexpansioninto accountoneobtainsthe value

a-035±5MeV. (D.6)

For (pI.csjp)= 0 thisresult is smallerthan thevaluesextractedfrom ITN-scatteringby aboutafactor two.Onemay of courseblamethe differenceon the expectationvalueof ss in the nucleonwhich is indeednot known. Since the vacuum expectationvalue (0I.csIO) is substantialone should not dismiss thepossibility that (pI~sIp)is alsoratherlargewithout further thought.Note however,that y would havetobe of order0.4 if this effect is maderesponsiblefor the discrepancywith the observedvalue. As notedabovethe valueof y determinesthevalueof the nucleonmassM0 in the chiral limit. Fory = 0 onehasM0= 870MeV andy = 0.2 correspondsto M,,= 670MeV. If y shouldturn out to be aslargeas0.4 thenthe valueof the nucleonmassin the chiral limit would be completelydifferent from what it is in thisworld. We concludethat u(2M~)= 65 MeV is not consistentwith QCD unlessvery strangethingshappenin the chiral limit (seealso appendixE).

Let us now come backto the questionof isospin asymmetries.The discussionapplies also to theu-termswhich occur in other channelsof ITN-scattering.Isospin-breakingeffects (from in,. � ind, thedataare assumedto be correctedfor electromagneticinteractions)in a--termsoccur in 3 places[300].First in the (theoretical)connectionbetweenphysical pion—nucleonscatteringamplitudesandu-terms. 1To the bestof our knowledgean analysissimilar to the onedoneby Brown, Pardeeand Peccei[297] 1

SiSiSiSiSiSiSiSiSiSiSiSiB

1 Gasser and H. Leutwyler, Quark masses 157

Table7a--termsin irN-scattering.Thequantity S measuresthe iso-spin asymmetries due to

m,�md.

Channel a--term

a-5~_±fl—~7J•±fl a-

IT°p-41T°p a-—~S~r°n—’ir°n a-+~5ir°p-~1T~n1v’25

~-p--~~~-°n—~‘T2s

andPagelsandPardee[298]hasnot yet beenperformedfor the casein,, ~ md. We havenothingto addto this problem.Second,isospin asymmetriesmust be takeninto accountin the data analysiswhich isusuallydonewith isospinsymmetricamplitudes.[SeeHöhler [299]andGensini[301]for a discussionoftheseproblems.] In the third place, in,, � in,,, producesisospin asymmetriesin the current algebraexpressionfor the u-terms.This point is easyto discuss.The u-termsfor the variouschannelsaregivenin table7, wherewe havemadeuseof isospinsymmetryto expressthe relevantmatrix elementsof theoperatorsñu, fld, du and dd betweenprotonsand neutronsin termsof the isospin symmetricpart a-definedin (D.1) and the asymmetry5 given by

= md—rn~(p~uu— ddjp). (D.7)

The asymmetryis relatedto the proton—neutronmassdifference.In fact, to lowestorder in the quarkmassexpansionin powersof in,, and in,,, the massdifferenceM,. — M~is given by 8:

S = Al,.— M~= (2.05±0.30)MeV (D.8)

(comparetable 3). This shows that the isospin asymmetriesare not very large. The u-term in ~0pscatteringis smallerthan the u-termin u-°nscatteringby about2 MeV.

This isospin violation is substantiallysmallerthanwhat onemayobtain from roughestimatesbasedon the additivity rule. In fact, this rule ((pIüu(p)=’(pIu~uJp)etc.) is too crudean approximationif onewantsto obtain agood estimateof both the size andthe asymmetriesof the u-terms(picking a valuelike m,= 150MeV, e.g., andusingthe quark massratiosobtainedfrom the mesonspectrumonefindsu 18 MeV and 5 3.3MeV rather than the values u 35 MeV and S = 2 MeV given above).Theadditivity rule doesgive an acceptableestimatefor the size of the F couplingin the massoperator,butit doesnot reproducethe observedisospin breakingsbecauseD couplingplaysan essentialrole here.

AppendixE. What if (0I4qJ0)vanishes?

The discussionof symmetry breakingreviewedin this paperis basedon the assumptionthat theexpectationvalueof 4q in the groundstatedoesnot vanishin the chiral limit. From the consistencyofthe variousdeterminationsof the chiral symmetry breakingquark massratiosonemight betemptedto

158 J. Gamer and H. Leutwyler, Quark masses

concludethat this analysisat the sametime alsochecksthe correctnessof the underlyingassumption.As hasbeendemonstratedby a numberof authors(seee.g. refs. [302—307]andthe referencesquotedinthesepapers)phenomenologyis howeverequally consistentwith an alternativepicture basedon theassumptionthat the quantity ~0Iqq~0)is of order ‘flq,,ark, i.e. vanishesin the chiral limit. Theoretically,this alternativeis not “natural” becausea quantity that hasno reasonto vanish in generalis indeeddifferent from zero. This argumentfor ruling the alternativeout may however fail for one of thefollowing reasons:

(i) For the analysisgiven in sections13, 14 and 15 to makesenseit is not sufficient to knowthat theorderparameter(014q10)is different from zeroin the chiral limit. It is importantthat this quantity is notsmall in units of the scaleof QCD. The theoreticalunderstandingof theinterrelationbetweenthechiralorder parameterand the renormalizationgroupinvariant scaleA is still ratherbleak. The only directevidencefor a sizeablevalue of (0~4q(0)comes from lattice calculationswhich break chiral symmetrythrough termsthat aretheoreticallyinessential,but might jeopardizethe numericalvaluesfoundfor theorderparameter.

(ii) It is conceivablethat a discretesubgroupof the U(1) group generatedby the axial currentis asymmetryof the groundstate,implying that (014qI0) vanishesin the chiral limit [308,307].

Before theseissuesareclarified oneshouldnot takethe standardframeworkfor granted,evenif thealternativebasedon ~0~4qI0)= O(inq,,ark)is not very appealing.Onedeficiencyof this alternativeis thatthe Gell-Mann—Okuboformula for the ~j mesonis lost. The mesonmassesareof order inq,,ark ratherthan of order (mq,,ark)

2 hencethe ratio ins: 01 assumesasmallervaluesuchas 5 or 10 insteadof 24.Themassesof the quarks u and d are larger than in the standardframework and one thereforeobtainsviolationsof chiral SU(2)x SU(2) that aresubstantiallylarger. Indicationsfor relatively largedeviationsfrom chiral SU(2)x SU(2)havelong beennoticed in pion—nucleonscatteringwhere the Goldberger—Treiman prediction for the pion—nucleoncoupling constantappearsto be off by as muchas 7%. Alsovalues given for the a--term both in pion—nucleon scattering(see appendix D) and in pion—pionscatteringappearto be too largeto beconsistentwith the standardframework, in which SU(2)x SU(2)is an almost perfect symmetryof the Hamiltonian becausein,, and ~1d are tiny in comparisonto thescaleof QCD. One may of course try to understandthesedeviationswithin the alternativeschemebasedon the assumption~0Iqq~O)= O(tflq,,ark). It is howeverclear that the questioncannotbe resolvedby lookingat smalldeviationsin the phenomenology;what onehasto do is to calculate(O~4q~0)in termsof A. We will thereforenot commenton ratiosand absolutevaluesof quark massesfound on the basisof the assumption~0~4q~0)= O(inq,,ark).

Appendix F. A quantum mechanicalmodel for p — ~ mixing

We set up a quantummechanicalmodelwhich describesp — w mixing; especiallywe shall show thatthe effect of p — w mixing in the processe~e~—~ IT~IT near the p — w resonancesis to replacethep-propagatorby

1 ~s—M~+iM~F~ s—M~+iM~F~)(s—M~,,+iMJ,,.) (F.1)

whereF~and~ arethe photon—vectormesoncouplingstrengths(OIj~mIp, h, i) = ~ i = p, w; ~

denotesthe nondiagonalpiece in the massmatrix of the p — w system(eq. (B.10)).To set up notation,we start with the exponentialdecaylaw for an unstableparticle.

5555SiaaSiSiSi

I Gasser and H. Leutwyler, Quark masses 159 1St

P.1. Decayof an unstableparticle I

We split the Hamiltonianinto two piecesH = H0 + V. The spectrumof H0 is assumedto consistof a Icontinuousanddiscretepart, I

H0!E)=EIE), (E’IE)=S(E’—E), E�~

H0!1)=E511), (1I1)=1, E5>~i

1)(1J+JdEIE)(EI=l

V causestransitionsfrom the state Ji) to the continuum, (iJ ViE) � 0. We shall use the notation(11VJE)= V(1,E), (EJ Vu) = V(E,1), (EJ VJE’) = V(E,E’), (ii Vjl) = V(1, 1).

Productionof the state 1) at t = 0 andsubsequentdecayis describedby the initial valueproblem

ic9,Jçli(t)) = H~i/i(t))

j~(t))= b(t)e~”I1)+JdEC(E, t) e~’JE)

b(0)= 1, C(E,0)=0.

TheFourier transformof b(t) (with b(t) = 0, t<0)

b(t) = ~— JdE’ exp{i(E1 — E’)t} G(E1,E’)

is foundto be

G(E,,E)1= E — M(E)+ ie

M(E) = E1 + V(1, 1) + J dE’ V(1,E’)U(E’, E)

The function U(E’, E) is the solutionof theintegralequation

U(E’, E) = V(E’, 1) + J dE” V(E~E”)U(E”, E)

Exponentialdecay of the state Ii) amountsin this pictureto the approximationM(E) M(E1). Thenonehas

(1~e~”~I1) = e_~Mw1)

M(E1)= ER—~iF


where

ER=El+ V(1,1)+~fdE~V(1~E’)L2+O(vs)

F/2= ITI V(1, E = E5)1

2+ 0(V3). (F.2)

Next we considerproductionof the stateIi) in a scatteringexperiment.

P2. Productionof the unstablestate Ii) in e~e-~ IT4~IT

Considerthe processe~e—~ IT~IT nearthe p — w complex.The transitionamplitudeis proportionalto the matrix element

~IT~IT out j~m.I0)

We mimick this matrix elementin our quantummechanicalmodel by

(EI~I~i)

whereIE) is the solutionof the Lippmann—Schwingerequation

IE)=IE)+~~~ VuE)~.

I~/’) is an arbitrary state, / maybe any operatorfor which the matrix element(EI~Içti)is a smoothfunction of E.

We expandE)_ in termsof the completesystemIi), IE), viz.,

IE)~= IE)+ Y(E)I1)+ JdE’ X(E, E’)IE’)

and solve for Y(E), ~(E, E’). (We shall set (E’I ViE) =0 for simplicity.) Insertingthe functions Y(E)

and~(E, E’) in the matrix element (EI4I~/~)onefinds

E - E + V(E, 1) (11*) f~+ 1 f dE’ V(1, E’) (E’j~)l-( I*)-( 1*) E-M(E) ~ (1I~)J E-E’+ie J

In the approximationM(E) = M(E5) (which leadsto the exponentialdecaylaw) we find

(EI*)=(EI*)+ V(E~1)(iI~{1~0(~} (F.3)

Thisshowsthat _(EI4I~i)has a Breit—Wignerpole at F = ER — ~iF.Mixing is consideredin the next paragraph.

asasasasasasas

J. Gasser and H. Leutwyler, Quark masses 161

P3. Productionof two resonanceswhich mix B

Assumenow that the spectrumof H0 hasthe form

HOjE, i) = EIE, i), E�

Holi)=E11i), E1>~1 1

(E’,ilE,j)=s(E’-E)s~ I

(i(j)=S~ i=1,2; E1—E2. I

Theinteraction V inducestransitionsbetweenthe boundstatesIi) andthe continuum,

(ii ViE, i) = ~ V(j, E), (E, ii Vii) = S~V(E,j)

and mixes the states 1) and 2),

(1jV12)= (2)Vl1)*

We shall again assume(E, ij ViE’, j) = 0 to simplify the calculation and use the notation (11 1/12) =

V(1, 2), (21 Vu) = V(2,1).Let f, k~)be arbitrary as before,with (E, iI~lt,~)a smoothfunction of E. Now considerthematrix

element

(E,1l~); E,1)=IE,1)+E~i VE,1)-.

We expecttwo typesof dominatingcontributionsin the vicinity of F -~F, =— E2:(i) a peakat E-~E, dueto the transitions

~ 1),

(ii) anotherpeakatE —~E2 dueto the transitions

l~i)—-H2)—-7l1)-—~HE,1).

Thefollowing calculationunravelsth~setwo effects. Let

E, 1) = E, 1) + Y1(E) 1)+ Y2(E)j2)+ JdE’ ~(E, E’) JE’, 1) + J dE’ D(E, E’) E’, 2).

In the approximation(F’, i$ ViE, j) = 0 wefind


(E—M~(E))Y1(E)= V(1,E)+ V(1,2) Y2(E)

(E — M~(E))Y2(E) V(2, 1) Y1(E)

~(E, E’) = (E — E’ — is)’ V(E’, 1) Y1(E)

D(E, F’) = (E — E’ — ie)1 V(E’, 2) Y

2(E)

with

M(E) = E1 + V(i, i) + J dE’IV(i,E’)12

Insertionof theseexpressionsinto the matrix element (El~I~I)leadsto

_(E, 1l44~fr)= (E, 1l~1~i)+ V(E,1)(iIc~!i)+ (‘M)) + corrections. (F.4)

Now set M1(E)= M~(E,),i = 1, 2 and identify 1) ~ Ip°), 2)*-~ fw), V(1,2) +-~ M~,(lIdiIifr) = F,,M,,,

(2IcbI’!’) = F0,M,,,.Thenit is evident from eq. (F.4) that the dominantcontributionsto the matrix element(Elq5I~r)at E=- E1 —= E2 aredescribedby termsof the structuregiven in eq. (F.!). In fact therelativistic

treatmentis an immediategeneralizationof this quantummechanicalsystemand leadsdirectly to thismodified form of the p-propagator.

References

Section 1

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Section 3

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164 J. Gasser and H. Leutwvler, Quark masses

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Section 4

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Section 5

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Section 6

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Section 7

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Section 10

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Section 12

[94]W.N. Cottingham,Ann. of Phys. 25 (1963)424.[95] H. Leutwylerand P. Otterson,in: Scaleandconformalsymmetryin hadronphysics,ed. R. Gatto(Wiley, New York, 1973) p. 3.[96]iC. Collins, NucI. Phys.B149 (1979) 90.[97] 1. GasserandH. Leutwyler,Mud. Phys. B94 (1975) 269.[98] A. Zee,Phys. Reports3C (1972) 129.

IsIsIsIsIsIs


[99] M. Cmi and P. Stichel,Text bookon elementaryparticlephysics,electromagneticinteractions,Sept. 1972, unpublished.[100]iF. Gunion, Phys. Rev. D8 (1973)517.[101]S. Colemanand S.L. Glashow,Phys.Rev. Lett. 6 (1961)423.[102]S. Colemanand H. Schnitzer,Phys.Rev. 136B (1964) 223.[103]Y. Miyamoto, Progr.Theor.Phys.35 (1966) 175.[104]A. Dc Riijula, H. Georgi and S.L. Glashow,Phys. Rev.DI2 (1975) 147.[105]K. LaneandS. Weinberg,Phys. Rev.Lett. 37 (1976) 717.[106]H. Fritzsch, (a) Phys. Lett. 63B (1976) 419; (b) Phys. Lett. 7!B (1977)429.[107]5. Ono, Phys.Rev. Lett. 37 (1976) 655.[108]AD. Dolgov et al., Phys.Lett. 15 (1965) 84.[109]W. mining, Acta PhysicaAustriaca,Suppl. II (1965) 205.(110] J. ArafuneandY. Iwasaki, Progr.Theor. Phys. 35 (1966) 339.[111]G. Barton and D. Dare, NuovoCim.46A (1966) 433.[112]5. Ishida,K. Konno andH. Shimodaira,NuovoCim. 46A (1966) 194.[113]T. Minamikawaet al., Suppl. Progr.Theor.Phys.37/38 (1966)56.[114]HR. Rubinstein,Phys.Rev. Lett. 17 (1966) 41.[1151A. Gal and F. Scheck,NucI. Phys.B2 (1967) 110.[116]5. Ishidaet al., NucI. Phys. B2 (1967)307.[117]H.J. Lipkin, Nucl. Phys. BI (1967) 597.[118]HR. Rubinstein,F. Scheck andRH. Socolow,Phys.Rev. 154 (1967) 1608.[119]R.P.Feynman,Photonhadroninteractions(WA. Benjamin,Inc., Reading,Mass., 1972).[120]A. De ROjula, H. Georgi and S.L. Glashow,Phys. Rev.D12 (1975) 147.[121]D.B. Lichtenberg,Phys.Rev. D12 (1975) 3760.(122] D.B. Lichtenberg,Phys. Rev.DI4 (1976) 1412.[123]W. Celmaster,Phys.Rev.Lets.37 (1976) 1042.[124]C. Itoh et al., Progr.Theor. Phys. 61(1979)548.[125] N. Isgur, Phys. Rev.D21 (1980)779.[126]T. Das, G. Guralnik, V. Mathur, F. Low andJ. Young,Phys. Rev.Lett. 18 (1967)759.[127]D. Gross,S. Treimanand F. Wilczek, Phys. Rev.D19 (1979) 2188.[128]R. Socolow,Phys. Rev. 137B (1965)1221.[129]C.J. Bebeket al., Phys.Rev. DI7 (1978) 1693.[130]R. Dashen,Phys.Rev. 183 (1969) 1245.[131]P. LangackerandH. Pagels,Phys. Rev.D8 (1973)4020.[132] D.H. Boal andA.C.D. Wright, Phys. Rev. D16 (1977) 1505.(133] VS. Zidell, R.A. Arndt andL.D. Roper, Phys. Rev.D21 (1980) 1255, 1289.

Section 13

The materialpresentedin this section is essentiallytakenfrom[134]J. Gasser,Ann. of Phys. 136 (1981) 62.

[135]H. Leutwyler, in: Proc.Topical Meeting on IntermediateEnergyPhysics,April 2—12, Zuoz 1975 (SIN-DocumentationGroup, CH-5234Villigen, Switzerland).

[136]A. Halprin, B.W. LeeandP. Sorba,Phys. Rev.DI4 (1976) 2343.[137]S. Weinberg, in: A Festschriftfor 1.1. Rabi,ed. L. Motz (NewYork Academyof Sciences,New York, 1977) p. 185.(138] CA. Dominguezand A. Zepeda,Phys.Rev. D18 (1978) 884.[139]CA. Dominguez,Phys.Lett. 86B (1979) 171.[140]P. LangackerandH. Pagels,Phys. Rev.D19 (1979)2020.[141]P. Langacker,Phys. Rev.D20 (1979) 2983.[142]MA. Shifman,A.!. VainshteinandVI. Zakharov,Mud. Phys.B147 (1979) 385, 519.[143]P. Minkowski and A. Zepeda,NucI. Phys. B164 (1980)25.(144] H. Fritzsch andP. Minkowski, Phys.Reports73C (1981) 67.

Section 14

[145]S.L. Glashow,Phys. Rev.Lett. 7 (1961)469.[146]S. Colemanand S.L. Glashow,Phys.Rev. 134B (1964)671.[147]M. Gourdin.L. Stodolsky andF.M. Renard,Phys.Lett. 30B (1969)347.

166 J. Gasser and H. Leutwy/er. Quark ,nasses

(148] F.M. Renard,NucI. Phys. BI5 (1970) 118.(149] R.P. Feynman,Photonhadroninteractions(WA. Benjamin,Inc., Reading,Mass..1972).[150] AS. Goldhaberet al., Phys.Lett. 30B (1969)249:

S.A. Coonet al., NucI. Phys. A287 (1977)381.(151] MA. Shifman,Al. VainshteinandV.1. Zakharov,Mud. Phys. B147 (1979)519.[152] P. Langacker,Phys. Rev.D20 (1979)2983.[153] D. Benaksaset al., Phys. Lett. 39B (1972)289; Phys.Lett. 42B (1972)507.[154] A. Quenzeret al., Phys.Lett. 76B (1978) 512.(155] G. Segréand I. Weyers, Phys. LetI. 62B (1976) 91.[156] H. Genz,Lets. NuovoCim. 21(1978)270.[157] P. Langacker,Phys.LetS. 90B (1980) 447.[158] B.L. Ioffe, Yad. Fiz. 29 (1979)1611.[159] M. Voloshin, Proc.Workshopon physicalprogramof e’~e’storagerings (Novosibirsk, 1979).[160]B.L. loffe andMA. Shifman,Phys. Lett. 95B (1980) 99.[161]M. Oregliaet al., Phys.Rev. Lett. 45 (1980) 959.[162]TM. Himel et al., Phys. Rev.Lett. 44 (1980) 920.[163]D.G. Sutherland,Phys. Lett. 23 (1966) 384.[164]S.K. Boseand A.H. Zimmermann,NuovoCim. 43A (1966)1165.[165] P. Dittner,PH. Dondi andS.Eliezer, Phys.Rev. D8 (1973)2253.[166]A. Browman et al., Phys. Rev.Lett. 32 (1974) 1067.[167] C. Baglin et al., BAPS 12 (1967)567; Phys. Lett. 29B (1969)445.[168] F.W. Bullock et al., Phys.Lett. 27B (1968)402.[169] A. NeveuandJ. Scherk,Ann. of Phys. 57 (1970) 39.[170] H. Osbornand Di. Wallace, NucI. Phys.B20 (1970) 23.[171] N.G. Deshpandeand TN. Truong,Phys. Rev. Lett. 41(1978)1579.[172] P. LangackerandH. Pagels,Phys. Rev. 010 (1974) 2904; Phys.Rev. Dl9 (1979) 2070.[173] S. Weinberg,Phys.Rev. Dli (1975) 3583.[174] R.Crewther,CargeseLectures(Plenum Press,1975).[175] S. Raby,Phys. Rev.D13 (1976) 2594.(176] CA. Dominguezand A. Zepeda.Phys.Rev. DI8 (1978) 884.[177] C. Roiesneland TN. Truong,NucI. Phys.B187 (1981) 293.[178]WI. Brandeliket al., Phys.Lets. 70B (1977) 132; Phys. LeSt. 80B (1979)412; Phys.Lett. 76B (1978)361.

Section 15

[179]Al. Vainshteinet al., Soy. J.Nucl. Phys. 27 (1978) 274.[180]Al. Vainshtein,VI. ZakharovandMA. Shifman,JETPLett. 27 (1977)55.(181] MA. Shifman,Al. VainshteinandVI. Zakharov,Mud. Phys.B147 (1979)385, 448, 519.[182]R. Aron andR.S. Longacre.Phys. Rev.D24 (1981) 1207.[183]M. Bonesiniet al. Phys. Lett. 103B (1981) 75.[184] C. Becchi,S. Narison,E. de RafaelandF.J.Yndurain, Z. Physik CS(1981) 335.[185] S. Narison andE. de Rafael,Phys.Left. 103B (1981) 57.[186] 1. Schwinger,Particles,sourcesandfields,Vol. II. (Addison-Wesley,1973).[187] MA. Shifman,Al. Vainshtein,MB. Voloshin and V.1. Zakharov,Phys. Lett. 77B (1978)80.[188] E.G.Floratos,S. Narisonand E. deRafael,NucI. Phys.B155 (1979) 115.[189]Li. Reinders,HR. Rubinsteinand S. Yazaki, NucI. Phys. B186 (1981) 109.[190] K. Schilcher.Minh D. Tran andN.F. Nasrallah,NucI. Phys.BI8I (1981) 91.[191] D.J.Broadhurst.Phys. Lets.1O1B (1981)423.[192]W. Hubschmidand S. Mallik, Nudl. Phys. B193 (1981)368.[193]SM. Eidelman,L.M. Kurdadzeand A.!. Vainshtein,Phys.Lett. 82B (1979) 278.[194] H. Sazdjian,NucI. Phys.B129 (1977)319. a[195]CA. Dominguez,Phys.Rev. D15 (1977) 1350; Phys.Rev. DIO (1977) 2313.[196]CA. Dominguez, Chiral-symmetry breaking corrections to the Goldberger-Treimanrelation, Texas A + M University preprint. 1981.

DOE/ER/05223-45.[197] S. Mallik, preprintKarlsruhe 1982, to bepublished. a[198]N. CraigieandI. Stern,Triestepreprint 1981. to be published.[199]B.L. loffe, NucI. Phys.B188 (1981) 317. 55

(200] Y. Chung,HG. Dosch, M. Kremerand 0. Schall. NucI. Phys. B197(1982) S5.Si’‘St‘St‘St‘St‘St‘StSI;BSI;SI;asas

SiSiSiSiSiSiIt

I Gasser andH. Leutwyler, Quark masses 167

Section 16

[201]V.A. Novikov, LB. Okun,MA. Shifman,Al. Vainshtein,MB. Voloshinand VI. Zakharov,Phys.Rev.Lett. 38 (1977)626;Phys.Lett. 67B(1977) 409; Phys.Reports41C (1978) 1.

[202]J.S. Bell andR.A. Bertlmann,Z. PhysikC ParticlesandFields4 (1980) 11; Mud. Phys.B187 (1981)285; Mud. Phys.B177 (1981) 218.[203] R.A. Bertlmann,Duality betweenresonancesandasymptotia,ReportUWThPh-80-38,Vienna University,Austria (1980).[204]Li. Reinders,HR. RubinsteinandS. Yazaki,NucI. Phys. B186 (1981) 109.[205] Li. Reinders,HR. RubinsteinandS. Yazaki,Phys.Lett. 94B (1980) 203; Phys. Lett. 95B (1980) 103.[206]B. Guberina,R. Meckbach,RD. PecceiandR. Rückl, Mud. Phys. B184 (1981) 476.[207] L.J. Reinders,Lecturesgiven at theIntern. Schoolof ElementaryParticlePhysics,Kupari-Dubrovnik (Yugoslavia)1981,preprintRutherford

Lab. 81-078.[208] MA. Shifman, ReportIntern. Symp.on Lepton andPhotonInteractionsat High Energies,Bonn,Aug. 24—29, 1981, ed.W. Pfeil (Phys.Inst.,

University of Bonn, Bonn).(209] H. Georgi andD.H. Politzer,Phys.Rev.DI4 (1976) 1829.[210] R. Coquereaux,Ann. of Phys. 125 (1980) 401.[211] MB. Voloshin, Yad. Fiz. 29 (1979)1368[Soy.I. NucI. Phys.29 (1979) 703]; Mud. Phys. B154 (1979)365;ITEP preprint ITEP-21, 1980.[212] S. Iwao, Lett. NuovoCim. 29 (1981) 343.[213] A. Martin, Talk at theEPSIntern. Conf. on High EnergyPhysics,Lisbon 1981.[214] R.Tarrach,NucI. Phys.B183 (1981) 384.[215] H. Leutwyler, Phys.Lett. 98B (1981)447.[216] MB. Voloshin, Nonperturbativeeffects in pre-Coulombiclevelsof aheavyquarkonium,preprint ITEP-30, 1981.[217] Al. Vainshteinet al., Soy. i. Mud. Phys.27 (1978) 274.[218] R.A. Bertlmann,Heavyquark-antiquarksystemfrom exponentialmomentsin QCD,CERN preprintTH-3192, 1981.[2191B. Ovrut andH. Schnitzer,NucI. Phys. B179(1981) 381.[220] W. Bernreutherand W. Wetzel,Decouplingof heavyquarksin theMS scheme,preprintCornell, July 1981, CLNS 81/500.

Section 17

[221]M.K. Gaillard, B.W. LeeandJ.L. Rosner,Rev.Mod. Phys.47 (1975) 277.[222] S. Borchardt,VS. MathurandS. Okubo,Phys.Rev. Lett. 34 (1975)38, 236.[223] R.VergaraCaffarelli, Phys. Lett. 55B (1975)481; Phys.Lett. 58B (1975) 100.[224]A. Kazi, G. KramerandD.H. Schiller,Acta PhysicaAustriaca45 (1976)65.

[225] S. Iwao,Ann, Sci, (KanazawaUniv.) 13 (1976);NuovoCim. Lett. 15 (1976)331, 569; Progr. Theor. Phys.55 (1976) 943.[226]H. Hayashi,I. Ishiwata,S. Iwao, M. Shako andS. Takeshita,Ann. of Phys.101 (1976) 394; Progr.Theor. Phys. 55 (1976)1912.[227]Z. Maki andI. Umemura,Progr.Theor.Phys.59 (1978) 507.[228]Z. Maki 1. TeshimaandI. Umemura,Progr.Theor. Phys. 60 (1978)1127.[229]1. Kaudaswamy,i. SchechterandM. Singer,Phys.Rev.D17 (1978) 1430.[230]K. Yamawaki,NuovoCim. 51A (1979)159.

For an analysisof theeffectivespin—spininteractionsee[231] AD. Sakharov,JETPLett. 21(1975) 9; JETPLeSt. 78 (1980) 2115.

[232]H. Leutwyler, in: Proc.Topical Meetingon IntermediateEnergyPhysics,March29—April 8, Zuoz 1978(SIN-DocumentationGroup, CH-5234Villigen, Switzerland).

Section 18

For a reviewof grandunified theoriessee[233] P. Langacker,Phys. Reports72C (1981) 185.

[234]iC. Pati andA. Salam,Phys.Rev. D8 (1973) 1240; Phys. Rev.Lett. 31(1973)661; Phys. Rev. D10 (1974)275.[235] H. Georgi andS.L. Glashow,Phys. Rev. Lett. 32 (1974) 438.[236] H. Georgi,H. Quinn andS. Weinberg,Phys.Rev. Lets.32 (1974)451.[237]Al. Buras,i. Ellis, M.K. Gaillard andDV. Nanopoulos,NucI. Phys.B135 (1978) 66.[238] I. Antoniadis,C. KounnasandC. Roiesnel,Symmetrybreakingeffects in grandunified theories,preprintCentrede physiquetheorique,Ecole

polytechnique,Palaiseau,1981.[239]E. Gildener,Phys.Rev. DI4 (1976) 1667.[240]J. Ellis, Lecturesgivenat the21st ScottishUniversitiesSummerSchool in Physics,St Andrews,Scotland1980.[241]MS. Chanowitz,J. Ellis andM.K. Gaillard, NucI. Phys. B128 (1977)506.

168 I Gasser and H. Leutwy/er, Quark masses

[242] DV. Nanopoulosand D.A. Ross.Nucl. Phys.B157 (1979) 273.(243] P. Binétruyand T. SchOcker,Nucl. Phys. B178(1981) 293, 307.[244] J. Ellis and M.K. Gaillard, Phys. Lett. 88B (1979) 315.[245] H. Georgi andC. Jarlskog,Phys. Lett. 86B (1979) 297.[246] R. Gatto,G. Sartoriand M. Tonin, Phys. Lett. 28B (1968) 128.[247] N. GabibboandL. Maiani, Phys.Lett. 28B (1968)131.[248] R. Gatto, G. Morchio and F. Strocchi,Phys. Lett. 80B (1979) 265; Phys. Lett. 83B (1979)348.[249] G. Sartori,Phys, Lett. 82B (1979) 255.[250]H. Georgi and DV. Nanopoulos,NucI. Phys. B155 (1979)52; NucI. Phys.BI59 (1979) 16; Phys. Lett.82B (1979) 329.[251] RN. MohapatraandD.Wyler, Phys.Lett. 89B (1980)81.[252] RN. MohapatraandB. Sakita,Phys. Rev. D21 (1980) 1062.[253] R. Barbieri, DV. NanopoulosandD. Wyler, Phys.Lett. 106B (1981)303.[254] G. ‘t Hooft, NucI.Phys. B79 (1974)276.[255] AM. Polyakov, JETPLett. 20 (1974) 194.[256] M.A.B. Beg andA. Sirlin, Ann. Rev.Nucl. Sci. 24 (1974) 379.[257] S.Weinberg,Phys. Rev.D13 (1976) 974; Phys.Rev. D19 (1979) 1277.[258]L. Susskind,Phys. Rev. D20 (1979) 2619.[259]M. Gell-Mann. P. RamondandR. Slansky,in: Supergravity,eds.P. van Nieuwenhuizenand D.Z. Freedman(North-Holland, 1979)p. 315.[260] 5. DimopoulosandL. Susskind,NucI. Phys. B155 (1979)237.[261]E. Eichtenand K. Lane, Phys. LeSt. 90B (1980) 125.[262] S. ColemanandE. Weinberg,Phys. Rev.07 (1973) 1888.[263] S.Weinberg, in: A Festschriftfor 1.1. Rabi, ed. L. Motz (New York Academyof Sciences,New York, 1977) p. 185.[264]S. Weinberg,Phys. Lett. 82B (1979) 387.

Section 19

[265]P. Hasenfratz,Talk given at theEPS Intern. Conf. on High Energy Physics,Lisbon, 1981.[266] P. Hasenfratz,1981 Intern. Symp.on Lepton and PhotonInteractionsat High Energies,Bonn, Aug. 24—29, 1981, ed. W. Pfeil (Phys.Inst.,

Universityof Bonn, Bonn).[267] E. Marinan,G. Parisi andC. Rebbi, Phys.Rev. Lett.47 (1981) 1795.[268]H. HamberandG. Parisi, Phys.Rev. Lett.47 (1981) 1792.[2691D.H. Weingarten,Phys.Lett. 109B (1982) 57.[270]H. Hamber,E. Marinari, G. Parisi andC. Rebbi, Phys.Lett. 108B (1982) 314.[271] A. Hasenfratz,Z. Kunszt,P. HasenfratzandC.B. Lang, Hopping parameterexpansionfor themesonspectrumin SU(3) latticeQCD,preprint

CERN TH-3220, 1981.[272] A. Casher,Phys.Lett. 83B (1979) 395.[273]T. Banksand A. Casher,NucI. Phys. Bl69 (1980) 103.[274] J. Smit, NucI. Phys. B175 (1980)307.[275] H. PagelsandS. Stokar,Phys. Rev.D20 (1979)2947; Phys. Rev.D22 (1980) 2876.(276] iF. Donoghueand K. Johnson,Phys.Rev. D21 (1980) 1975.[277] M. Ida andi. Okada.Spontaneousbreakingof chirai flavour symmetry,preprintKobe University81-4 (1981).

Appendices

[278] S. Okubo,Phys. Rev. Dli (1975) 3261; 1. Math. Phys. 16 (1975) 528.[279] S. Borchardt,VS. Mathur and S. Okubo,Phys. Rev. Dli (1975) 2572;Phys. Rev.Lett. 34 (1975)236.[280] A. Kazi, G. Kramerand OH. Schiller,Acta PhysicaAustriaca45 (1976) 65.[281] S.Weinberg,Physics96A (1979) 327.[282] J. GasserandA. Zepeda,NucI. Phys. B174 (1980)445;

J. Gasser,Ann. of Phys. 136 (1981) 62.[283]MM. Nagelset ai., Mud. Phys. B147(1979) 189.[284]D.T. Cornwell.Nuci. Phys.B55 (1973)436.

For referencesto earlywork on thecr-termsee[285]E. Reya.Rev. Mod. Phys. 46 (1974) 545.

For more recentpaperssee[286]T.P. Cheng,Phys.Rev. 013 (1976) 2161.[287]R.L. Jaffe.Phys. Rev. 021 (1980) 3215.[288]CA. DominguezandP. Langacker,Phys.Rev. D24 (1981) 1905 andthereferencesquotedtherein.

asasasasasasasas

I Gasser and H. Leutwyler, Quark masses 169as

Numericalvaluesaretabulatedin[289]Ri. Oakes,Acta PhysicaAustriacaSuppl. 9 (1972) 518. B

[290]H. Pilkuhnet al., Mud. Phys.B65 (1973) 460. as(291] MM. Nagelset al., (a) NucI. Phys.B109 (1976) 1; (b) BI47 (1979) 189. I[292] M.K. Banerjeeand J.B. Cammarata,Phys. Rev.DI6 (1977) 1334; Phys.Rev. Ci7 (1978) 1125; Phys.Rev. D18 (1978)4078;Phys. Rev.D19 IS

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(294] W. Langbein,NuovoCimento51A (1979)219. Is

[295]MG. Olssonand E.T. Osypowski,i. Phys. G6 (1980)423. 1[296]MG. 0!sson,I. Phys. G6(1980) 431. Si

(297] L.S. Brown,Wi. Pardeeand RD. Peccei,Phys. Rev. D4 (1971) 2801. 1[298] H. PagelsandWi. Pardee,Phys. Rev. D4(1971) 3335. Si[299]G. Höhler, Pion—nucleonscatteringandquark models,Lecturegivenat the3rd Adriatic Meeting on ParticlePhysics,Dubrovnik, September 1

1980.[300]S. Weinberg,in: A Festschriftfor 1.1. Rabi, ed. L. Motz (NewYork Academyof Sciences,New York, 1977)p. 185. I[3011P.M. Gensini, Lectures given at the Workshop on Low- and IntermediateEnergy Kaon Physics, Rome, March 24—28, 1980, Report

UL/IF-81-79~’80,Universitàdi Lecce.J302J iF. Gunion, P.C. McNameeandM.D. Scadron,Mud. Phys.B123 (1977) 445.[303]H. Sazdjian,NucI. Phys. B129 (1977) 319.(304] N.H. Fuchsand H. Sazdjian,Phys. Rev. D18 (1978) 889.(305] N.H. Fuchsand M.D. Scadron,Phys.Rev. D20 (1979) 2421.(306] M.D. Scadron,Rep.Progr.Phys.44 (1981) 213; i. Phys. G7 (1981) 1325.(307] H. Sazdjianandi. Stern,OrsayPreprint 1982, to appear.(308] R. Dashen,Phys.Rev. 183 (1969) 1245.

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