Lecture 2 Quark Model The Eight Fold Wayfcfm.uas.edu.mx/eff2014/docs/CULIACAN-Lecture-2.pdf ·...

Lecture Lecture –– 22 Quark Model Quark Model –– The Eight Fold WayThe Eight Fold Way

AdnanAdnan BashirBashir, IFM, UMSNH, Mexico, IFM, UMSNH, Mexico August August 20142014

CuliacánCuliacán SinaloaSinaloa

ContentsContents

•• GellGell--Mann Okubo Mass FormulaeMann Okubo Mass Formulae •• GellGell--Mann Okubo Mass FormulaeMann Okubo Mass Formulae

•• The SU(3) GroupThe SU(3) Group •• The SU(3) GroupThe SU(3) Group

•• SU(3) and MesonsSU(3) and Mesons •• SU(3) and MesonsSU(3) and Mesons

•• Quarks and AntiQuarks and Anti--quarksquarks •• Quarks and AntiQuarks and Anti--quarksquarks

•• Masses and Symmetry BreakingMasses and Symmetry Breaking •• Masses and Symmetry BreakingMasses and Symmetry Breaking

•• SU(3) and BaryonsSU(3) and Baryons •• SU(3) and BaryonsSU(3) and Baryons

•• QuarkQuark--Model TestedModel Tested •• QuarkQuark--Model TestedModel Tested

•• The Color Quantum NumberThe Color Quantum Number •• The Color Quantum NumberThe Color Quantum Number

•• The Charm and SU(4)The Charm and SU(4) •• The Charm and SU(4)The Charm and SU(4)

•• Beyond SU(4)Beyond SU(4) •• Beyond SU(4)Beyond SU(4)

•• PionsPions and and ChiralChiral Symmetry BreakingSymmetry Breaking •• PionsPions and and ChiralChiral Symmetry BreakingSymmetry Breaking

The SU(3) GroupThe SU(3) Group

•• TheThe groupgroup SU(SU(33)) hashas 3322--11==88 generatorsgenerators.. •• TheThe groupgroup SU(SU(33)) hashas 3322--11==88 generatorsgenerators..

•• TheThe numbernumber ofof generatorsgenerators forfor SU(N)SU(N) isis NN22--11.. •• TheThe numbernumber ofof generatorsgenerators forfor SU(N)SU(N) isis NN22--11..

•• TheThe generatorsgenerators cancan bebe takentaken toto bebe tracelesstraceless andand HermitionHermition whichwhich impliesimplies thatthat thethe elementselements ofof thethe groupgroup areare unitaryunitary andand havehave unitunit determinantdeterminant..

•• TheThe generatorsgenerators cancan bebe takentaken toto bebe tracelesstraceless andand HermitionHermition whichwhich impliesimplies thatthat thethe elementselements ofof thethe groupgroup areare unitaryunitary andand havehave unitunit determinantdeterminant..

•• OutOut ofof 88 generators,generators, atat mostmost twotwo areare diagonaldiagonal.. •• OutOut ofof 88 generators,generators, atat mostmost twotwo areare diagonaldiagonal..

•• TheThe numbernumber ofof commutingcommuting generatorsgenerators isis equalequal toto thethe rankrank ofof thethe groupgroup.. TheThe rankrank ofof SU(N)SU(N) isis NN--11.. ForFor SU(SU(33),), itit isis 22..

•• TheThe numbernumber ofof commutingcommuting generatorsgenerators isis equalequal toto thethe rankrank ofof thethe groupgroup.. TheThe rankrank ofof SU(N)SU(N) isis NN--11.. ForFor SU(SU(33),), itit isis 22..

•• ThereThere areare 22 CasimirCasimir operatorsoperators.. ForFor LieLie groups,groups, numbernumber ofof CasmirCasmir operatorsoperators isis equalequal toto thethe rankrank ofof thethe groupgroup.. •• ThereThere areare 22 CasimirCasimir operatorsoperators.. ForFor LieLie groups,groups, numbernumber ofof CasmirCasmir operatorsoperators isis equalequal toto thethe rankrank ofof thethe groupgroup..


•• StandardStandard choicechoice ofof thethe generatorsgenerators ofof thethe fundamentalfundamental representationsrepresentations areare FFii== ½½ ii,, ii areare GellGell--MannMann matricesmatrices.. •• StandardStandard choicechoice ofof thethe generatorsgenerators ofof thethe fundamentalfundamental representationsrepresentations areare FFii== ½½ ii,, ii areare GellGell--MannMann matricesmatrices..

•• 33 andand 88 areare diagonaldiagonal matricesmatrices.. NoteNote thatthat 11,, 22 andand 33 generategenerate thethe SU(SU(22)) groupgroup.. •• 33 andand 88 areare diagonaldiagonal matricesmatrices.. NoteNote thatthat 11,, 22 andand 33 generategenerate thethe SU(SU(22)) groupgroup..


•• TheThe propertiesproperties ofof thethe SU(SU(33)) groupgroup areare defineddefined byby thethe commutatorcommutator:: •• TheThe propertiesproperties ofof thethe SU(SU(33)) groupgroup areare defineddefined byby thethe commutatorcommutator::

•• TheThe structurestructure constantsconstants ffijkijk areare completelycompletely antianti--symmetricsymmetric inin thethe indicesindices.. •• TheThe structurestructure constantsconstants ffijkijk areare completelycompletely antianti--symmetricsymmetric inin thethe indicesindices..

•• SinceSince [F[F33 ,,FF88]=]=00,, wewe cancan labellabel thethe statesstates withwith thethe eigeneigen-- valuesvalues ofof FF33 andand FF88.. •• SinceSince [F[F33 ,,FF88]=]=00,, wewe cancan labellabel thethe statesstates withwith thethe eigeneigen-- valuesvalues ofof FF33 andand FF88..

•• TheThe groupgroup elementselements ofof thethe ofof SU(SU(33)) areare ofof thethe formform:: •• TheThe groupgroup elementselements ofof thethe ofof SU(SU(33)) areare ofof thethe formform::


•• AsAs aa basisbasis forfor thethe fundamentalfundamental representationrepresentation inin SU(SU(33)) wewe choosechoose thethe eigenstateseigenstates ofof λλ33 andand λλ88.. •• AsAs aa basisbasis forfor thethe fundamentalfundamental representationrepresentation inin SU(SU(33)) wewe choosechoose thethe eigenstateseigenstates ofof λλ33 andand λλ88..

•• TheThe eigenvalueseigenvalues areare:: •• TheThe eigenvalueseigenvalues areare::

The SU(3) GroupThe SU(3) Group •• WhatWhat cancan SU(SU(33)) possiblypossibly havehave toto dodo withwith mesonsmesons andand baryonsbaryons discovereddiscovered soso far?far? •• WhatWhat cancan SU(SU(33)) possiblypossibly havehave toto dodo withwith mesonsmesons andand baryonsbaryons discovereddiscovered soso far?far?

•• LetLet usus definedefine thethe matricesmatrices forfor isospinisospin andand hyperchargehypercharge asas:: •• LetLet usus definedefine thethe matricesmatrices forfor isospinisospin andand hyperchargehypercharge asas::

•• ThenThen:: •• ThenThen::

QuarksQuarks

•• DueDue toto thethe relationrelation:: •• DueDue toto thethe relationrelation::

wewe cancan assignassign chargecharge QQ toto thethe threethree basisbasis statesstates:: wewe cancan assignassign chargecharge QQ toto thethe threethree basisbasis statesstates::

LetLet usus alsoalso definedefine thethe ladderladder operatorsoperators:: LetLet usus alsoalso definedefine thethe ladderladder operatorsoperators::

ForFor thethe fundamentalfundamental representationrepresentation ofof SU(SU(22)):: ForFor thethe fundamentalfundamental representationrepresentation ofof SU(SU(22))::

QuarksQuarks •• WeWe definedefine:: •• WeWe definedefine::

•• WeWe thusthus havehave forfor allall basisbasis statesstates:: •• WeWe thusthus havehave forfor allall basisbasis statesstates::

•• ThusThus SU(SU(33)) triplettriplet inin isospinisospin--hyperchargehypercharge planeplane isis:: •• ThusThus SU(SU(33)) triplettriplet inin isospinisospin--hyperchargehypercharge planeplane isis::

AntiAnti--QuarksQuarks

•• SU(SU(33)) antianti--triplettriplet inin thethe isospinisospin--hyperchargehypercharge planeplane:: •• SU(SU(33)) antianti--triplettriplet inin thethe isospinisospin--hyperchargehypercharge planeplane::

AntiAnti--QuarksQuarks

•• TheThe generatorsgenerators inin thethe conjugateconjugate representationrepresentation:: •• TheThe generatorsgenerators inin thethe conjugateconjugate representationrepresentation::

•• AsAs thethe isospinisospin andand hyperchargehypercharge ofof quarksquarks andand antianti--quarksquarks areare reversedreversed whilewhile statesstates areare representedrepresented byby samesame vectorsvectors •• AsAs thethe isospinisospin andand hyperchargehypercharge ofof quarksquarks andand antianti--quarksquarks areare reversedreversed whilewhile statesstates areare representedrepresented byby samesame vectorsvectors

•• WeWe choosechoose thethe basisbasis statesstates inin thethe antianti--triplettriplet spacespace asas:: •• WeWe choosechoose thethe basisbasis statesstates inin thethe antianti--triplettriplet spacespace asas::

•• NoteNote thatthat theythey shouldshould bebe distinguisheddistinguished fromfrom thethe basisbasis statesstates inin thethe triplettriplet spacespace.. •• NoteNote thatthat theythey shouldshould bebe distinguisheddistinguished fromfrom thethe basisbasis statesstates inin thethe triplettriplet spacespace..

•• SU(SU(33)) algebraalgebra fixesfixes thethe remainingremaining matricesmatrices.. •• SU(SU(33)) algebraalgebra fixesfixes thethe remainingremaining matricesmatrices..

AntiAnti--QuarksQuarks •• ThusThus inin thethe antianti--triplettriplet representationrepresentation:: •• ThusThus inin thethe antianti--triplettriplet representationrepresentation::

•• AndAnd thethe isospinisospin ladderladder operatorsoperators areare:: •• AndAnd thethe isospinisospin ladderladder operatorsoperators areare::

•• WeWe thusthus havehave thethe actionsactions:: •• WeWe thusthus havehave thethe actionsactions::

•• WeWe needneed antianti--quarksquarks toto constructconstruct mesonsmesons •• WeWe needneed antianti--quarksquarks toto constructconstruct mesonsmesons

•• InIn thethe quarkquark model,model, mesonsmesons areare constructedconstructed fromfrom quarkquark && antianti--quarkquark statesstates.. •• InIn thethe quarkquark model,model, mesonsmesons areare constructedconstructed fromfrom quarkquark && antianti--quarkquark statesstates..

MesonsMesons

•• TheThe correspondingcorresponding multipletsmultiplets areare obtainedobtained byby combiningcombining aa 33 andand aa 33.. •• TheThe correspondingcorresponding multipletsmultiplets areare obtainedobtained byby combiningcombining aa 33 andand aa 33..

•• TheThe baryonbaryon numbernumber ofof aa mesonmeson isis 11//33 –– 11//33 ==00.. •• TheThe baryonbaryon numbernumber ofof aa mesonmeson isis 11//33 –– 11//33 ==00..

•• TheThe totaltotal spinspin isis J=L+SJ=L+S.. SinceSince S=S=00,,11,, thethe mesonsmesons havehave integerinteger spinspin andand hencehence areare bosonsbosons.. •• TheThe totaltotal spinspin isis J=L+SJ=L+S.. SinceSince S=S=00,,11,, thethe mesonsmesons havehave integerinteger spinspin andand hencehence areare bosonsbosons..

•• Let us start with SU(2) where we just have u and dLet us start with SU(2) where we just have u and d--quarks. quarks. •• Let us start with SU(2) where we just have u and dLet us start with SU(2) where we just have u and d--quarks. quarks.

•• WeWe cancan constructconstruct aa mesonmeson withwith chargecharge 11 byby combiningcombining aa uu andand aa dd.. •• WeWe cancan constructconstruct aa mesonmeson withwith chargecharge 11 byby combiningcombining aa uu andand aa dd..

•• TheThe isospinisospin projectionprojection isis givengiven byby:: •• TheThe isospinisospin projectionprojection isis givengiven byby::

SU(3) and MesonsSU(3) and Mesons

•• ThusThus |u|u d>d> belongsbelongs toto T=T=11 isotripletisotriplet ((pionspions)):: •• ThusThus |u|u d>d> belongsbelongs toto T=T=11 isotripletisotriplet ((pionspions))::

•• T=T=00 statestate isis orthogonalorthogonal toto T=T=11 andand TT33==00 statestate:: •• T=T=00 statestate isis orthogonalorthogonal toto T=T=11 andand TT33==00 statestate::

•• T=T=11 statesstates areare transformedtransformed intointo eacheach otherother throughthrough isospinisospin transformationstransformations.. T=T=00 isospinisospin statestate isis invariantinvariant.. •• T=T=11 statesstates areare transformedtransformed intointo eacheach otherother throughthrough isospinisospin transformationstransformations.. T=T=00 isospinisospin statestate isis invariantinvariant..


ForFor SU(SU(33)) group,group, wewe havehave UU andand VV spinsspins inin additionaddition toto thethe isospinisospin TT:: ForFor SU(SU(33)) group,group, wewe havehave UU andand VV spinsspins inin additionaddition toto thethe isospinisospin TT::

MoreMore mesonsmesons areare formedformed fromfrom quarkquark antianti--quarkquark statesstates:: MoreMore mesonsmesons areare formedformed fromfrom quarkquark antianti--quarkquark statesstates::

TT++ raisesraises TT33 byby 11 && YY isis unchangedunchanged UU++ lowerslowers TT33 byby ½½ && raisesraises YY byby 11 VV++ raisesraises TT33 byby ½½ && raisesraises YY byby 11..

TT++ raisesraises TT33 byby 11 && YY isis unchangedunchanged UU++ lowerslowers TT33 byby ½½ && raisesraises YY byby 11 VV++ raisesraises TT33 byby ½½ && raisesraises YY byby 11..


GroundGround statestate MesonsMesons inin SU(SU(33)) GroundGround statestate MesonsMesons inin SU(SU(33))


JJPP==11----MesonsMesons InIn SU(SU(33)) JJPP==11----MesonsMesons InIn SU(SU(33))

SU(3) and BaryonsSU(3) and Baryons •• TheThe irreducibleirreducible representationsrepresentations forfor BaryonsBaryons:: •• TheThe irreducibleirreducible representationsrepresentations forfor BaryonsBaryons::

SU(3) and BaryonsSU(3) and Baryons •• TheThe EightEight FoldFold WayWay:: •• TheThe EightEight FoldFold WayWay::

SU(3) and BaryonsSU(3) and Baryons

•• J=J=33//22 baryonbaryon decoupletdecouplet::

•• J=J=33//22 baryonbaryon decoupletdecouplet::

Masses and Symmetry BreakingMasses and Symmetry Breaking •• TheThe SU(SU(22)) symmetrysymmetry:: •• TheThe SU(SU(22)) symmetrysymmetry::

•• MMnn--MMpp== 11..293293 MeVMeV.. ExcellentExcellent isospinisospin symmetrysymmetry isis exhibitedexhibited becausebecause upup andand downdown massesmasses areare nearlynearly equalequal.. •• MMnn--MMpp== 11..293293 MeVMeV.. ExcellentExcellent isospinisospin symmetrysymmetry isis exhibitedexhibited becausebecause upup andand downdown massesmasses areare nearlynearly equalequal..

Masses and Symmetry BreakingMasses and Symmetry Breaking •• TheThe massmass differencesdifferences alongalong YY--axisaxis ~~ 200200 MeVMeV.. •• TheThe massmass differencesdifferences alongalong YY--axisaxis ~~ 200200 MeVMeV..

•• ItIt isis stillstill muchmuch smallersmaller thanthan thethe massesmasses themselvesthemselves.. SoSo wewe stillstill havehave anan SU(SU(33)) flavorflavor symmetrysymmetry thoughthough notnot asas goodgood asas SU(SU(22)) isospinisospin symmetrysymmetry..

•• ItIt isis stillstill muchmuch smallersmaller thanthan thethe massesmasses themselvesthemselves.. SoSo wewe stillstill havehave anan SU(SU(33)) flavorflavor symmetrysymmetry thoughthough notnot asas goodgood asas SU(SU(22)) isospinisospin symmetrysymmetry..

Masses and Symmetry BreakingMasses and Symmetry Breaking

•• TheThe massmass differencesdifferences ofof baryonsbaryons inin J=J=33//22 decoupletdecouplet.. •• TheThe massmass differencesdifferences ofof baryonsbaryons inin J=J=33//22 decoupletdecouplet..

•• DiscoveryDiscovery ofof thethe ΩΩ.. 19691969 NobelNobel prizeprize toto GellGell--MannMann.. •• DiscoveryDiscovery ofof thethe ΩΩ.. 19691969 NobelNobel prizeprize toto GellGell--MannMann..

GellGell--Mann Okubo Mass FormulaeMann Okubo Mass Formulae

•• WeWe estimateestimate thethe hadronhadron massesmasses inin thethe quarkquark modelmodel byby assumingassuming thatthat aa partpart HHII ofof thethe totaltotal HamiltonianHamiltonian HH00+H+HII

breaksbreaks SU(SU(33)) symmetrysymmetry..

•• WeWe estimateestimate thethe hadronhadron massesmasses inin thethe quarkquark modelmodel byby assumingassuming thatthat aa partpart HHII ofof thethe totaltotal HamiltonianHamiltonian HH00+H+HII

breaksbreaks SU(SU(33)) symmetrysymmetry..

•• LetLet usus assumeassume thatthat thethe isospinisospin symmetrysymmetry remainsremains intactintact.. ThusThus mmuu==mmdd.. AssumeAssume thatthat HHII ~~ YY.. •• LetLet usus assumeassume thatthat thethe isospinisospin symmetrysymmetry remainsremains intactintact.. ThusThus mmuu==mmdd.. AssumeAssume thatthat HHII ~~ YY..

•• LetLet usus alsoalso assumeassume thatthat thethe bindingbinding energyenergy ofof hadronshadrons isis independentindependent ofof quarkquark flavorflavor andand thethe massmass differencedifference isis entirelyentirely duedue toto quarkquark massmass differencesdifferences..

•• LetLet usus alsoalso assumeassume thatthat thethe bindingbinding energyenergy ofof hadronshadrons isis independentindependent ofof quarkquark flavorflavor andand thethe massmass differencedifference isis entirelyentirely duedue toto quarkquark massmass differencesdifferences..

•• ConsiderConsider 00-- mesonmeson multipletmultiplet:: •• ConsiderConsider 00-- mesonmeson multipletmultiplet::

•• NoteNote thatthat wewe havehave usedused thethe quadraticquadratic massmass forfor mesonsmesons.. ItIt fitsfits muchmuch betterbetter thanthan linearlinear massmass andand itit isis relatedrelated toto chiralchiral symmetrysymmetry breakingbreaking inin QCDQCD.. ThusThus

•• NoteNote thatthat wewe havehave usedused thethe quadraticquadratic massmass forfor mesonsmesons.. ItIt fitsfits muchmuch betterbetter thanthan linearlinear massmass andand itit isis relatedrelated toto chiralchiral symmetrysymmetry breakingbreaking inin QCDQCD.. ThusThus

•• ExperimentallyExperimentally:: RHSRHS ~~ 00..9292 GeVGeV22,, LHSLHS ~~ 00..9898 GeVGeV22.. ThusThus thethe relationrelation isis goodgood toto aa fewfew percentpercent.. •• ExperimentallyExperimentally:: RHSRHS ~~ 00..9292 GeVGeV22,, LHSLHS ~~ 00..9898 GeVGeV22.. ThusThus thethe relationrelation isis goodgood toto aa fewfew percentpercent.. •• WeWe cancan repeatrepeat thethe exerciseexercise forfor ½½++ baryonsbaryons:: •• WeWe cancan repeatrepeat thethe exerciseexercise forfor ½½++ baryonsbaryons::


•• Experimentally,Experimentally, LHSLHS isis 22..2323 GeVGeV andand RHSRHS isis 22..2525 GeVGeV.. •• Experimentally,Experimentally, LHSLHS isis 22..2323 GeVGeV andand RHSRHS isis 22..2525 GeVGeV..

•• WeWe alsoalso havehave thethe relationrelation:: •• WeWe alsoalso havehave thethe relationrelation::


•• ForFor 33//22 baryonbaryon decoupletdecouplet,, oneone cancan alsoalso derivederive thethe equalequal spacingspacing relationsrelations:: •• ForFor 33//22 baryonbaryon decoupletdecouplet,, oneone cancan alsoalso derivederive thethe equalequal spacingspacing relationsrelations::

•• ThisThis relationrelation waswas usedused byby GellGell--MannMann toto predictpredict thethe existence,existence, naturenature andand massmass ofof thethe particleparticle.. •• ThisThis relationrelation waswas usedused byby GellGell--MannMann toto predictpredict thethe existence,existence, naturenature andand massmass ofof thethe particleparticle..

GellGell--MannMann hashas alsoalso foundfound thatthat "The"The EightfoldEightfold Way"Way" cancan bebe describeddescribed veryvery simplysimply byby assumingassuming thatthat allall particlesparticles whichwhich interactinteract stronglystrongly withwith eacheach otherother areare composedcomposed ofof onlyonly threethree kindskinds ofof particlesparticles whichwhich hehe calledcalled quarksquarks andand ofof thethe correspondingcorresponding antiparticlesantiparticles.. TheThe quarksquarks areare peculiarpeculiar inin particularparticular becausebecause theirtheir chargescharges areare fractionsfractions ofof thethe protonproton chargecharge whichwhich accordingaccording toto allall experienceexperience upup toto nownow isis thethe indivisibleindivisible elementaryelementary chargecharge.. ItIt hashas notnot yetyet beenbeen possiblepossible toto findfind individualindividual quarksquarks althoughalthough theythey havehave beenbeen eagerlyeagerly lookedlooked forfor.. GellGell--Mann'sMann's ideaidea isis nonenone thethe lessless ofof greatgreat heuristicheuristic valuevalue..


NobelNobel PrizePrize ((19691969)):: NobelNobel PrizePrize ((19691969))::

The Quark Model The Quark Model –– TestedTested

•• IfIf thethe quarkquark modelmodel werewere notnot correct,correct, oneone couldcould trytry toto looklook forfor exoticexotic mesonsmesons (say(say ofof strangenessstrangeness --33,, likelike thethe baryonbaryon ΩΩ-- oror ofof chargecharge ++22 likelike thethe ΔΔ++++)) oror exoticexotic baryonsbaryons (S=(S=00 andand Q=Q=--22)).. NoneNone ofof thesethese exoticsexotics werewere foundfound..

•• IfIf thethe quarkquark modelmodel werewere notnot correct,correct, oneone couldcould trytry toto looklook forfor exoticexotic mesonsmesons (say(say ofof strangenessstrangeness --33,, likelike thethe baryonbaryon ΩΩ-- oror ofof chargecharge ++22 likelike thethe ΔΔ++++)) oror exoticexotic baryonsbaryons (S=(S=00 andand Q=Q=--22)).. NoneNone ofof thesethese exoticsexotics werewere foundfound..

•• TheThe failurefailure toto produceproduce isolatedisolated quarksquarks inin experimentsexperiments mademade peoplepeople veryvery skepticalskeptical aboutabout thethe quarkquark modelmodel.. •• TheThe failurefailure toto produceproduce isolatedisolated quarksquarks inin experimentsexperiments mademade peoplepeople veryvery skepticalskeptical aboutabout thethe quarkquark modelmodel..

•• ThoseThose whowho wantedwanted toto stickstick toto thethe quarkquark modelmodel inventedinvented confinementconfinement ofof quarksquarks.. •• ThoseThose whowho wantedwanted toto stickstick toto thethe quarkquark modelmodel inventedinvented confinementconfinement ofof quarksquarks..

•• CouldCould thethe experimentsexperiments likelike thethe RutherfordRutherford scatteringscattering indicateindicate thethe existenceexistence ofof quarksquarks?? •• CouldCould thethe experimentsexperiments likelike thethe RutherfordRutherford scatteringscattering indicateindicate thethe existenceexistence ofof quarksquarks??


•• TheThe inelasticinelastic electronelectron--nucleusnucleus scatteringscattering experimentsexperiments conductedconducted betweenbetween 19671967 andand 19731973 atat thethe StanfordStanford LinearLinear AcceleratorAccelerator CenterCenter providedprovided aa keykey evidenceevidence forfor thethe existenceexistence ofof quarksquarks..

•• TheThe inelasticinelastic electronelectron--nucleusnucleus scatteringscattering experimentsexperiments conductedconducted betweenbetween 19671967 andand 19731973 atat thethe StanfordStanford LinearLinear AcceleratorAccelerator CenterCenter providedprovided aa keykey evidenceevidence forfor thethe existenceexistence ofof quarksquarks..

•• TheThe leadersleaders ofof thesethese experimentsexperiments werewere JJ.. FriedmanFriedman andand HH.. KendallKendall ofof MITMIT && RR.. TaylorTaylor ofof SLACSLAC.. TheyThey werewere awardedawarded thethe 19901990 NobelNobel PrizePrize inin PhysicsPhysics..

•• TheThe leadersleaders ofof thesethese experimentsexperiments werewere JJ.. FriedmanFriedman andand HH.. KendallKendall ofof MITMIT && RR.. TaylorTaylor ofof SLACSLAC.. TheyThey werewere awardedawarded thethe 19901990 NobelNobel PrizePrize inin PhysicsPhysics..

•• TheThe electronselectrons ofof 2020 GeVGeV werewere scatteredscattered offoff thethe protonsprotons inin aa deepdeep inelasticinelastic scatteringscattering processprocess.. •• TheThe electronselectrons ofof 2020 GeVGeV werewere scatteredscattered offoff thethe protonsprotons inin aa deepdeep inelasticinelastic scatteringscattering processprocess..

•• TheThe plotplot ofof crosscross--sectionsection versusversus invariantinvariant momentummomentum transfertransfer toto thethe protonproton QQ22==22EE’(EE’(11--coscosθθ)) showedshowed thatthat itit droppeddropped muchmuch moremore slowlyslowly thanthan thethe samesame quantityquantity forfor elasticelastic scatteringscattering..

•• TheThe plotplot ofof crosscross--sectionsection versusversus invariantinvariant momentummomentum transfertransfer toto thethe protonproton QQ22==22EE’(EE’(11--coscosθθ)) showedshowed thatthat itit droppeddropped muchmuch moremore slowlyslowly thanthan thethe samesame quantityquantity forfor elasticelastic scatteringscattering..


CrossCross--sectionsection forfor inelasticinelastic electronelectron--protonproton scatteringscattering measuredmeasured atat 66oo inin thethe firstfirst MITMIT--SLACSLAC experiment,experiment, normalizednormalized byby thosethose expectedexpected forfor MottMott scatteringscattering.. TheThe datadata pointspoints areare givengiven forfor twotwo valuesvalues ofof W,W, thethe invariantinvariant massmass ofof thethe unobservedunobserved finalfinal statestate ofof hadronshadrons..

CrossCross--sectionsection forfor inelasticinelastic electronelectron--protonproton scatteringscattering measuredmeasured atat 66oo inin thethe firstfirst MITMIT--SLACSLAC experiment,experiment, normalizednormalized byby thosethose expectedexpected forfor MottMott scatteringscattering.. TheThe datadata pointspoints areare givengiven forfor twotwo valuesvalues ofof W,W, thethe invariantinvariant massmass ofof thethe unobservedunobserved finalfinal statestate ofof hadronshadrons..

The Color Quantum NumberThe Color Quantum Number

•• ThereThere waswas anotheranother objectionobjection toto thethe QuarkQuark ModelModel.. ItIt seemedseemed toto violateviolate PauliPauli exclusionexclusion principleprinciple.. •• ThereThere waswas anotheranother objectionobjection toto thethe QuarkQuark ModelModel.. ItIt seemedseemed toto violateviolate PauliPauli exclusionexclusion principleprinciple..

•• TheThe hadronhadron wavewave--functionfunction cancan bebe decomposeddecomposed intointo 33 partsparts,, thethe space,space, spinspin andand isospinisospin partsparts:: •• TheThe hadronhadron wavewave--functionfunction cancan bebe decomposeddecomposed intointo 33 partsparts,, thethe space,space, spinspin andand isospinisospin partsparts::

•• LetLet usus considerconsider thethe ΔΔ++++ whosewhose quarkquark contentcontent isis uuuuuu withwith thethe spinsspins alignedaligned toto givegive aa totaltotal spinspin 33//22.. SoSo thethe spinspin-- isospinisospin partpart isis::

•• LetLet usus considerconsider thethe ΔΔ++++ whosewhose quarkquark contentcontent isis uuuuuu withwith thethe spinsspins alignedaligned toto givegive aa totaltotal spinspin 33//22.. SoSo thethe spinspin-- isospinisospin partpart isis::

•• InIn thethe lowestlowest state,state, spacespace partpart mustmust bebe symmetricsymmetric tootoo.. ThisThis isis inin contradictioncontradiction withwith spinspin--statisticsstatistics asas quarksquarks havehave spinspin ½½..

•• InIn thethe lowestlowest state,state, spacespace partpart mustmust bebe symmetricsymmetric tootoo.. ThisThis isis inin contradictioncontradiction withwith spinspin--statisticsstatistics asas quarksquarks havehave spinspin ½½..


•• ThisThis problemproblem waswas solvedsolved byby thethe worksworks ofof GreenbergGreenberg ((19641964)) andand HanHan && NambuNambu ((19651965)).. •• ThisThis problemproblem waswas solvedsolved byby thethe worksworks ofof GreenbergGreenberg ((19641964)) andand HanHan && NambuNambu ((19651965))..

•• AA newnew quantumquantum numbernumber colorcolor associatedassociated withwith aa symmetrysymmetry groupgroup SU(SU(33))cc waswas proposedproposed.. •• AA newnew quantumquantum numbernumber colorcolor associatedassociated withwith aa symmetrysymmetry groupgroup SU(SU(33))cc waswas proposedproposed..

•• TheThe quarksquarks lielie inin thethe fundamentalfundamental representationrepresentation ofof thisthis groupgroup.. TheyThey cancan havehave anyany ofof thethe threethree colors,colors, saysay green,green, redred andand blueblue..

•• TheThe quarksquarks lielie inin thethe fundamentalfundamental representationrepresentation ofof thisthis groupgroup.. TheyThey cancan havehave anyany ofof thethe threethree colors,colors, saysay green,green, redred andand blueblue..

•• AsAs wewe cancan differentiatedifferentiate betweenbetween thethe quarksquarks throughthrough theirtheir color,color, PauliPauli exclusionexclusion principleprinciple doesdoes notnot applyapply toto themthem.. •• AsAs wewe cancan differentiatedifferentiate betweenbetween thethe quarksquarks throughthrough theirtheir color,color, PauliPauli exclusionexclusion principleprinciple doesdoes notnot applyapply toto themthem..


•• WithWith thethe hypothesishypothesis ofof thethe color,color, thethe numbernumber ofof quarksquarks hashas increasedincreased fromfrom 33 toto 99.. However,However, thethe numbernumber ofof observedobserved hadronshadrons remainsremains thethe same!same!

•• WithWith thethe hypothesishypothesis ofof thethe color,color, thethe numbernumber ofof quarksquarks hashas increasedincreased fromfrom 33 toto 99.. However,However, thethe numbernumber ofof observedobserved hadronshadrons remainsremains thethe same!same!

•• ItIt isis becausebecause itit isis furtherfurther assumedassumed thatthat allall hadronshadrons mustmust bebe colorcolor singletsinglet.. ThusThus ΔΔ++++ isis aa boundbound statestate ofof threethree upup quarksquarks withwith differentdifferent colorscolors ((uurr,, uugg,, uubb)).. WeWe nevernever findfind aa ΔΔ++++ withwith ((uurr,, uurr,, uubb)) forfor exampleexample..

•• ItIt isis becausebecause itit isis furtherfurther assumedassumed thatthat allall hadronshadrons mustmust bebe colorcolor singletsinglet.. ThusThus ΔΔ++++ isis aa boundbound statestate ofof threethree upup quarksquarks withwith differentdifferent colorscolors ((uurr,, uugg,, uubb)).. WeWe nevernever findfind aa ΔΔ++++ withwith ((uurr,, uurr,, uubb)) forfor exampleexample..

•• TheThe onlyonly colorlesscolorless combibationscombibations ofof quarksquarks thatthat cancan bebe mademade areare qq qq,, qq qq qq oror qq qq qq.. •• TheThe onlyonly colorlesscolorless combibationscombibations ofof quarksquarks thatthat cancan bebe mademade areare qq qq,, qq qq qq oror qq qq qq..

•• OfOf coursecourse wewe cancan alsoalso havehave boundbound statestate ofof sixsix quarksquarks butbut itit cancan bebe interpretedinterpreted asas aa boundbound statestate ofof twotwo baryonsbaryons.. •• OfOf coursecourse wewe cancan alsoalso havehave boundbound statestate ofof sixsix quarksquarks butbut itit cancan bebe interpretedinterpreted asas aa boundbound statestate ofof twotwo baryonsbaryons..

The Charm and SU(4)The Charm and SU(4) •• BjorkenBjorken andand GlashowGlashow observedobserved aa mismatchmismatch betweenbetween leptonsleptons andand quarksquarks:: e,e, ννee,, μμ,, ννμμ u,u, d,d, ss

•• BjorkenBjorken andand GlashowGlashow observedobserved aa mismatchmismatch betweenbetween leptonsleptons andand quarksquarks:: e,e, ννee,, μμ,, ννμμ u,u, d,d, ss

•• DemandingDemanding leptonlepton quarkquark symmetrysymmetry,, theythey expectedexpected thethe

existenceexistence ofof aa newnew charmcharm quarkquark..

•• DemandingDemanding leptonlepton quarkquark symmetrysymmetry,, theythey expectedexpected thethe

existenceexistence ofof aa newnew charmcharm quarkquark..

•• AA newnew mesonmeson J/ψJ/ψ waswas discovereddiscovered inin 19741974.. ItIt hadhad aa massmass

33 GeVGeV,, 33 timestimes asas heavyheavy asas thatthat ofof aa protonproton..

•• AA newnew mesonmeson J/ψJ/ψ waswas discovereddiscovered inin 19741974.. ItIt hadhad aa massmass

33 GeVGeV,, 33 timestimes asas heavyheavy asas thatthat ofof aa protonproton..

•• ItIt waswas electricallyelectrically neutralneutral andand hadhad aa lifelife--timetime ofof aboutabout 1010--2020 secondsseconds whilewhile allall otherother hadronshadrons ofof thatthat massmass rangerange hadhad aa lifelife timetime ofof aboutabout 1010--2323 secondsseconds..

•• ItIt waswas electricallyelectrically neutralneutral andand hadhad aa lifelife--timetime ofof aboutabout 1010--2020 secondsseconds whilewhile allall otherother hadronshadrons ofof thatthat massmass rangerange hadhad aa lifelife timetime ofof aboutabout 1010--2323 secondsseconds..

•• ItIt camecame outout toto bebe thethe boundbound statestate ofof quarksquarks charmcharm andand antianti--charm,charm, aa charmoniumcharmonium.. RichterRichter andand TingTing werewere awardedawarded NobelNobel prizeprize ofof 19761976 forfor itsits discoverydiscovery..

•• ItIt camecame outout toto bebe thethe boundbound statestate ofof quarksquarks charmcharm andand antianti--charm,charm, aa charmoniumcharmonium.. RichterRichter andand TingTing werewere awardedawarded NobelNobel prizeprize ofof 19761976 forfor itsits discoverydiscovery..

The Charm and SU(4)The Charm and SU(4)

•• TheThe multipletsmultiplets ofof S(S(44)) mesonsmesons andand baryonsbaryons cancan readilyreadily bebe constructedconstructed.. TheThe nonetsnonets ofof lightlight mesonsmesons occupyoccupy thethe centralcentral planeplane..

•• TheThe multipletsmultiplets ofof S(S(44)) mesonsmesons andand baryonsbaryons cancan readilyreadily bebe constructedconstructed.. TheThe nonetsnonets ofof lightlight mesonsmesons occupyoccupy thethe centralcentral planeplane..

The Charm and SU(4)The Charm and SU(4) Similarly,Similarly, oneone cancan constructconstruct thethe SU(SU(44)) baryonbaryon multipletsmultiplets Similarly,Similarly, oneone cancan constructconstruct thethe SU(SU(44)) baryonbaryon multipletsmultiplets

Beyond SU(4)Beyond SU(4)

•• WeWe nownow knowknow therethere areare sixsix quarkquark flavorsflavors:: u,d,s,c,b,tu,d,s,c,b,t •• WeWe nownow knowknow therethere areare sixsix quarkquark flavorsflavors:: u,d,s,c,b,tu,d,s,c,b,t

PionsPions and and ChiralChiral Symmetry Breaking Symmetry Breaking

PionsPions werewere predictedpredicted byby YukawaYukawa toto bebe thethe mediatorsmediators ofof thethe strongstrong nuclearnuclear forceforce betweenbetween protonsprotons andand neutronsneutrons whosewhose rangerange isis onlyonly 1010--1414 metersmeters..

PionsPions werewere predictedpredicted byby YukawaYukawa toto bebe thethe mediatorsmediators ofof thethe strongstrong nuclearnuclear forceforce betweenbetween protonsprotons andand neutronsneutrons whosewhose rangerange isis onlyonly 1010--1414 metersmeters..

PionsPions areare thethe lightestlightest ofof hadronshadrons.. TheyThey dodo notnot havehave zerozero massmass.. OtherwiseOtherwise strongstrong forceforce willwill bebe aa longlong rangerange forceforce.. PionsPions areare thethe lightestlightest ofof hadronshadrons.. TheyThey dodo notnot havehave zerozero massmass.. OtherwiseOtherwise strongstrong forceforce willwill bebe aa longlong rangerange forceforce..

AA typicaltypical mesonmeson likelike aa ρρ hashas aa massmass ofof 770770 MeVMeV whilewhile thethe nucleonnucleon hashas aa massmass ofof 940940 MeVMeV.. ThisThis isis consistentconsistent withwith aa constituentconstituent u,du,d,, massmass ofof aroundaround 300300 MeVMeV..

AA typicaltypical mesonmeson likelike aa ρρ hashas aa massmass ofof 770770 MeVMeV whilewhile thethe nucleonnucleon hashas aa massmass ofof 940940 MeVMeV.. ThisThis isis consistentconsistent withwith aa constituentconstituent u,du,d,, massmass ofof aroundaround 300300 MeVMeV..

However,However, pionspions onlyonly weighweigh aboutabout 140140 MeVMeV,, whichwhich isis 11//55thth ofof thethe massmass ofof thethe ρρ.. However,However, pionspions onlyonly weighweigh aboutabout 140140 MeVMeV,, whichwhich isis 11//55thth ofof thethe massmass ofof thethe ρρ..

ThisThis cannotcannot bebe anan accidentaccident.. ThisThis cannotcannot bebe anan accidentaccident..


QuarksQuarks havehave helicityhelicity ++11//22 oror --11//22.. HelicityHelicity dependsdepends onon thethe observerobserver.. ForFor masslessmassless quarks,quarks, itit becomesbecomes aa goodgood quantumquantum numbernumber andand cancan bebe identifiedidentified withwith chiralitychirality..

QuarksQuarks havehave helicityhelicity ++11//22 oror --11//22.. HelicityHelicity dependsdepends onon thethe observerobserver.. ForFor masslessmassless quarks,quarks, itit becomesbecomes aa goodgood quantumquantum numbernumber andand cancan bebe identifiedidentified withwith chiralitychirality..

DiracDirac equationequation forfor masslessmassless quarksquarks isis:: DiracDirac equationequation forfor masslessmassless quarksquarks isis::

RecallRecall thethe γγ55 operatoroperator:: RecallRecall thethe γγ55 operatoroperator::

ItIt antianti--commutescommutes withwith allall thethe γγ matricesmatrices.. ThusThus:: ItIt antianti--commutescommutes withwith allall thethe γγ matricesmatrices.. ThusThus::

PionsPions and and ChiralChiral Symmetry Breaking Symmetry Breaking ThusThus wewe cancan workwork withwith linearlinear combinationscombinations ofof ψψ andand γγ55ψψ:: ThusThus wewe cancan workwork withwith linearlinear combinationscombinations ofof ψψ andand γγ55ψψ::

LL andand RR meanmean helicityhelicity --11//22 andand ++11//22 respectivelyrespectively.. LL andand RR meanmean helicityhelicity --11//22 andand ++11//22 respectivelyrespectively..

TheThe masslessmassless LagrangianLagrangian:: cancan alsoalso bebe writtenwritten asas TheThe masslessmassless LagrangianLagrangian:: cancan alsoalso bebe writtenwritten asas

whichwhich hashas enhancedenhanced symmetrysymmetry:: whichwhich hashas enhancedenhanced symmetrysymmetry::

TheThe massmass termterm mixesmixes thethe chiralchiral partnerspartners:: TheThe massmass termterm mixesmixes thethe chiralchiral partnerspartners::

Nobel Prize 2008:Nobel Prize 2008: “for“for thethe discoverydiscovery ofof thethe mechanismmechanism ofof spontaneousspontaneous brokenbroken symmetrysymmetry inin subatomicsubatomic physics”physics”

Nobel Prize 2008:Nobel Prize 2008: “for“for thethe discoverydiscovery ofof thethe mechanismmechanism ofof spontaneousspontaneous brokenbroken symmetrysymmetry inin subatomicsubatomic physics”physics” quark-anti-quark quark-anti-quark


ItsIts spontaneousspontaneous breakingbreaking leadsleads toto largelarge effectiveeffective quarkquark massesmasses andand thethe existenceexistence ofof GoldstoneGoldstone bosonsbosons:: pionspions.. ItsIts spontaneousspontaneous breakingbreaking leadsleads toto largelarge effectiveeffective quarkquark massesmasses andand thethe existenceexistence ofof GoldstoneGoldstone bosonsbosons:: pionspions..

PionsPions and and ChiralChiral Symmetry Breaking Symmetry Breaking •• ExplicitExplicit SymmetrySymmetry BreakingBreaking:: •• ExplicitExplicit SymmetrySymmetry BreakingBreaking::

•• SpontaneousSpontaneous SymmetrySymmetry BreakingBreaking:: •• SpontaneousSpontaneous SymmetrySymmetry BreakingBreaking::

ExampleExample:: ForFor explicitexplicit chiralchiral symmetrysymmetry breakingbreaking inin QCD,QCD, HH11 containscontains massmass termsterms forfor lightlight quarksquarks.. ExampleExample:: ForFor explicitexplicit chiralchiral symmetrysymmetry breakingbreaking inin QCD,QCD, HH11 containscontains massmass termsterms forfor lightlight quarksquarks..

AA brokenbroken generatorgenerator resultsresults inin aa masslessmassless GoldstoneGoldstone modemode.. AA brokenbroken generatorgenerator resultsresults inin aa masslessmassless GoldstoneGoldstone modemode..

ExampleExample:: PionsPions areare masslessmassless GoldstoneGoldstone bosonsbosons ofof chiralchiral symmetrysymmetry breakingbreaking.. TheirTheir lightlight massmass isis duedue toto explicitexplicit chiralchiral symmetrysymmetry breakingbreaking inin thethe QCDQCD LagrangianLagrangian.. ItIt alsoalso givesgives largelarge effectiveeffective massmass toto constituentconstituent quarksquarks..

ExampleExample:: PionsPions areare masslessmassless GoldstoneGoldstone bosonsbosons ofof chiralchiral symmetrysymmetry breakingbreaking.. TheirTheir lightlight massmass isis duedue toto explicitexplicit chiralchiral symmetrysymmetry breakingbreaking inin thethe QCDQCD LagrangianLagrangian.. ItIt alsoalso givesgives largelarge effectiveeffective massmass toto constituentconstituent quarksquarks..

What Next?What Next?

•• AccordingAccording toto Yukawa,Yukawa, pionspions areare exchangedexchanged betweenbetween protonsprotons andand neutronsneutrons givinggiving riserise toto shortshort rangerange strongstrong interactionsinteractions.. HowHow cancan wewe constructconstruct aa LagrangianLagrangian inin termsterms ofof quarksquarks degreesdegrees ofof freedomfreedom??

•• AccordingAccording toto Yukawa,Yukawa, pionspions areare exchangedexchanged betweenbetween protonsprotons andand neutronsneutrons givinggiving riserise toto shortshort rangerange strongstrong interactionsinteractions.. HowHow cancan wewe constructconstruct aa LagrangianLagrangian inin termsterms ofof quarksquarks degreesdegrees ofof freedomfreedom??

•• ChiralChiral symmetrysymmetry andand itsits breakingbreaking seemseem toto havehave farfar reachingreaching consequencesconsequences forfor hadronhadron physicsphysics.. HowHow cancan wewe understandunderstand it?it? Models?Models? Theory?Theory?

•• ChiralChiral symmetrysymmetry andand itsits breakingbreaking seemseem toto havehave farfar reachingreaching consequencesconsequences forfor hadronhadron physicsphysics.. HowHow cancan wewe understandunderstand it?it? Models?Models? Theory?Theory?

•• WhatWhat isis thethe naturenature andand originorigin ofof fundamentalfundamental forceforce betweenbetween quarksquarks whichwhich isis responsibleresponsible forfor aa multitudemultitude ofof hadronshadrons andand theirtheir varyingvarying propertiesproperties,, confinementconfinement asas wellwell asas chiralchiral symmetrysymmetry breakingbreaking??

•• WhatWhat isis thethe naturenature andand originorigin ofof fundamentalfundamental forceforce betweenbetween quarksquarks whichwhich isis responsibleresponsible forfor aa multitudemultitude ofof hadronshadrons andand theirtheir varyingvarying propertiesproperties,, confinementconfinement asas wellwell asas chiralchiral symmetrysymmetry breakingbreaking??

Lecture 2 Quark Model The Eight Fold Wayfcfm.uas.edu.mx/eff2014/docs/CULIACAN-Lecture-2.pdf ·...

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Transcript of Lecture 2 Quark Model The Eight Fold Wayfcfm.uas.edu.mx/eff2014/docs/CULIACAN-Lecture-2.pdf ·...