Chapter 7 - folk.uio.nofolk.uio.no/farido/fys3510/HadronInteractionsQuarkModel7.pdf · ¡A hadron is made of quarks and has dimensions ~1 fm § Mesons – hadrons with integer spin

Chapter7

¡  7.1HadronsandQuarks§  7.1.1TheYukawaModel

¡  7.2Proton-NeutronSymmetryandtheIsotopicSpin¡  7.3TheStrongInteractionCross-Section

§  7.3.1MeanFreePath

¡  7.4LowEnergyHadron-HadronCollisions§  7.4.1Antibaryons 7.4.2HadronResonances

¡  7.5Breit–WignerEquationforResonances§  7.5.1The_CC.1232/Resonance 7.5.2ResonanceFormationandProduction

§  7.5.3AngularDistributionofResonanceDecayProducts

¡  7.6ProductionandDecayofStrangeParticles¡  7.7ClassificationofHadronsMadeofu;d;sQuarks¡  7.8TheJP=3/2CBaryonicDecuplet

§  7.8.1FirstIndicationsfortheColorQuantumNumber

¡  7.9TheJP=1/2CBaryonicOctet ¡  7.10PseudoscalarMesons 7.11TheVectorMesons¡  7.12StrangenessandIsospinConservation 7.13TheSixQuarks¡  7.14ExperimentalTestsontheStaticQuarkModel

§  7.14.1LeptonicDecaysofNeutralVectorMesons 7.14.2LeptonPairProduction§  7.14.3Hadron-HadronCross-SectionsatHighEnergies7.14.4BaryonMagneticMoments 7.14.5RelationsBetweenMasses

¡  7.15SearchesforFreeQuarksandLimitsoftheModel2

¡  Ahadronismadeofquarksandhasdimensions~1fm§  Mesons–hadronswithintegerspin§  Baryons–semi-integerspin§  Hyperonsare“strange”baryons,i.e.,composedofatleastonesquark.

¡  Hadron“spectroscopy”§  canbeexplainedusingthesimplestaticquarkmodelofhadrons

¡  Constituentquarks–valencequarks–§  explaintheregularitiesofthehadronspectra§  Quarkswerefirstmathematicalfictionasnofreequarkshaveeverbeen

observed.¡  Evidenceforquarks

§  lepton-hadroncollisionswithhightransferredmomentum¡  Deepinelasticscatteringexperiments

§  hadronsalsocontaingluonsandvirtualqqbarpairs(seaquarks),rapidlycreatedandannihilated

07/03/16 F. Ould-Saada 3

¡  Klein-Gordonequation§  E,p,mrelation§  Correspondenceprinciple§  φinterpretedaspotentialUincoordinate

space§  staticpotentialU(r)


−!2 ∂2ϕ("r, t)∂t2

= −!2c2∇2ϕ(!r, t)+m2c4ϕ(!r, t)

∇2ϕ =m2c2

!2ϕ ⇒ ϕ = K

re−ra

E 2 =m2c4+!p2c2

E→ih ∂∂t

; "p→− ih"∇

%

&'

('⇒

1c2

∂2

∂t2 −"∇2*

+,

-

./φ+

m2c2

!2 φ = 0

¡  1stattempttoexplaininteractionbetweennucleonsinnuclei§  quantummechanicalmodel–Yukawa1930s.▪  InanalogywithQED–exchangeofmasslessγ,potentialV~1/r▪  Strongforces–maximumrange~1fm,exchangebosonmustbeheavy~150MeV▪  SochangeQEDpotentialsuchthatititquicklyvanisheswithdistanceduetoexchangeof

massiveparticles

¡  Huntingafterameson(me<mπ<mp)opened§  3chargestatestoaccommodatepp,pn,nninteractions


¡  Wasbelievedthatthemuonfoundincosmicrayswasthepion§  MuoncannotbeYukawa’smeson…doesnotinteractstrongly!§  Pionsdiscoveredlater…

φ(!r, t)→U(r)⇒∇2U =m2c2

"2U =

1r2

∂∂r

r2 ∂U∂r

%

&'

(

)* U(r) = g

4π1re−r/R ; R = !

mc

R ≈ 1.2fm ⇒ mc2 =!cR≈150MeV ≈ mπ

¡  Heisenberg,1932§  Neutron&protonastwostatesofasingle

particle,thenucleonN,withrespecttostronginteractions

§  Analogywithspin,isospinofN:I=1/2▪  pandnareeigenstatesofI3

§  StronginteractiondependsonI,notonI3▪  I3behavesas“Q”

§  IsopinnotconservedindecaysinducedbyweakandEMinteractions▪  EMconservesI3


Nucleon≡ N ≡ pn( ) ; I = 1

2p ≡ 1

2,+1

2≡ 1

0( ) ; n ≡ 12

,−12≡ 0

1( )Proton and neutron are eigenstates of I3

I3 p = +12 p ; I3 n = −

12 n

Ii =σ i2 ;!I =

I1I2I3

⎛

⎝⎜⎞

⎠⎟∈ Isospinspace

¡  Isospinhassamemathematicalbehaviourasspin§  Spinisaphysicalquantity▪  measuredinunitsofh▪  withdimensionofEnergy.Time

§  Isospinisadimensionlessquantity▪  helpsclassifyinghadronsintomultiplets


Ii =σ i2 ;!I =

I1I2I3

!

"#$

%&∈ Isospinspace

σ1= 0 11 0( ) ; σ 2 = 0 −i

i 0( ) ; σ 3 = 1 00 −1( ) ; 1= 1 0

0 1( )

Ladder operators: transformations p ↔ n

I+ ≡ I1 + iI2 = 0 10 0( ) ; I− ≡ I1 − iI2 = 0 0

1 0( )I1

2 + I22 + I3

2 = I 2 → eigenvalues I(I +1)I = 0,1 / 2,1,3 / 2,...−I ≤ I3 ≤ I→ multiplet (2I +1)

¡  ConservationofI§  ConnectedtoobservationthatnuclearstateswithsameN(nucleons)

butdifferentN(protons)havesameenergy,spinandparity.

¡  InvarianceoftheHamiltonianofnuclearinteractions§  =nuclearinteractionindependencefromtheelectriccharge

¡  Groundstatesof7Beand7LihavesameE,s,P§  7Be(I3=+1/2):pppair;7Li(I3=-1/2):nnpair▪  ExcludingEMeffects,impliestheequalityofforcesbetweenthennandpp

pairs


¡  NucleiwithA=14haveJP=0+andnearlysameenergy:§  14C(I3=-1)):nnpair§  14N(I3=0)):nppair§  14O(I3=+1)):pppair§ 

¡  ReactionprohibitedinstronginteractionsandallowedforEMinteraction

▪  ProcessmustproceedwithcrosssectionstypicalofEM▪  Confirmedthatcrosssectionis100timessmallerthantypicalstrongcross

sections


d + d→4 He+π 0

I 0 0 0 1 ← I not conservedI3 0 0 0 0 ← I3 conserved

¡  Mirrornuclei:samevalueofAbutvaluesofNandZinterchanged

¡  Chargesymmetryapproximatelyverifiedà¡  Ifdifferentnp-pairs,effectduetofactthat

nppairnotsubjecttoPauliprinciple

§  testofchargeindependence¡  Butevidencethatstrongforcestrongerfor

S=1thanS=0state07/03/16 F. Ould-Saada 10

511B A =11,Z = 5,N = 6( )10pp,15nn pairs

611C A =11,Z = 6,N = 5( )15pp,10nn pairs

!

"

##

$

##

same np− pairs

Lowlyingenergylevelsofmirrornuclei

BeCB 114

116

115 ,,

¡  ConservationofIsospininSI§  SIdependsonlyonIandisindependentonI3andQ§  TotalIisconservedinSI§  àselectionrulesandpreciserelationbetweenproductioncrosssectionsofrelatedprocesses

¡  2-dimensionalspacerepresentation§  Nucleon:(p,n)orquarks(u,d):I=1/2§  Generatorsare2x2Paulimatrices

¡  3-dimensionalspacerepresentation§  Basis:(π+, π0, π-)withisospinI=1andI3=+1,0,-1§  Generatorsare3x3matrices


Q = I3 +B2

σ pp→ dπ +( )σ pn→ dπ 0( )

= 2 p + n→ d +π 0

I 1 / 2 1 / 20,1

!"# $# 0 11!

p + p→ d +π +

I 1 / 2 1 / 21

!"# $# 0 11!

¡  4-dimensionalspacerepresentation§  Δ(1232):I=3/2àI3=3/2,1/2,-1/2,-3/2§  Basis(Δ++,Δ-,Δ0,Δ-)

¡  Chargemultiplets:§  Particles~samemass,samequantumnumbers(J,C,P,B,S,C,B’,...)but

differentQ§  Ex:(π+, π-, π0),(p,n)àm(d)-m(u)=(3±1)MeV/c2

§  àp(938)=uud;n(940)=udd

¡  Flavourindependenceofcolourforcebetween2quarksatsamedistance–u,d,s,c,b,t§  us=ds;u-ubar=d-dbar§  Strongisospindoublet:I=1/2:(u,d)=(+1/2,-1/2)§  Strongisospinsinglets:I=0:s,c,b,tàm>>mu,d


¡  πNinteractionsèΔ&N*resonances

§  Next3slides:remindercouplingof2or3(iso)spins


1(π )× 12(N ) : 3⊗ 2 = 4⊕ 2

π ≡ (π +,π 0,π − ) ; N ≡ (p,n)I = 3 / 2 :Δ ≡ (Δ++,Δ+,Δ0,Δ− )I =1/ 2 :N *


1

2×1

2: 2⊗ 2 = 3⊕ 1

1× 12: 3⊗ 2 = 4⊕ 2

1

2×1

2×1

2: 2⊗ 2⊗ 2 = 4S ⊕ 2MS⊕2MA

15

πN States → πN; I, I3Δ++ ≡ 3

2 , +32 = π +p

Δ+ ≡ 32 , +

12 = +

13π +n +

23π 0p

Δ0 ≡ 32 , −

12 = 1

3π −p +

23π 0n

Δ− ≡ 32 , −

32 = π −n

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

N*+ ≡ 12 , +

12 = +

23π +n −

13π 0p

N*0 ≡ 12 , −

12 = 2

3π −p −

13π 0n

1(π )× 12(N ) : 3⊗ 2 = 4⊕ 2

π ≡ (π +,π 0,π − ) ; N ≡ (p,n)

Γ Δ+ → π +n( )Γ Δ+ → π 0p( )

=12


⇒M π −p→ π −p( ) = 13M3/2 +

23M1/2

M π −p→ π 0n( ) = 23 M3/2 −

23 M1/2

$%&

'&

I(Δ) = 3 / 2⇒M3/2 >>M1/2 @ Δ(1232)

Δ0 ≡ 32 , −

12 = 1

3π −p +

23π 0n

N*0 ≡ 12 , −

12 = 2

3π −p −

13π 0n

⎫⎬⎪

⎭⎪⇒

π −p =1332 , −

12 − 2

312 , −

12

π 0n =2332 , −

12 + 1

312 , −

12

σ total π−p( ) =σ π −p→ π −p( )+σ π −p→ π 0n( )∝ 1

3 M3/22

Δ++ ≡ 32 , +

32 = π +p

σ total π+p( )∝ M3/2

2

&

'((

)((

σ total π+p( )

σ total π−p( )

= 3


( )( ) 3=−

+

pp

total

total

πσπσ

¡  Crosssectionmeasurement§  Informationontheinteractionpotential§  Caseofshort-rangepotential,negligibleforr>Ro(Yukawa)▪  ExpectpuregeometricCS:σ=πR0

2validforprojectilesizeλ<<targetsize

▪  P>>1GeVàpointlikeprojectilewrtnucleardimensions

¡  TotalCS§  ElasticCS▪  Projectile&targetunchanged&sameenergyincomsystem

§  InelasticCSdominantatHE▪  àparticleexcitationsandnewparticleproduction

18

σ tot =σ el +σ inel

λ =2π !cpc

= 1.24p GeV[ ]

fm[ ]

plab >>1GeV ⇒σ pp ≅ 40mb

σπ p ≅ 25mb; σ = πR0

2 ⇒⎧⎨⎪

⎩⎪

R0 ≅1.1 fmR0 ≅ 0.9 fm

⎧⎨⎪

⎩⎪¡  Nuclearforceofshortrange

§  seeπpandppcrosssections

¡  σ=f(plab)¡  Nuclearforcesareshort-

range

¡  π-p,π-d§  Athigh(plab),σ variesonly

slowlywithenergy

§  Atlowp,resonanceformations


plab >>1GeVσπ p ≅ 25mbR0 ≅ 0.9 fm

20

¡  σ=f(plab)¡  Nuclearforcesareshort-

range¡  p-p,pbar-p

§  Athigh(plab),σ variesonlyslowlywithenergy

plab >>1GeVσ pp ≅ 40mbR0 ≅1.1 fm

¡  Beforecontinuingwithlowenergyhadron-hadroncollisions,let’sintroducestrangehadrons

¡  Hadrons§  Baryonsandanti-baryons:half-integralspin§  Mesonsandanti-mesons:integralspin

¡  Strangeparticles§  Firstdiscoveredin1947incloudchamberphotographsofcosmicrays.§  K-mesonsorKaonsfirstcalledV-particles§  Hyperonsinvolveprotonsintheirdecays▪  ProducedinSIs(t~10-23s)butdecaythroughWIs(t~10-10s)

07/03/16 F. Ould-Saada

dudduuduuddnuudp

qqMqqqBqqqB

===

==

≡

≡≡

−+ πππ ;,;;

;

0

21

K 0 → π + +π −

π − + p→Λ0 + K 0

Λ0→π −+p

¡  Associatedproduction§  Pais,1952:strangeparticlesproducedinpairs§  Newquantumnumber,strangenessS,wasintroducedin1953byGell-

MannandNishijima.

§  Sisconservedinstrong&EMinteractionsbutnotinweakinteractions.


1001:

0

1001:

0

01100:

00

=Δ→+

−+

=Δ→−

−

=Δ→+−

−

+→

+→Λ

+Λ→+

SS

SS

SS

K

p

Kp

ππ

π

π€

S ≡ −Ns = − N(s) − N(s )( )

mΛ =1115.7MeVm

K 0= 497.7MeV

K − + p→Ω− +K + + K 0

S: −1 0 −3 +1 +1 → ΔS=0

¡  Kaons§  K+andK-(m~494MeV)andK0(m~498MeV)§  Spin0andoddparity,closetopionbut

Isospin½§  K0:Particle(S=1)distinctfromantiparticle

(S=-1)▪  Leadstoimportanteffects

¡  Hyperons§  Spin½likeproton:Λ0(M~1115MeV,I=0)§  Σ+,-,0(M~1190MeV,I=1):Ξ0,-,(M~1190MeV,

I=½,S=-2)§  moremassivehyperonswithspin3/2,5/2,...


⎩⎨⎧

−=

+==−=

⎩⎨⎧

−=

+===

−

+

KIKI

IS

KIKI

IS

21

3

021

321

021

3

21

321

,1

,1

K −p→Λπ 0

π−p→Σ0K 0 ; Σ0→Λγ

π±p→Σ±K+

Σ±→nπ± ; Σ+→pπ 0

K−p→Ξ−π+K 0 ; Ξ−→Λπ−

Ξ0→Λπ 0 ; Λ→pπ−

¡  Featuresofhadron-hadroncollisionsobtainedbyanalyzingmeasurementsofσtotofchargedhadronsonhydrogenanddeuterium,§  (seepreviousfiguresaswell)

¡  Peaks&structures?


E ≤ 3GeVYes :π ±p,K −p,K −nless :K +p, pp, pp

¡  Tobeexplainedlater¡  Let’slookatK-p


(1) K −p→Λπ +π −, Λ→ π −p(2) K −p→Σ+*π −, Σ+*→Λπ +

(2 ')K −p→Σ−*π +, Σ−*→Λπ −

¡  DirectproductionofΛ(1)orthrougharesonance(2,2’)?

¡  Dalitzplot–2Dplot§  K-(p=1.22GeV)p

interaction§  Σ*±resonance

throughreactions(2),2’)at1.385GeV

¡  Effectiveorinvariant

mass:


(1) K −p→Λπ +π −, Λ→ π −p(2) K −p→Σ+*π −, Σ+*→Λπ +

(2 ')K −p→Σ−*π +, Σ−*→Λπ − mΛπ+2 = E

Λπ+2 − p

Λπ+2

= (EΛ +Eπ+)2 − ( !pΛ +

!pπ+)2

= EΛ2 +E

π+2 + 2EΛEπ+

− pΛ2 − p

π+2 − 2 !pΛ

!pπ+

=mΛ2 +m

π+2 + 2EΛEπ+

− 2pΛpπ+ cosθΛπ

Γ~100MeV

⇒ τ =!Γ~10−23s


Breit–Wigner shape Γ=width, ER=mass of resonnance

a+ b→ R→ a '+ b '

σ (E, J ) = 4πλ 2 2J +12sa +1( ) 2sb +1( )

Γ2 / 4ER −E( )2 +Γ2 / 4

¡  Resonanceischaracterizedbydefined§  AngularmomentumJ=l(forspinlessparticles)§  ParityP§  IsospinI§  Mass=ER,totalenergyincenterofmassat

resonancemaximum.§  Lifetime(τ),asdeterminedbywidthathalf-

maximum(Γ)ofcurve.

¡  ShapeofBreit-Wignercurve§  Fromformalismofamplitudesandphasesof

matterwaves§  E-dependenceofamplitude=Fourier

transformofawavefunctiondescribingasurvivalprobabilitydecreasingexponentiallyovertime,withlifetimeτ.

§  ReadinbookthederivationoftheBreit-Wignerequation(seenextslideaswell)

¡  UnstableresonanceRdescribedbythefreeparticlewavefunctionmultipliedbyarealfunctiondescribingitsdecayprobabilityasafunctionoftime

¡  Probabilityoffindingtheparticleatatimet

28

ψ(t) =ψ0e−iωRte

−t2τ =ψ0e

−iER!te−Γ2!t

I(t) =ψ02e

−tτ = I0e

−tτ

σ (E) =σ 0χ*(E)χ (E) =σ 0

K 2

(ER −E)2 +

Γ2

4

; σ 0 = π (2λ)2

1= χ *(ER )χ (ER ) = 4K2 / Γ2 ⇒ K 2 = Γ2 / 4

χ (E) = ψ(t)eiEt dt =∫ ψ0 e− i(ER−E )+

Γ2

⎡

⎣⎢⎤

⎦⎥tdt∫ =

K

(ER −E)− iΓ2

¡  Fouriertransform

σ (E, J ) = 4πλ 2 2J +12sa +1( ) 2sb +1( )

Γ2 / 4ER −E( )2 +Γ2 / 4

¡  Addspinmultiplicities

¡  π+ptotalcrosssection§  Largepeakat

Tπlab=191MeV,Ecm=1232MeV,Γ=120MeV

§  Δ++:B=+1,I=3/2,J=3/2

¡  Check§  puttingthese

numbersintotheBWequationàσmax~188mbcompatiblewithfigure

§  J=1/2à94mb…29

¡  (a)Resonanceformationins-channel§  π+pàΔ++àπ+p§  (t-channel:pbarpàπ+π-)


¡  (b)Resonanceproduction§  π+pàΔ++π0àpπ+π0

¡  (c)Resonanceformationintermsofquarkconstitutents§  π+pàΔ++àπ+p

§  π-pcrosssection

§  centreofmassenergy1.2-2.4GeV

§  2(+2)enhancementsontopofnon-resonantcontributions

§  ResonancewidthsΓ~100MeV§  Interactionstimesτ~10-23s§  àStronginteraction

§  àConsistentwithtimetakenforarelativisticpiontotransitthedimensionofaproton


¡  TodeterminespinofΔ++(àpπ+)§  Studyangulardistributionatresonanceofπ+incentreofmass(c.m)§  J=3/2,l=1èI(θ*)=1/(8π)(1+3cos2θ*))§  Others,lconfigurationswouldleadtoverydifferentangulardistributions



¡  GNNrelation¡  ElectricchargeQ,BaryonnumberB,StrangenessS,3rd

componentofisospinI3,HyperchargeY=B+S¡  Withthehandofthisrelation2hyperonspredictedin

1953Σ0, Ξ- discoveredin1958and1959


Q = I3 +B+ S2

= I3 +Y2

−= 0PJ+

= 21PJ

¡  Stronginteraction§  I,Sconserved


¡  EMinteraction§  S,I3conserved,notI

¡  Weakinteraction§  S,Iviolated

¡  Meson=quark-antiquarkcombination

07/03/16 36

2S+1lJ ; JPC 1S0 ; 0–+ 3S1 ; 1–– 1P1 ; 1+– 3P0 ; 0++ 3P1 ; 1++ 3P2 ; 2++ ……..

8133 ⊕=⊗

2 ⊗ 2 =1 ⊕3

9 vectors (JPC = 1– –)

9 pseudoscalars JPC = 0– +

3⊗ 3 = 1 ⊕ 8

F. Ould-Saada

¡  Totalspin:§  S=0 (antisymmetric spin wave function) and S=1 (symmetric)

P=(–1)l+1 C=(–1)l+SàPossible (allowed) states:

¡  GroundstatemesonsL=0,S=0,1

07/03/16 37

Baryons JP=1/2+

€

! S = 1

2 + 12 + 1

2 →S = 12 ,32

L = 0→! J =! S

u,d,s

Notsymmetricenough

àsomethingismissing!

F. Ould-Saada

*

*

*

*

*

HyperonsJP=3/2+

07/03/16 38

The Ω- (S=-3) predicted to have a mass of 1680 MeV, discovered in 1964 at mΩ = 1674 ±3 MeV, τΩ=82 ps

F. Ould-Saada

¡  u,d/charged,neutralhadronmassdifferenceverysmall§  Atmost%level

§  DuetoEMinteraction§  StrongIsospingoodsymmetry

¡  s-quarkmass>>mu,d


¡  u,dquarksà22combinations

¡  u,d,sà32combinations

intotal§  η1andη8mixtogiveη

andη’ mesonsobservedinnature

40

¡  u,dquarksà22combinations§  I=1:ρ+,ρ0,ρ-

§  I=0:ω¡  u,d,sà32combinationsin

total§  K*mesons§  φ, ω mesonsobservedinnature

41

¡  Experimentally§  φmostlytostrangeparticles§  Althoughitisenergeticallypossibleto

decayto3pions,comparedto2kaons§  Why?à“Zweigrule”vshigherorderQCD

¡  Zweigrule§  Quarkflow–notreallyFeynmandiagrams§  c)disconnectedlinesbetweeninitialandfinalstates

àSeveresuppression


¡  Fromthetotalangularmomentum

§  Accountforspin-spacecoupling

§  With


!J =

!L +!S( )

2=!L2 +!S 2 + 2

!L ⋅!S

!L ⋅!S =

J(J +1)− L(L +1)− S(S +1)2

L − S ≤ J ≤ L + S

¡  Therearenoq-qbarmesonswithexoticquantumnumberssuchas

JPC=0+-,1-+,…¡  Ontheotherhandexoticstates(non-q-

qbar)withnormalquantumnumbersarenotforbidden:gluballs,hybrids,pentaquarks…

¡  Itisthedutyofhadronsspectroscopytoconfirmtheexistinghadronsandtofindthemissingones.

¡  SymmetrizedWFs§  Explainingthe10completelysymmetricstates

mΣ −mΔ = 152MeV 1s − 0s

mΞ −mΣ = 149MeV 2s −1s

mΩ −mΞ = 139MeV 3s − 2s

I = 3 / 2; S = 0; Y = +1 Δ−(I3 = −3 / 2),Δ0 (−1 / 2),Δ+(+1 / 2),Δ++(+3 / 2)

I =1; S = −1; Y = 0 Σ*−(I3 = −1),Σ*0 (0),Σ*+(1)

I =1/ 2; S = −2; Y = −1 Ξ*−(I3 = −1 / 2),Ξ*0 (+1 / 2)

I = 0; S = −3; Y = −2 Ω−(I3 = 0)

¡  Baryonoctet§  I=1/2:p,n§  I=1:Σ-,Σ0,Σ+§  I=0:Λ0§  I=1/2:Ξ-,Ξ0

¡  Masses§  mΛ-mN=177MeV§  mΞ-mΛ=203MeV


¡  p=uud§  Spinwf2quarksà

antisymmetric§  Flavourwfu,dàantisymmetric


↑↓−↓↑( )2

ud − du( )2

$

%&&

'&&

⇒ A = u↑d↓ −u↓d↑ − d↑u↓+ d↓u↑

¡  N=ddu§  Replaceuudbyddu!

¡  Exercise:usetheprotonandneutronWFstoreproducethechargeandmagneticmomentofpandn.

!µ =

q!2mc

"σ =

q2m!σ

µ f ≡ f !µ f =q2m

µu = u !µ u =qu2mu

, µd =qd2md

mu ≈ md;qu = −2qd ⇒ µu ≈ −2µd

¡  Diractheory§  Point-likefermionwithelectric

chargeq,massm,spin½hasmagneticdipolemoment

§  Magneticmoments§  Useconstituentquarkmasses


Λ(1116) = udss(ud) = 0

"#$⇒ µΛ = µs

"bare" :mu ≈ md ≈ 5MeV; ms =md +150MeV"constituent": mu ≈ md ≈ mp / 3; ms =md +150MeV

µp = p !µ p =8µu − 2µd

6=32µu

µn =4µd −µu

3= −µu

"

#$$

%$$

⇒

µn

µp

= −23

exp = −0.685

¡  Predictionagreeswithexperimentifconstituentmassesused

¡  Quarkmodelisnotbadatreproducingmagneticmoments…§  Theagreementis

howevernotperfect


µN =e!2mpc

⇒ µu = 2µN ; µd = −1µN ; µs = −0.67µN

µp,n,Λ ⇒ µu = +1.852µN ; µd = −1.972µN ; µu = −0.613µN

¡  Hadronmasses§  1/2+Baryonoctet§  3/2+Baryondecuplet§  àms-mu,d~120-200MeV/c2


MΞ −MΣ =MΞ −MΛ =MΛ −MN =ms −mu,d

MΩ −MΞ*=M

Ξ*−M

Σ*=M

Σ*−MΔ =ms −mu,d

ΔE∝!µi ⋅!µ j

rij3 ; if pointlike

!µi =

eimi

$

%&

'

()!Si

ΔE = 8π3

eiejmimj

ψ(0) 2 !Si ⋅!Sj

ψ(0) :wave function at the origin rij = 0

Chromomagnetic interaction ⇒ΔM ∝

!Si ⋅!Sj

mimj

§  L=0àdifferencesinmassesduetospinstructureofstates

§  interactionenergybetween2spin½particleswithmagneticmoments

Supplement


!S 2 ≡

!S1 +!S2( )

2=!S12 +!S22 + 2!S1 ⋅!S2 ⇒

!S1 ⋅!S2 =

−34"2 (S = 0)

+14"2 (S =1)

%

&''

(''

M (meson) =m1 +m2 +ΔM

ΔM ∝

!Si ⋅!Sj

mimj

ΔM JP = 0−( ) = − 3a41

m1m2

ΔM JP =1−( ) = a41

m1m2

%

&

''

(

''

a :constant

MK =m+ms −3a4mms

Supplement


¡  DifficultiesinstaticQuarkModelàintroductionofcolourquantumnumber§  Non-observationoffreequarks§  Disagreementbetweentheoryandexperiment§  Wavefunctionofbaryonswithidenticalquarks


↑↑↑−

↑↑↑++

≡Ω

≡Δ

sssuuu

¡  Problemwithspinstatistics¡  BaryonsarefermionsandtheirWFmustbe

globallyantisymmetricundertheexchangeofany2ofthe3quarks

07/03/16 F. Ould-Saada

54

Way out: 3 colors à Anti-symmetric WF

Symmetric WF

Pauli principle!

07/03/16 55

3⊗3=8a ⊕1s

g1 = RGg2 = RBg3 = GRg4 = GBg5 = BRg6 = BG

g7 =12RR −GG( )

g8 =16RR +GG − 2BB( )

g0 =13RR + BB +GG( )

¡  Colourcountingine+e-àsection9.2


Experimentally:

¡  Protonmass=sumofquarksmasses,plusworktobedoneonsystemtobringconstituentsintominimumenergyconfiguration§  M(u+u+d)<2%mp:restcomesfromthekineticenergyofgluonsandvirtualparticles

¡  Energyscalerelatedtoquarkconfinedwithin1fm:200-300MeV§  For3quarks:~1GeV

¡  Nucleonradius§  Quarks:r<10-16cm§  Hadrons:r~10-13cm–3quarks“choose”averagedistancestominimizeenergy


¡  Quarkswereintroducedinordertoexplaintheregularityandthesymmetrypropertiesofhadronspectroscopy

¡  Dofreequarksexist?§  Fractionalcharges…notobservedinnature…yet

¡  Charmedandbottomhadrons


c ≡ Nc ≡ N(c)− N(c )( )b ≡ −Nb ≡ − N(b)− N(b )( )

Q = I3 +B+ S + c+ b+ (t)

2= I3 +

Y2

c: Charm; b: Bottom; (t :Top)Y = B+S+ c+ b+ (t)

Top does not form bound-states, simply too heavy

¡  Hypercharge


JPC = 0– + JPC = 1– –

JPC = 1/2 + JPC = 3/2 +

¡  Charmedmesons



CS

CS

(I,I3) S C

¡  Afterthediscoveryoftheτleptonin1975,nodoubtfora3rdquarkfamily

¡  Anomaly-freetheory§  Sumofchargeswithinallfamilies

is0§  Existenceof6quarkflavours

requiredwithintheCKMquarkmixingmatrixevenbeforethediscoveryofcharmquarkin1974!

¡  b-quarkdiscoveredin1976§  JPC=0-+B-mesons§  2isospindoubletsand4isospin

singlets


Q(l− )+Q(ν l )+3× Q(qu )+Q(qd )[ ] = 0Nc = 3; l = e,µ,τ ; qu = u,c, t; qd = d, s,b

B+

Bo

!

"##

$

%&& ≡

bubd

!

"##

$

%&& ;

Bo

B−

!

"##

$

%&& ≡

bdbu

!

"##

$

%&&

Bso ≡ bs( ) ; Bs

o ≡ bs( )

Bc+ ≡ bc( ) ; Bc

− ≡ bc( )

Symmetryagainandagain

¡  Hadronspectroscopy§  Studyofhadronproperties:masses,lifetimes,decaymodes,spins,charges,

andotherQNs,ledtoQuarkModel(Gell-Mann,Zweig1964)

¡  Leptonscattering§  ledtonucleonsubstructureandpoint-likeconstituents

¡  “Jet”production


jetjetqqee +→+→−+

JADE@PETRA,DESY

€

u(Q = +23

) ; d(Q = −13

) ⇒ p ≡ uud ; n ≡ uddπ + ≡ ud ; π− ≡ u d

& ' (

€

Baryons ≡ qqqMesons ≡ qq

¡  WeakInteractionsresponsibleforup-downtransitions

¡  InSIandEMIquarkscanonlybecreatedanddestroyedinpairs

¡  Quarknumberconservation:§  SI,EMI

¡  Totalquarknumberconserved§  InWI

¡  BaryonNumberconserved§  Nucleonisbaryon,notpion


eνpen −→

uceeccee

→/→

→→−+

−+

γ

γ Nc = +1 for c; Nc = −1 for c; Nc = 0 for others

p ≡ uud→ Nu = 2,Nd =1,Ns,c,b,t

= 0

Nq ≡ N(q)− N(q ) ; c(1)→ s(1)+u(1)+ d (−1)

3)()(

3qNqNN

B q −=≡

N f ≡ N( f )− N( f ) ( f = u,d, s,c,b, t)

Documents

Chapter 7 - folk.uio.nofolk.uio.no/farido/fys3510/HadronInteractionsQuarkModel7.pdf · ¡A hadron is made of quarks and has dimensions ~1 fm § Mesons – hadrons with integer spin