Chapter 7 - folk.uio.nofolk.uio.no/farido/fys3510/HadronInteractionsQuarkModel7.pdf · ¡A hadron...
Transcript of Chapter 7 - folk.uio.nofolk.uio.no/farido/fys3510/HadronInteractionsQuarkModel7.pdf · ¡A hadron...
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)
07/03/16 F. Ould-Saada 4
−!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
07/03/16 F. Ould-Saada 5
¡ 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
07/03/16 F. Ould-Saada 6
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
07/03/16 F. Ould-Saada 7
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
07/03/16 F. Ould-Saada 8
¡ NucleiwithA=14haveJP=0+andnearlysameenergy:§ 14C(I3=-1)):nnpair§ 14N(I3=0)):nppair§ 14O(I3=+1)):pppair§
¡ ReactionprohibitedinstronginteractionsandallowedforEMinteraction
▪ ProcessmustproceedwithcrosssectionstypicalofEM▪ Confirmedthatcrosssectionis100timessmallerthantypicalstrongcross
sections
07/03/16 F. Ould-Saada 9
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
07/03/16 F. Ould-Saada 11
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
07/03/16 F. Ould-Saada 12
¡ πNinteractionsèΔ&N*resonances
§ Next3slides:remindercouplingof2or3(iso)spins
07/03/16 F. Ould-Saada 13
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 *
07/03/16 F. Ould-Saada 14
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
07/03/16 F. Ould-Saada 16
⇒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
07/03/16 F. Ould-Saada 17
( )( ) 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
07/03/16 F. Ould-Saada 19
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.
07/03/16 F. Ould-Saada 22
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,...
07/03/16 F. Ould-Saada 23
⎩⎨⎧
−=
+==−=
⎩⎨⎧
−=
+===
−
+
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?
07/03/16 F. Ould-Saada 24
E ≤ 3GeVYes :π ±p,K −p,K −nless :K +p, pp, pp
¡ Tobeexplainedlater¡ Let’slookatK-p
07/03/16 F. Ould-Saada 25
(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:
07/03/16 F. Ould-Saada 26
(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
07/03/16 F. Ould-Saada 27
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-)
07/03/16 F. Ould-Saada 30
¡ (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
07/03/16 F. Ould-Saada 31
¡ TodeterminespinofΔ++(àpπ+)§ Studyangulardistributionatresonanceofπ+incentreofmass(c.m)§ J=3/2,l=1èI(θ*)=1/(8π)(1+3cos2θ*))§ Others,lconfigurationswouldleadtoverydifferentangulardistributions
07/03/16 F. Ould-Saada 32
07/03/16 F. Ould-Saada 33
¡ GNNrelation¡ ElectricchargeQ,BaryonnumberB,StrangenessS,3rd
componentofisospinI3,HyperchargeY=B+S¡ Withthehandofthisrelation2hyperonspredictedin
1953Σ0, Ξ- discoveredin1958and1959
07/03/16 F. Ould-Saada 34
Q = I3 +B+ S2
= I3 +Y2
−= 0PJ+
= 21PJ
¡ Stronginteraction§ I,Sconserved
07/03/16 F. Ould-Saada 35
¡ 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
07/03/16 F. Ould-Saada 39
¡ 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
07/03/16 F. Ould-Saada 42
¡ Fromthetotalangularmomentum
§ Accountforspin-spacecoupling
§ With
07/03/16 F. Ould-Saada 43
!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
07/03/16 F. Ould-Saada 45
¡ p=uud§ Spinwf2quarksà
antisymmetric§ Flavourwfu,dàantisymmetric
07/03/16 F. Ould-Saada 46
↑↓−↓↑( )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
07/03/16 F. Ould-Saada 48
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
07/03/16 F. Ould-Saada 49
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
07/03/16 F. Ould-Saada 50
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
07/03/16 F. Ould-Saada 51
!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
07/03/16 F. Ould-Saada 52
¡ DifficultiesinstaticQuarkModelàintroductionofcolourquantumnumber§ Non-observationoffreequarks§ Disagreementbetweentheoryandexperiment§ Wavefunctionofbaryonswithidenticalquarks
07/03/16 F. Ould-Saada 53
↑↑↑−
↑↑↑++
≡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
07/03/16 F. Ould-Saada 56
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
07/03/16 F. Ould-Saada 57
¡ Quarkswereintroducedinordertoexplaintheregularityandthesymmetrypropertiesofhadronspectroscopy
¡ Dofreequarksexist?§ Fractionalcharges…notobservedinnature…yet
¡ Charmedandbottomhadrons
07/03/16 F. Ould-Saada 58
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
07/03/16 F. Ould-Saada 59
JPC = 0– + JPC = 1– –
JPC = 1/2 + JPC = 3/2 +
¡ Charmedmesons
07/03/16 F. Ould-Saada 60
07/03/16 F. Ould-Saada 61
CS
CS
(I,I3) S C
¡ Afterthediscoveryoftheτleptonin1975,nodoubtfora3rdquarkfamily
¡ Anomaly-freetheory§ Sumofchargeswithinallfamilies
is0§ Existenceof6quarkflavours
requiredwithintheCKMquarkmixingmatrixevenbeforethediscoveryofcharmquarkin1974!
¡ b-quarkdiscoveredin1976§ JPC=0-+B-mesons§ 2isospindoubletsand4isospin
singlets
07/03/16 F. Ould-Saada 62
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
07/03/16 F. Ould-Saada 63
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
07/03/16 F. Ould-Saada 64
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)