Lecture 5 QCD Symmetries & Their Breaking From Quarks to...

Lecture Lecture –– 55 QCD Symmetries & Their BreakingQCD Symmetries & Their Breaking

From Quarks to HadronsFrom Quarks to Hadrons

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

HermosilloHermosillo SonoraSonora

ContentsContents •• ΛΛQCDQCD and Light Quarks and Light Quarks •• ΛΛQCDQCD and Light Quarks and Light Quarks

•• Symmetries of the QCD Symmetries of the QCD LagrangianLagrangian •• Symmetries of the QCD Symmetries of the QCD LagrangianLagrangian

•• Explicit Explicit ChiralChiral Symmetry Breaking Symmetry Breaking •• Explicit Explicit ChiralChiral Symmetry Breaking Symmetry Breaking

•• GellGell--MannMann--OakesOakes--Renner Formula Renner Formula •• GellGell--MannMann--OakesOakes--Renner Formula Renner Formula

•• GellGell--MannMann--Okubo Mass FormulaOkubo Mass Formula •• GellGell--MannMann--Okubo Mass FormulaOkubo Mass Formula

•• On Quark MassesOn Quark Masses •• On Quark MassesOn Quark Masses

•• ChiralChiral Symmetry and Its Breaking Symmetry and Its Breaking •• ChiralChiral Symmetry and Its Breaking Symmetry and Its Breaking

•• Parity and Handedness Parity and Handedness •• Parity and Handedness Parity and Handedness

•• Parity Doubling Parity Doubling •• Parity Doubling Parity Doubling

•• Charting out The QCharting out The Q22 EvolutionEvolution •• Charting out The QCharting out The Q22 EvolutionEvolution

ΛΛQCDQCD and Light Quarksand Light Quarks

WeWe analyzeanalyze thethe expressionexpression forfor thethe runningrunning QCDQCD couplingcoupling:: WeWe analyzeanalyze thethe expressionexpression forfor thethe runningrunning QCDQCD couplingcoupling::

AsAs BB isis positive,positive, whenwhen Q’Q’22 decreasesdecreases fromfrom Q’Q’22 >>>> QQ22,, ααss(Q’(Q’22)) increasesincreases.. AsAs BB isis positive,positive, whenwhen Q’Q’22 decreasesdecreases fromfrom Q’Q’22 >>>> QQ22,, ααss(Q’(Q’22)) increasesincreases..

WeWe definedefine Q’Q’ ==ΛΛQCDQCD,, soso thatthat ααss(Q’(Q’22)) -->> ∞∞.. ThenThen wewe expectexpect perturbationperturbation theorytheory toto breakbreak downdown muchmuch aboveabove thatthat scalescale.. WeWe definedefine Q’Q’ ==ΛΛQCDQCD,, soso thatthat ααss(Q’(Q’22)) -->> ∞∞.. ThenThen wewe expectexpect perturbationperturbation theorytheory toto breakbreak downdown muchmuch aboveabove thatthat scalescale..

LetLet usus assumeassume thatthat perturbationperturbation theorytheory breaksbreaks downdown atat:: LetLet usus assumeassume thatthat perturbationperturbation theorytheory breaksbreaks downdown atat::


LetLet usus looklook atat thethe quarkquark massmass termsterms.. LightLight quarkquark massesmasses:: LetLet usus looklook atat thethe quarkquark massmass termsterms.. LightLight quarkquark massesmasses::

HeavyHeavy quarkquark massesmasses:: HeavyHeavy quarkquark massesmasses::


LookLook atat thethe QCDQCD LagrangianLagrangian withwith u,u, dd andand ss quarksquarks:: LookLook atat thethe QCDQCD LagrangianLagrangian withwith u,u, dd andand ss quarksquarks::

TheThe massmass termterm isis nownow:: TheThe massmass termterm isis nownow::

Symmetries of the QCD Symmetries of the QCD LagrangianLagrangian

Symmetries of the QCD Symmetries of the QCD LagrangianLagrangian U(U(11)) VectorVector SymmetrySymmetry:: U(U(11)) VectorVector SymmetrySymmetry::

ThisThis impliesimplies conservationconservation ofof individualindividual flavors,flavors, ofof baryonbaryon numbernumber andand electromagneticelectromagnetic chargecharge inin strongstrong interactionsinteractions.. ThisThis impliesimplies conservationconservation ofof individualindividual flavors,flavors, ofof baryonbaryon numbernumber andand electromagneticelectromagnetic chargecharge inin strongstrong interactionsinteractions..

SU(SU(22)) VectorVector SymmetrySymmetry:: SU(SU(22)) VectorVector SymmetrySymmetry::

ItIt isis exactlyexactly conservedconserved forfor equalequal uu andand dd massesmasses.. ItIt impliesimplies thatthat thethe hadronshadrons mademade outout ofof uu andand dd quarksquarks withinwithin aa givengiven multipletmultiplet shouldshould havehave equalequal massesmasses..

ItIt isis exactlyexactly conservedconserved forfor equalequal uu andand dd massesmasses.. ItIt impliesimplies thatthat thethe hadronshadrons mademade outout ofof uu andand dd quarksquarks withinwithin aa givengiven multipletmultiplet shouldshould havehave equalequal massesmasses..

SU(SU(33)) VectorVector SymmetrySymmetry:: SU(SU(33)) VectorVector SymmetrySymmetry::

ItIt isis exactlyexactly conservedconserved forfor equalequal u,u, dd andand ss massesmasses.. ItIt iimpliesmplies thatthat thethe hadronshadrons mademade outout ofof u,u, dd && ss quarksquarks withinwithin aa givengiven multipletmultiplet shouldshould havehave equalequal massesmasses.. TheThe SU(SU(33)) hashas threethree subgroupssubgroups ofof SU(SU(22)) naturenature..

ItIt isis exactlyexactly conservedconserved forfor equalequal u,u, dd andand ss massesmasses.. ItIt iimpliesmplies thatthat thethe hadronshadrons mademade outout ofof u,u, dd && ss quarksquarks withinwithin aa givengiven multipletmultiplet shouldshould havehave equalequal massesmasses.. TheThe SU(SU(33)) hashas threethree subgroupssubgroups ofof SU(SU(22)) naturenature..

AAxialxial symmetriessymmetries forfor masslessmassless quarksquarks cancan alsoalso bebe defineddefined AAxialxial symmetriessymmetries forfor masslessmassless quarksquarks cancan alsoalso bebe defineddefined

MasslessMassless LagrangianLagrangian:: MasslessMassless LagrangianLagrangian::

SU(SU(22))RR XX SU(SU(22))LL:: SU(SU(22))RR XX SU(SU(22))LL::

TheThe LagrangianLagrangian remainsremains invariantinvariant.. ItIt isis chiralchiral symmetrysymmetry involvinginvolving onlyonly upup andand downdown quarksquarks.. SS quarksquarks cancan readilyreadily bebe includedincluded toto studystudy SU(SU(33))RR XX SU(SU(33))LL chiralchiral symmetrysymmetry..

TheThe LagrangianLagrangian remainsremains invariantinvariant.. ItIt isis chiralchiral symmetrysymmetry involvinginvolving onlyonly upup andand downdown quarksquarks.. SS quarksquarks cancan readilyreadily bebe includedincluded toto studystudy SU(SU(33))RR XX SU(SU(33))LL chiralchiral symmetrysymmetry..

GlobalGlobal ChiralChiral SU(SU(22))RR XX SU(SU(22))LL SymmetrySymmetry:: GlobalGlobal ChiralChiral SU(SU(22))RR XX SU(SU(22))LL SymmetrySymmetry::

Symmetries of the QCD Symmetries of the QCD LagrangianLagrangian

WeWe cancan workwork eithereither withwith thethe symmetrysymmetry transformationstransformations SU(SU(22))VV XX SU(SU(22))AA oror SU(SU(22))LL XX SU(SU(22))RR.. WeWe cancan workwork eithereither withwith thethe symmetrysymmetry transformationstransformations SU(SU(22))VV XX SU(SU(22))AA oror SU(SU(22))LL XX SU(SU(22))RR..

WhatWhat isis chiralitychirality?? WhatWhat isis chiralitychirality??

ChiralChiral Symmetry and Its BreakingSymmetry and Its Breaking

LetLet usus considerconsider combinationscombinations ofof quarkquark fields,fields, whichwhich carrycarry thethe quantumquantum numbersnumbers ofof thethe mesonsmesons underunder considerationconsideration:: LetLet usus considerconsider combinationscombinations ofof quarkquark fields,fields, whichwhich carrycarry thethe quantumquantum numbersnumbers ofof thethe mesonsmesons underunder considerationconsideration::

HowHow pionspions,, rhosrhos andand otherother mesonsmesons transformtransform underunder chiralchiral transformations?transformations? HowHow pionspions,, rhosrhos andand otherother mesonsmesons transformtransform underunder chiralchiral transformations?transformations?

TheThe vectorvector signsign indicatesindicates thethe isoiso vectorvector naturenature ofof thethe particleparticle.. TheThe μμ indexindex isis thethe LorentzLorentz indexindex (vector(vector particle)particle).. TheThe vectorvector signsign indicatesindicates thethe isoiso vectorvector naturenature ofof thethe particleparticle.. TheThe μμ indexindex isis thethe LorentzLorentz indexindex (vector(vector particle)particle)..


SeeSee howhow aa pionpion transformstransforms underunder SU(SU(22))VV transformationstransformations.. SeeSee howhow aa pionpion transformstransforms underunder SU(SU(22))VV transformationstransformations..


TheThe rhorho:: TheThe rhorho::


TheThe fermionsfermions:: TheThe fermionsfermions::

TheThe mesonsmesons:: TheThe mesonsmesons::

SeeSee howhow mesonsmesons transformstransforms underunder SU(SU(22))AA transformationstransformations.. SeeSee howhow mesonsmesons transformstransforms underunder SU(SU(22))AA transformationstransformations..

RecallRecall:: RecallRecall::

Parity and HandednessParity and Handedness

ParityParity ofof aa spinorspinor wavewave functionsfunctions isis determineddetermined fromfrom:: ParityParity ofof aa spinorspinor wavewave functionsfunctions isis determineddetermined fromfrom::

DiracDirac spinorsspinors tansformtansform underunder parityparity asas:: DiracDirac spinorsspinors tansformtansform underunder parityparity asas::

Parity and HandednessParity and Handedness LeftLeft andand rightright handedhanded spinorsspinors dodo notnot havehave specificspecific parityparity:: LeftLeft andand rightright handedhanded spinorsspinors dodo notnot havehave specificspecific parityparity::

HpweverHpwever,, followingfollowing combinationscombinations havehave specificspecific parityparity:: HpweverHpwever,, followingfollowing combinationscombinations havehave specificspecific parityparity::

RecallRecall thatthat axialaxial symmetrysymmetry isis aa symmetrysymmetry ofof thethe masslessmassless QCDQCD HamiltonianHamiltonian.. RecallRecall thatthat axialaxial symmetrysymmetry isis aa symmetrysymmetry ofof thethe masslessmassless QCDQCD HamiltonianHamiltonian..

ThisThis shouldshould implyimply thatthat thethe statesstates whichwhich cancan bebe rotatedrotated intointo eacheach otherother byby thisthis symmetrysymmetry operationoperation shouldshould havehave thethe samesame eigenvalueseigenvalues,, i,ei,e..,, thethe samesame massesmasses..

ThisThis shouldshould implyimply thatthat thethe statesstates whichwhich cancan bebe rotatedrotated intointo eacheach otherother byby thisthis symmetrysymmetry operationoperation shouldshould havehave thethe samesame eigenvalueseigenvalues,, i,ei,e..,, thethe samesame massesmasses..

TheThe linearlinear combinationscombinations ofof TheThe linearlinear combinationscombinations ofof

ofof thethe leftleft andand rightright handedhanded chargecharge operatorsoperators commutecommute withwith tthehe masslessmassless QCDQCD HamiltonianHamiltonian.. TheyThey havehave oppositeopposite parityparity.. ofof thethe leftleft andand rightright handedhanded chargecharge operatorsoperators commutecommute withwith tthehe masslessmassless QCDQCD HamiltonianHamiltonian.. TheyThey havehave oppositeopposite parityparity..

ThusThus forfor anyany statestate ofof positivepositive parity,parity, oneone wouldwould expectexpect thethe existenceexistence ofof aa degeneratedegenerate statestate ofof negativenegative parityparity ((parityparity doublingdoubling))..

ThusThus forfor anyany statestate ofof positivepositive parity,parity, oneone wouldwould expectexpect thethe existenceexistence ofof aa degeneratedegenerate statestate ofof negativenegative parityparity ((parityparity doublingdoubling))..

Parity DoublingParity Doubling


DegeneracyDegeneracy ofof parityparity partnerspartners cancan alsoalso bebe shownshown asas followsfollows:: DegeneracyDegeneracy ofof parityparity partnerspartners cancan alsoalso bebe shownshown asas followsfollows::

LetLet usus looklook atat parityparity partnerspartners:: LetLet usus looklook atat parityparity partnerspartners::

GlobalGlobal ChiralChiral SU(SU(22))VV XX SU(SU(22))AA SymmetrySymmetry:: GlobalGlobal ChiralChiral SU(SU(22))VV XX SU(SU(22))AA SymmetrySymmetry::

Explicit Explicit ChiralChiral Symmetry BreakingSymmetry Breaking

VectorVector andand axialaxial vectorvector currentscurrents:: VectorVector andand axialaxial vectorvector currentscurrents::

ExplicitExplicit transformationstransformations && currentscurrents areare::

ExplicitExplicit transformationstransformations && currentscurrents areare::

MassMass termterm breaksbreaks ChiralChiral oror axialaxial symmetrysymmetry:: MassMass termterm breaksbreaks ChiralChiral oror axialaxial symmetrysymmetry::

AsAs longlong asas massesmasses areare smallsmall asas comparedcompared toto aa relevantrelevant massmass scale,scale, thethe symmetrysymmetry isis almostalmost (partially)(partially) conservedconserved.. uu andand dd massesmasses areare 55--1010 MeVMeV whichwhich isis muchmuch smallersmaller thanthan ΛΛQCDQCD..

AsAs longlong asas massesmasses areare smallsmall asas comparedcompared toto aa relevantrelevant massmass scale,scale, thethe symmetrysymmetry isis almostalmost (partially)(partially) conservedconserved.. uu andand dd massesmasses areare 55--1010 MeVMeV whichwhich isis muchmuch smallersmaller thanthan ΛΛQCDQCD..

Explicit Explicit ChiralChiral Symmetry BreakingSymmetry Breaking

LetLet usus taketake mmuu≠m≠mdd,, ii..ee,, letlet usus breakbreak thethe isospinisospin symmetrysymmetry LetLet usus taketake mmuu≠m≠mdd,, ii..ee,, letlet usus breakbreak thethe isospinisospin symmetrysymmetry

LetLet usus definedefine:: LetLet usus definedefine::

ThenThen:: ThenThen::

GellGell--MannMann--OakesOakes--Renner FormulaRenner Formula

GellGell--MannMann--OakesOakes--Renner FormulaRenner Formula WeWe wouldwould likelike toto relaterelate observableobservable quantitiesquantities likelike pionpion decaydecay constantconstant andand pionpion massmass toto QCDQCD quantitiesquantities likelike thethe currentcurrent massesmasses ofof thethe quarksquarks andand thethe condensatescondensates..

WeWe wouldwould likelike toto relaterelate observableobservable quantitiesquantities likelike pionpion decaydecay constantconstant andand pionpion massmass toto QCDQCD quantitiesquantities likelike thethe currentcurrent massesmasses ofof thethe quarksquarks andand thethe condensatescondensates..

WeWe startstart fromfrom thethe generalgeneral expressionexpression forfor SU(SU(33)):: WeWe startstart fromfrom thethe generalgeneral expressionexpression forfor SU(SU(33))::

LetLet usus looklook atat a=a=11:: LetLet usus looklook atat a=a=11::


NoteNote thatthat:: NoteNote thatthat::

ThusThus:: ThusThus::

CompareCompare thisthis withwith currentcurrent algebraalgebra formulaformula whichwhich isis derivedderived usingusing antianti--commutationcommutation relationsrelations betweenbetween thethe quarkquark fieldsfields:: CompareCompare thisthis withwith currentcurrent algebraalgebra formulaformula whichwhich isis derivedderived usingusing antianti--commutationcommutation relationsrelations betweenbetween thethe quarkquark fieldsfields::

WeWe thusthus havehave:: WeWe thusthus havehave::


ToTo calculatecalculate thethe leftleft handhand side,side, wewe insertinsert:: ToTo calculatecalculate thethe leftleft handhand side,side, wewe insertinsert::

NoteNote thatthat forfor thethe casecase ofof spontaneousspontaneous symmetrysymmetry breaking,breaking, completecomplete setset ofof statesstates isis exhaustedexhausted byby thethe GoldstoneGoldstone BosonsBosons alonealone (nothing(nothing elseelse contributes)contributes).. InIn casecase ofof SU(SU(33),), thesethese areare pionspions,, kaonskaons andand etaeta..

NoteNote thatthat forfor thethe casecase ofof spontaneousspontaneous symmetrysymmetry breaking,breaking, completecomplete setset ofof statesstates isis exhaustedexhausted byby thethe GoldstoneGoldstone BosonsBosons alonealone (nothing(nothing elseelse contributes)contributes).. InIn casecase ofof SU(SU(33),), thesethese areare pionspions,, kaonskaons andand etaeta..

NormalizationNormalization confirmedconfirmed:: NormalizationNormalization confirmedconfirmed::


WeWe cancan nownow evaluateevaluate thethe LHSLHS atat x=x=00:: WeWe cancan nownow evaluateevaluate thethe LHSLHS atat x=x=00::

RecallRecall thatthat:: RecallRecall thatthat::



SimilarlySimilarly:: SimilarlySimilarly::



ThusThus wewe finallyfinally arrivearrive atat:: ThusThus wewe finallyfinally arrivearrive atat::

ItIt helpshelps estimateestimate thethe quarkquark--antianti--quarkquark condensatecondensate:: ItIt helpshelps estimateestimate thethe quarkquark--antianti--quarkquark condensatecondensate::

GellGell--MannMann--Okubo Mass FormulaOkubo Mass Formula

InIn generalgeneral:: InIn generalgeneral::

EmployingEmploying:: EmployingEmploying::

AndAnd usingusing currentcurrent algebraalgebra:: AndAnd usingusing currentcurrent algebraalgebra::

GellGell--MannMann--Okubo Mass FormulaOkubo Mass Formula

LetLet usus assumeassume thethe relationsrelations whichwhich areare strictlystrictly validvalid onlyonly inin thethe equalequal massmass limitlimit:: LetLet usus assumeassume thethe relationsrelations whichwhich areare strictlystrictly validvalid onlyonly inin thethe equalequal massmass limitlimit::

ItIt impliesimplies GellGell--MannMann--OkuboOkubo massmass formulaformula:: ItIt impliesimplies GellGell--MannMann--OkuboOkubo massmass formulaformula::

UseUse phenomenologicalphenomenological valuesvalues:: UseUse phenomenologicalphenomenological valuesvalues::

GoodGood predictionprediction:: GoodGood predictionprediction::

On Quark MassesOn Quark Masses

WeWe cancan alsoalso obtainobtain quarkquark massmass ratiosratios:: WeWe cancan alsoalso obtainobtain quarkquark massmass ratiosratios::

ThisThis impliesimplies thatthat thethe strangestrange quarkquark massmass isis muchmuch largerlarger thanthan thethe upup andand downdown quarkquark massesmasses.. ThisThis impliesimplies thatthat thethe strangestrange quarkquark massmass isis muchmuch largerlarger thanthan thethe upup andand downdown quarkquark massesmasses..

ThisThis relationrelation isis approximatelyapproximately satisfiedsatisfied ifif:: ThisThis relationrelation isis approximatelyapproximately satisfiedsatisfied ifif::

What Next?What Next?

•• TheThe staticstatic propertiesproperties ofof lowlow lyinglying hadronshadrons cancan bebe wellwell describeddescribed byby symmetrysymmetry principles,principles, theirtheir breakingbreaking andand thethe lowlow energyenergy theoremstheorems..

•• TheThe staticstatic propertiesproperties ofof lowlow lyinglying hadronshadrons cancan bebe wellwell describeddescribed byby symmetrysymmetry principles,principles, theirtheir breakingbreaking andand thethe lowlow energyenergy theoremstheorems..

•• LowLow energyenergy effectiveeffective QCDQCD Models,Models, SDESSDES andand latticelattice describedescribe thethe propertiesproperties ofof lightlight hadronshadrons wellwell.. •• LowLow energyenergy effectiveeffective QCDQCD Models,Models, SDESSDES andand latticelattice describedescribe thethe propertiesproperties ofof lightlight hadronshadrons wellwell..

•• FormForm factorsfactors ofof hadronshadrons provideprovide aa modernmodern testingtesting groundground toto studystudy QCDQCD fromfrom firstfirst principlesprinciples andand seesee ifif itit cancan enableenable usus toto studystudy thethe transitiontransition regionregion wherewhere hadronshadrons gogo fromfrom aa nonnon perturbativeperturbative descriptiondescription toto themthem beingbeing mademade ofof thethe valencevalence quarksquarks alonealone..

•• FormForm factorsfactors ofof hadronshadrons provideprovide aa modernmodern testingtesting groundground toto studystudy QCDQCD fromfrom firstfirst principlesprinciples andand seesee ifif itit cancan enableenable usus toto studystudy thethe transitiontransition regionregion wherewhere hadronshadrons gogo fromfrom aa nonnon perturbativeperturbative descriptiondescription toto themthem beingbeing mademade ofof thethe valencevalence quarksquarks alonealone..

•• ThusThus isis anan activeactive fieldfield ofof currentcurrent experimentalexperimental andand theoreticaltheoretical researchresearch.. •• ThusThus isis anan activeactive fieldfield ofof currentcurrent experimentalexperimental andand theoreticaltheoretical researchresearch..

ObservingObserving thethe transitiontransition ofof thethe hadronhadron fromfrom aa seasea ofof quarksquarks andand gluonsgluons toto thethe oneone withwith valencevalence quarksquarks alonealone isis anan experimentalexperimental andand theoreticaltheoretical challengechallenge..

ObservingObserving thethe transitiontransition ofof thethe hadronhadron fromfrom aa seasea ofof quarksquarks andand gluonsgluons toto thethe oneone withwith valencevalence quarksquarks alonealone isis anan experimentalexperimental andand theoreticaltheoretical challengechallenge..

Charting out the QCharting out the Q22 EvolutionEvolution Charting out the QCharting out the Q22 EvolutionEvolution

TheThe transitiontransition formform factorfactor:: TheThe transitiontransition formform factorfactor::

CELLOCELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2

CLEOCLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV2

BaBarBaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2

BelleBelle S. Uehara et. al., arXiv:1205.3249 [hep-ex] (2012). 4.0 – 40.0 GeV2

H.L.L. H.L.L. RobertesRobertes, C.D. Roberts, AB, L.X. , C.D. Roberts, AB, L.X. GutiérrezGutiérrez and P.C. Tandy, and P.C. Tandy, PhysPhys. Rev. C82, . Rev. C82, (065202:1(065202:1--11) 2010.11) 2010.



CELLOCELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2

CLEOCLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV2

BaBarBaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2

BelleBelle S. Uehara et. al., arXiv:1205.3249 [hep-ex] (2012). 4.0 – 40.0 GeV2



TheThe leadingleading twisttwist pQDCpQDC calculationcalculation:: TheThe leadingleading twisttwist pQDCpQDC calculationcalculation::

G.P. G.P. LepageLepage, and S.J. Brodsky,, and S.J. Brodsky, PhysPhys. Rev. D22, 2157 (1980).. Rev. D22, 2157 (1980).




•• BelleBelle IIII willwill havehave 4040 timestimes moremore luminosityluminosity.. •• BelleBelle IIII willwill havehave 4040 timestimes moremore luminosityluminosity..

PrecisePrecise measurementsmeasurements atat largelarge QQ22 willwill provideprovide aa stringentstringent constraintconstraint onon thethe patternpattern ofof chiralchiral symmetrysymmetry breakingbreaking.. PrecisePrecise measurementsmeasurements atat largelarge QQ22 willwill provideprovide aa stringentstringent constraintconstraint onon thethe patternpattern ofof chiralchiral symmetrysymmetry breakingbreaking..

Vladimir Vladimir SavinovSavinov:: 55thth Workshop of the APSWorkshop of the APS Topical Group on Topical Group on HadronicHadronic PhysicsPhysics

Vladimir Vladimir SavinovSavinov:: 55thth Workshop of the APSWorkshop of the APS Topical Group on Topical Group on HadronicHadronic PhysicsPhysics

Lecture 5 QCD Symmetries & Their Breaking From Quarks to...

Documents

Transcript of Lecture 5 QCD Symmetries & Their Breaking From Quarks to...