lecture 4 qcd as a gauge theory - universidad de...

Lecture Lecture –– 44 QCD as a Gauge TheoryQCD as a Gauge Theory

AdnanAdnan BashirBashir, IFM, UMSNH, Mexico, IFM, UMSNH, Mexico August 2013August 2013

HermosilloHermosillo SonoraSonora

ContentsContents

•• The Gauge Principle The Gauge Principle •• The Gauge Principle The Gauge Principle

•• Quantum Electrodynamics Quantum Electrodynamics •• Quantum Electrodynamics Quantum Electrodynamics

•• SU(N) Gauge Theory SU(N) Gauge Theory •• SU(N) Gauge Theory SU(N) Gauge Theory

•• Global Gauge Transformations Global Gauge Transformations •• Global Gauge Transformations Global Gauge Transformations

•• Local Gauge Transformations Local Gauge Transformations •• Local Gauge Transformations Local Gauge Transformations

•• Dynamics of Field Tensor Dynamics of Field Tensor •• Dynamics of Field Tensor Dynamics of Field Tensor

•• The QCD The QCD LagrangianLagrangian •• The QCD The QCD LagrangianLagrangian

•• Feynman Rules in Covariant Gauges Feynman Rules in Covariant Gauges •• Feynman Rules in Covariant Gauges Feynman Rules in Covariant Gauges

•• Running Coupling in QEDRunning Coupling in QED •• Running Coupling in QEDRunning Coupling in QED

•• Running Coupling in QCDRunning Coupling in QCD •• Running Coupling in QCDRunning Coupling in QCD

•• Asymptotic FreedomAsymptotic Freedom •• Asymptotic FreedomAsymptotic Freedom

The Gauge PrincipalThe Gauge Principal


Maxwell’sMaxwell’s formulationformulation ofof electrodynamicselectrodynamics waswas perhapsperhaps thethe firstfirst fieldfield theorytheory andand aa gaugegauge theorytheory inin physicsphysics.. Maxwell’sMaxwell’s formulationformulation ofof electrodynamicselectrodynamics waswas perhapsperhaps thethe firstfirst fieldfield theorytheory andand aa gaugegauge theorytheory inin physicsphysics..

InIn 19291929,, WeylWeyl showedshowed thatthat electrodynamicselectrodynamics waswas invariantinvariant underunder thethe gaugegauge transformationtransformation ofof thethe gaugegauge fieldfield andand thethe wavewave--functionfunction ofof thethe chargedcharged fieldfield..

InIn 19291929,, WeylWeyl showedshowed thatthat electrodynamicselectrodynamics waswas invariantinvariant underunder thethe gaugegauge transformationtransformation ofof thethe gaugegauge fieldfield andand thethe wavewave--functionfunction ofof thethe chargedcharged fieldfield..

InIn 19541954,, YangYang andand MillsMills studiedstudied thethe gaugegauge principleprinciple inin nonnon--abelianabelian fieldfield theoriestheories.. InIn 19541954,, YangYang andand MillsMills studiedstudied thethe gaugegauge principleprinciple inin nonnon--abelianabelian fieldfield theoriestheories..

GaugeGauge principleprinciple providesprovides usus withwith aa reciperecipe toto constructconstruct fieldfield theoriestheories describingdescribing thethe forcesforces ofof naturenature.. GaugeGauge principleprinciple providesprovides usus withwith aa reciperecipe toto constructconstruct fieldfield theoriestheories describingdescribing thethe forcesforces ofof naturenature..

WeWe shallshall firstfirst studystudy thisthis forfor QEDQED toto seesee howhow itit worksworks andand howhow interactionsinteractions betweenbetween electronselectrons andand photonsphotons emergeemerge naturallynaturally throughthrough employingemploying thethe gaugegauge principleprinciple..

WeWe shallshall firstfirst studystudy thisthis forfor QEDQED toto seesee howhow itit worksworks andand howhow interactionsinteractions betweenbetween electronselectrons andand photonsphotons emergeemerge naturallynaturally throughthrough employingemploying thethe gaugegauge principleprinciple..

WeWe shallshall thenthen employemploy itit toto QCD,QCD, aa nonnon abelianabelian theorytheory describingdescribing interactionsinteractions betweenbetween quarksquarks andand gluonsgluons (gauge(gauge bosons)bosons) andand howhow theythey bindbind togethertogether toto makemake hadronshadrons..

WeWe shallshall thenthen employemploy itit toto QCD,QCD, aa nonnon abelianabelian theorytheory describingdescribing interactionsinteractions betweenbetween quarksquarks andand gluonsgluons (gauge(gauge bosons)bosons) andand howhow theythey bindbind togethertogether toto makemake hadronshadrons..

NoteNote thatthat SalamSalam--WeinbergWeinberg--GlashowGlashow theorytheory ofof weakweak interactionsinteractions isis alsoalso aa gaugegauge theorytheory andand thethe interactionsinteractions cancan bebe derivedderived throughthrough gauginggauging thethe forceforce freefree LagrangianLagrangian..

NoteNote thatthat SalamSalam--WeinbergWeinberg--GlashowGlashow theorytheory ofof weakweak interactionsinteractions isis alsoalso aa gaugegauge theorytheory andand thethe interactionsinteractions cancan bebe derivedderived throughthrough gauginggauging thethe forceforce freefree LagrangianLagrangian..


Quantum ElectrodynamicsQuantum Electrodynamics

TheThe freefree DiracDirac LagrangianLagrangian:: TheThe freefree DiracDirac LagrangianLagrangian::

ThisThis isis invariantinvariant underunder globalglobal U(U(11)) gaugegauge transformationstransformations:: ThisThis isis invariantinvariant underunder globalglobal U(U(11)) gaugegauge transformationstransformations::

wherewhere αα isis aa realreal constantconstant.. ThisThis isis anan abelianabelian symmetrysymmetry becausebecause thethe twotwo successivesuccessive transformationstransformations commutecommute.. wherewhere αα isis aa realreal constantconstant.. ThisThis isis anan abelianabelian symmetrysymmetry becausebecause thethe twotwo successivesuccessive transformationstransformations commutecommute..

ThisThis isis aa continuouscontinuous symmetrysymmetry.. ElectricElectric currentcurrent isis thethe conservedconserved NoetherNoether currentcurrent andand thethe fermionfermion electricelectric chargecharge isis conservedconserved..

ThisThis isis aa continuouscontinuous symmetrysymmetry.. ElectricElectric currentcurrent isis thethe conservedconserved NoetherNoether currentcurrent andand thethe fermionfermion electricelectric chargecharge isis conservedconserved..

Quantum ElectrodynamicsQuantum Electrodynamics TheThe wavewave--functionfunction ofof anan electronelectron carriescarries aa phasephase withwith itit.. TheThe wavewave--functionfunction ofof anan electronelectron carriescarries aa phasephase withwith itit..

InIn globalglobal gaugegauge transformation,transformation, thisthis phasephase isis changedchanged byby thethe samesame amountamount atat everyevery spacespace--timetime pointpoint.. InIn globalglobal gaugegauge transformation,transformation, thisthis phasephase isis changedchanged byby thethe samesame amountamount atat everyevery spacespace--timetime pointpoint..


WeWe nownow wantwant toto generalizegeneralize thisthis symmetrysymmetry intointo aa locallocal gaugegauge symmetrysymmetry byby demandingdemanding thatthat thethe LagrangianLagrangian shouldshould bebe invariantinvariant underunder thisthis symmetrysymmetry eveneven ifif αα(x)(x) isis anan arbitraryarbitrary differentiabledifferentiable functionfunction ofof xx..

WeWe nownow wantwant toto generalizegeneralize thisthis symmetrysymmetry intointo aa locallocal gaugegauge symmetrysymmetry byby demandingdemanding thatthat thethe LagrangianLagrangian shouldshould bebe invariantinvariant underunder thisthis symmetrysymmetry eveneven ifif αα(x)(x) isis anan arbitraryarbitrary differentiabledifferentiable functionfunction ofof xx..

ThisThis isis aa strongstrong demanddemand.. ItIt sayssays thatthat wewe cancan changechange thethe phasephase ofof aa DiracDirac fieldfield atat anyany spacespace--timetime pointpoint inin anan arbitraryarbitrary wayway andand thethe Lagrangian,Lagrangian, ii..ee..,, thethe dynamicsdynamics ofof thethe DiracDirac fieldfield shouldshould remainremain thethe samesame..

ThisThis isis aa strongstrong demanddemand.. ItIt sayssays thatthat wewe cancan changechange thethe phasephase ofof aa DiracDirac fieldfield atat anyany spacespace--timetime pointpoint inin anan arbitraryarbitrary wayway andand thethe Lagrangian,Lagrangian, ii..ee..,, thethe dynamicsdynamics ofof thethe DiracDirac fieldfield shouldshould remainremain thethe samesame..

QED,QED, QCDQCD andand thethe StandardStandard ModelModel areare constructedconstructed thisthis wayway.. QED,QED, QCDQCD andand thethe StandardStandard ModelModel areare constructedconstructed thisthis wayway..

Quantum ElectrodynamicsQuantum Electrodynamics TheThe massmass termterm isis stillstill invariantinvariant.. TheThe massmass termterm isis stillstill invariantinvariant..

HoweverHowever thethe kinetickinetic energyenergy termterm isis nono longerlonger invariantinvariant.. HoweverHowever thethe kinetickinetic energyenergy termterm isis nono longerlonger invariantinvariant..

InIn orderorder toto fulfillfulfill ourour demand,demand, wewe introduceintroduce aa gaugegauge fieldfield andand definedefine thethe soso--calledcalled gaugegauge covariantcovariant derivativederivative.. InIn orderorder toto fulfillfulfill ourour demand,demand, wewe introduceintroduce aa gaugegauge fieldfield andand definedefine thethe soso--calledcalled gaugegauge covariantcovariant derivativederivative..

ItIt containscontains thethe couplingcoupling constantconstant ee ofof thethe fermionfermion fieldfield toto thethe gaugegauge fieldfield.. ItIt containscontains thethe couplingcoupling constantconstant ee ofof thethe fermionfermion fieldfield toto thethe gaugegauge fieldfield..


TheThe startingstarting pointpoint isis:: TheThe startingstarting pointpoint isis::

TheThe locallocal gaugegauge transformationtransformation nownow alsoalso involvesinvolves thethe transformationtransformation propertyproperty ofof thethe newnew gaugegauge fieldfield:: TheThe locallocal gaugegauge transformationtransformation nownow alsoalso involvesinvolves thethe transformationtransformation propertyproperty ofof thethe newnew gaugegauge fieldfield::

TheThe aboveabove LagrangianLagrangian isis invariantinvariant underunder thisthis locallocal U(U(11)) gaugegauge transformationtransformation.. TheThe aboveabove LagrangianLagrangian isis invariantinvariant underunder thisthis locallocal U(U(11)) gaugegauge transformationtransformation..


GaugeGauge fieldfield withwith compensatingcompensating transformationtransformation isis introducedintroduced.. GaugeGauge fieldfield withwith compensatingcompensating transformationtransformation isis introducedintroduced..


TheThe crucialcrucial pointpoint isis thatthat thethe covariantcovariant derivativederivative nownow transformstransforms asas:: TheThe crucialcrucial pointpoint isis thatthat thethe covariantcovariant derivativederivative nownow transformstransforms asas::

ThusThus thethe kinetickinetic energyenergy termterm isis nownow invariant,invariant, ensuringensuring thatthat thethe LagrangianLagrangian asas aa wholewhole isis alsoalso invariantinvariant.. ThusThus thethe kinetickinetic energyenergy termterm isis nownow invariant,invariant, ensuringensuring thatthat thethe LagrangianLagrangian asas aa wholewhole isis alsoalso invariantinvariant..

TheThe AA--fieldfield isis stillstill notnot dynamicaldynamical soso farfar.. TheThe AA--fieldfield isis stillstill notnot dynamicaldynamical soso farfar..

AA candidatecandidate forfor thethe dynamicsdynamics ofof thethe gaugegauge fieldfield AA isis thethe vectorvector potentialpotential ofof thethe Maxwell’sMaxwell’s equationsequations:: AA candidatecandidate forfor thethe dynamicsdynamics ofof thethe gaugegauge fieldfield AA isis thethe vectorvector potentialpotential ofof thethe Maxwell’sMaxwell’s equationsequations::

FFμνμν isis invariantinvariant underunder thethe gaugegauge transformationstransformations:: FFμνμν isis invariantinvariant underunder thethe gaugegauge transformationstransformations::


SU(N) Gauge TheorySU(N) Gauge Theory

AssumeAssume thethe ψψ--fieldfield toto bebe anan nn--tupletuple:: AssumeAssume thethe ψψ--fieldfield toto bebe anan nn--tupletuple::

TheThe freefree fieldfield isis describeddescribed byby thethe DiracDirac LagrangianLagrangian:: TheThe freefree fieldfield isis describeddescribed byby thethe DiracDirac LagrangianLagrangian::

AssumeAssume mm toto bebe aa diagonaldiagonal nn xx nn matrixmatrix withwith identicalidentical massesmasses asas itsits entriesentries.. DiracDirac matricesmatrices areare alsoalso embeddedembedded inin anan nn xx nn unitunit matrixmatrix inin aa diagonaldiagonal mannermanner..

AssumeAssume mm toto bebe aa diagonaldiagonal nn xx nn matrixmatrix withwith identicalidentical massesmasses asas itsits entriesentries.. DiracDirac matricesmatrices areare alsoalso embeddedembedded inin anan nn xx nn unitunit matrixmatrix inin aa diagonaldiagonal mannermanner..

WeWe considerconsider transformationtransformation propertiesproperties ofof thethe LagrangianLagrangian underunder thethe SU(N)SU(N) transformationstransformations:: WeWe considerconsider transformationtransformation propertiesproperties ofof thethe LagrangianLagrangian underunder thethe SU(N)SU(N) transformationstransformations::

Global Gauge TransformationsGlobal Gauge Transformations

ForFor fermionfermion fields,fields, wewe cancan writewrite:: ForFor fermionfermion fields,fields, wewe cancan writewrite::

HereHere θθaa areare thethe realreal constants,constants, independentindependent ofof xx.. ThatThat isis whywhy thesethese areare calledcalled thethe globalglobal SU(N)SU(N) transformationstransformations.. HereHere θθaa areare thethe realreal constants,constants, independentindependent ofof xx.. ThatThat isis whywhy thesethese areare calledcalled thethe globalglobal SU(N)SU(N) transformationstransformations..


AsAs ττaa dodo notnot commutecommute withwith eacheach other,other, SU(N)SU(N) LieLie groupgroup isis nonnon--abelianabelian.. ThisThis meansmeans thatthat 22 successivesuccessive transformationstransformations givegive aa differentdifferent resultresult whenwhen oneone interchangesinterchanges thethe orderorder..

AsAs ττaa dodo notnot commutecommute withwith eacheach other,other, SU(N)SU(N) LieLie groupgroup isis nonnon--abelianabelian.. ThisThis meansmeans thatthat 22 successivesuccessive transformationstransformations givegive aa differentdifferent resultresult whenwhen oneone interchangesinterchanges thethe orderorder..

SU(N)SU(N) DiracDirac LagrangianLagrangian isis invariantinvariant underunder thesethese globalglobal transformationstransformations.. ThereThere areare conservedconserved currentscurrents andand correspondinglycorrespondingly conservedconserved chargescharges::

SU(N)SU(N) DiracDirac LagrangianLagrangian isis invariantinvariant underunder thesethese globalglobal transformationstransformations.. ThereThere areare conservedconserved currentscurrents andand correspondinglycorrespondingly conservedconserved chargescharges::


ThereThere areare differentdifferent colorcolor phasesphases associatedassociated withwith quarksquarks ThereThere areare differentdifferent colorcolor phasesphases associatedassociated withwith quarksquarks

InIn globalglobal gaugegauge transformations,transformations, theythey areare transformedtransformed independentlyindependently ofof xx.. InIn globalglobal gaugegauge transformations,transformations, theythey areare transformedtransformed independentlyindependently ofof xx..

Local Gauge TransformationsLocal Gauge Transformations

WeWe nownow wantwant toto generalizegeneralize thisthis symmetrysymmetry intointo aa locallocal gaugegauge symmetrysymmetry byby demandingdemanding thatthat thethe LagrangianLagrangian shouldshould bebe invariantinvariant underunder thisthis symmetrysymmetry eveneven ifif θθ(x)(x) isis anan arbitraryarbitrary differentiabledifferentiable functionfunction ofof xx..

WeWe nownow wantwant toto generalizegeneralize thisthis symmetrysymmetry intointo aa locallocal gaugegauge symmetrysymmetry byby demandingdemanding thatthat thethe LagrangianLagrangian shouldshould bebe invariantinvariant underunder thisthis symmetrysymmetry eveneven ifif θθ(x)(x) isis anan arbitraryarbitrary differentiabledifferentiable functionfunction ofof xx..

ThisThis isis aa strongstrong demanddemand.. ItIt sayssays thatthat wewe cancan changechange thethe quarkquark phasesphases “a”“a” ofof aa fieldfield atat anyany spacespace--timetime pointpoint inin anan arbitraryarbitrary wayway andand thethe LagrangianLagrangian,, ii..ee..,, thethe dynamicsdynamics ofof thethe quarkquark fieldfield shouldshould remainremain thethe samesame..

ThisThis isis aa strongstrong demanddemand.. ItIt sayssays thatthat wewe cancan changechange thethe quarkquark phasesphases “a”“a” ofof aa fieldfield atat anyany spacespace--timetime pointpoint inin anan arbitraryarbitrary wayway andand thethe LagrangianLagrangian,, ii..ee..,, thethe dynamicsdynamics ofof thethe quarkquark fieldfield shouldshould remainremain thethe samesame..

QCDQCD andand thethe StandardStandard ModelModel areare constructedconstructed thisthis wayway.. QCDQCD andand thethe StandardStandard ModelModel areare constructedconstructed thisthis wayway..

Local Gauge TransformationsLocal Gauge Transformations InIn orderorder toto fulfillfulfill ourour demanddemand,, introduceintroduce 88 gaugegauge fieldsfields andand definedefine thethe covariantcovariant derivativederivative.. InIn orderorder toto fulfillfulfill ourour demanddemand,, introduceintroduce 88 gaugegauge fieldsfields andand definedefine thethe covariantcovariant derivativederivative..

NoteNote thatthat againagain itit containscontains thethe couplingcoupling constantconstant gg ofof thethe quarkquark fieldfield toto thethe gaugegauge fieldfield.. NoteNote thatthat againagain itit containscontains thethe couplingcoupling constantconstant gg ofof thethe quarkquark fieldfield toto thethe gaugegauge fieldfield..

WeWe havehave asas manymany gaugegauge fieldsfields asas thethe numbernumber ofof generatorsgenerators.. However,However, wewe havehave onlyonly oneone couplingcoupling.. TheThe samesame couplingcoupling existsexists betweenbetween veryvery quarkquark fieldfield andand everyevery gaugegauge fieldfield..

WeWe havehave asas manymany gaugegauge fieldsfields asas thethe numbernumber ofof generatorsgenerators.. However,However, wewe havehave onlyonly oneone couplingcoupling.. TheThe samesame couplingcoupling existsexists betweenbetween veryvery quarkquark fieldfield andand everyevery gaugegauge fieldfield..

WeWe demanddemand thatthat thethe gaugegauge fieldsfields transformtransform underunder thethe locallocal transformationstransformations suchsuch thatthat:: WeWe demanddemand thatthat thethe gaugegauge fieldsfields transformtransform underunder thethe locallocal transformationstransformations suchsuch thatthat::

Local Gauge TransformationsLocal Gauge Transformations

ThisThis isis achievedachieved ifif::

ThisThis isis achievedachieved ifif::

ThusThus thethe kinetickinetic energyenergy termterm isis nownow invariant,invariant, ensuringensuring thatthat thethe LagrangianLagrangian asas aa wholewhole isis alsoalso invariantinvariant.. ThusThus thethe kinetickinetic energyenergy termterm isis nownow invariant,invariant, ensuringensuring thatthat thethe LagrangianLagrangian asas aa wholewhole isis alsoalso invariantinvariant..

LetLet usus looklook atat itit pictoriallypictorially.. LetLet usus looklook atat itit pictoriallypictorially..

Local Gauge TransformationsLocal Gauge Transformations InitiallyInitially:: InitiallyInitially::

LocalLocal GaugeGauge TransformationTransformation:: LocalLocal GaugeGauge TransformationTransformation::

CompensatingCompensating FieldsFields:: CompensatingCompensating FieldsFields::

WeWe cancan easilyeasily checkcheck thethe followingfollowing inin QEDQED:: WeWe cancan easilyeasily checkcheck thethe followingfollowing inin QEDQED::

Dynamics of Field TensorDynamics of Field Tensor

CombiningCombining thethe twotwo identitiesidentities:: CombiningCombining thethe twotwo identitiesidentities::

AsAs ψψ isis anan arbitraryarbitrary spinorspinor andand exp(exp(--iiαα(x))(x)) && FFμνμν commute,commute, thethe laterlater beingbeing aa function,function, wewe obtainobtain thethe gaugegauge invarianceinvariance ofof thethe fieldfield tensortensor..

AsAs ψψ isis anan arbitraryarbitrary spinorspinor andand exp(exp(--iiαα(x))(x)) && FFμνμν commute,commute, thethe laterlater beingbeing aa function,function, wewe obtainobtain thethe gaugegauge invarianceinvariance ofof thethe fieldfield tensortensor..

HowHow dodo wewe definedefine FFμνμν forfor SU(N)?SU(N)? HowHow dodo wewe definedefine FFμνμν forfor SU(N)?SU(N)?

TheThe followingfollowing definitiondefinition doesdoes notnot workwork:: TheThe followingfollowing definitiondefinition doesdoes notnot workwork::

WeWe cancan checkcheck thatthat withwith thisthis definition,definition, WeWe cancan checkcheck thatthat withwith thisthis definition,definition,

isis notnot gaugegauge invariantinvariant.. isis notnot gaugegauge invariantinvariant..

ThusThus wewe followfollow thethe procedureprocedure justjust outlinedoutlined.. DefineDefine FFμνμν asas followsfollows andand deducededuce whatwhat itit isis:: ThusThus wewe followfollow thethe procedureprocedure justjust outlinedoutlined.. DefineDefine FFμνμν asas followsfollows andand deducededuce whatwhat itit isis::


IsIs FFμνμνaa gaugegauge independent?independent? IsIs FFμνμν

aa gaugegauge independent?independent?

SU(N)SU(N) LagrangianLagrangian:: SU(N)SU(N) LagrangianLagrangian::

ButBut itsits LorentzLorentz invariantinvariant constructionconstruction isis.. ThusThus:: ButBut itsits LorentzLorentz invariantinvariant constructionconstruction isis.. ThusThus::

GaugeGauge fieldfield tensortensor:: GaugeGauge fieldfield tensortensor::

TheseThese termsterms correspondcorrespond toto selfself couplingcoupling ofof thethe gaugegauge fieldfield.. TheseThese termsterms correspondcorrespond toto selfself couplingcoupling ofof thethe gaugegauge fieldfield..


The QCD The QCD LagrangianLagrangian

QCDQCD isis obtainedobtained byby demandingdemanding locallocal SU(SU(33)) gaugegauge invarianceinvariance ofof thethe followingfollowing freefree quarksquarks LagrangianLagrangian:: QCDQCD isis obtainedobtained byby demandingdemanding locallocal SU(SU(33)) gaugegauge invarianceinvariance ofof thethe followingfollowing freefree quarksquarks LagrangianLagrangian::

QuarksQuarks havehave 33 internalinternal degreesdegrees ofof freedom,freedom, calledcalled colorcolor.. ItIt isis thesethese colorcolor degreesdegrees ofof freedomfreedom whichwhich areare gaugedgauged withwith anan SU(SU(33)) colorcolor gaugegauge transformationtransformation..

QuarksQuarks havehave 33 internalinternal degreesdegrees ofof freedom,freedom, calledcalled colorcolor.. ItIt isis thesethese colorcolor degreesdegrees ofof freedomfreedom whichwhich areare gaugedgauged withwith anan SU(SU(33)) colorcolor gaugegauge transformationtransformation..

The QCD The QCD LagrangianLagrangian

ThusThus thethe QCDQCD LagrangianLagrangian cancan bebe writtenwritten asas:: ThusThus thethe QCDQCD LagrangianLagrangian cancan bebe writtenwritten asas::

GluonGluon FieldsFields:: MasslessMassless,, spinspin 11 bosons,bosons, colorcolor octetoctet becausebecause thethe groupgroup containscontains eighteight generators,generators, flavorflavor singletsinglet (i(i..ee..,, gluonsgluons areare flavorflavor blind,blind, notnot distinguishingdistinguishing betweenbetween flavorsflavors ofof whateverwhatever theythey interactinteract withwith..

GluonGluon FieldsFields:: MasslessMassless,, spinspin 11 bosons,bosons, colorcolor octetoctet becausebecause thethe groupgroup containscontains eighteight generators,generators, flavorflavor singletsinglet (i(i..ee..,, gluonsgluons areare flavorflavor blind,blind, notnot distinguishingdistinguishing betweenbetween flavorsflavors ofof whateverwhatever theythey interactinteract withwith..

Feynman Rules in Covariant GaugesFeynman Rules in Covariant Gauges

TheThe PropagatorsPropagators:: TheThe PropagatorsPropagators::

TheThe GluonGluon SelfSelf InteractionsInteractions:: TheThe GluonGluon SelfSelf InteractionsInteractions::


TheThe GluonGluon VerticesVertices withwith GhostsGhosts andand QuarksQuarks:: TheThe GluonGluon VerticesVertices withwith GhostsGhosts andand QuarksQuarks::


Running Coupling in QEDRunning Coupling in QED


LandauLandau PolePole:: LandauLandau PolePole::


ExperimentalExperimental MeasurementMeasurement:: ExperimentalExperimental MeasurementMeasurement::

Running Coupling in QCDRunning Coupling in QCD

QCDQCD:: QCDQCD::


CompetitionCompetition betweenbetween colorcolor andand flavorflavor:: CompetitionCompetition betweenbetween colorcolor andand flavorflavor::

Running Coupling in QCDRunning Coupling in QCD AsymptoticAsymptotic FreedomFreedom:: AsymptoticAsymptotic FreedomFreedom::

What Next?What Next?

•• HowHow cancan wewe describedescribe hadrons,hadrons, startingstarting fromfrom QCDQCD dynamics?dynamics? •• HowHow cancan wewe describedescribe hadrons,hadrons, startingstarting fromfrom QCDQCD dynamics?dynamics?

•• HowHow cancan wewe invokeinvoke symmetrysymmetry (and(and itsits breaking)breaking) argumentsarguments toto generategenerate pionspions?? •• HowHow cancan wewe invokeinvoke symmetrysymmetry (and(and itsits breaking)breaking) argumentsarguments toto generategenerate pionspions??

•• HowHow cancan thethe massmass ofof thethe nucleonnucleon bebe relatedrelated toto chiralchiral symmetrysymmetry breaking?breaking? •• HowHow cancan thethe massmass ofof thethe nucleonnucleon bebe relatedrelated toto chiralchiral symmetrysymmetry breaking?breaking?

•• CanCan wewe derivederive GellGell--MannMann--OkuboOkubo massmass formulaeformulae andand otherother lowlow energyenergy relations?relations? •• CanCan wewe derivederive GellGell--MannMann--OkuboOkubo massmass formulaeformulae andand otherother lowlow energyenergy relations?relations?

•• WhatWhat rolerole doesdoes thethe vacuumvacuum havehave toto playplay inin thethe studystudy ofof chiralchiral dynamicsdynamics throughthrough thethe fundamentalfundamental degreesdegrees ofof freedomfreedom ofof quarksquarks andand gluons?gluons?

•• WhatWhat rolerole doesdoes thethe vacuumvacuum havehave toto playplay inin thethe studystudy ofof chiralchiral dynamicsdynamics throughthrough thethe fundamentalfundamental degreesdegrees ofof freedomfreedom ofof quarksquarks andand gluons?gluons?

lecture 4 qcd as a gauge theory - universidad de...

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