Chapter7
¡
7.1HadronsandQuarks
§
¡
¡
7.2Proton-NeutronSymmetryandtheIsotopicSpin
7.3TheStrongInteractionCross-Section
§
¡
§
7.5.1The_CC.1232/Resonance 7.5.2ResonanceFormationandProduction 7.5.3AngularDistributionofResonanceDecayProducts
7.8.1FirstIndicationsfortheColorQuantumNumber
7.9TheJP=1/2CBaryonicOctet
7.10PseudoscalarMesons 7.11TheVectorMesons
7.12StrangenessandIsospinConservation
7.13TheSixQuarks
7.14ExperimentalTestsontheStaticQuarkModel
§
§
¡
7.4.2HadronResonances
7.6ProductionandDecayofStrangeParticles
7.7ClassificationofHadronsMadeofu;d;sQuarks
7.8TheJP=3/2CBaryonicDecuplet
§
¡
¡
¡
¡
7.4.1Antibaryons
7.5Breit–WignerEquationforResonances
§
¡
¡
¡
7.3.1MeanFreePath
7.4LowEnergyHadron-HadronCollisions
§
¡
7.1.1TheYukawaModel
7.14.1LeptonicDecaysofNeutralVectorMesons
7.14.2LeptonPairProduction
7.14.3Hadron-HadronCross-SectionsatHighEnergies7.14.4BaryonMagneticMoments 7.14.5RelationsBetweenMasses
7.15SearchesforFreeQuarksandLimitsoftheModel
2
¡
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),rapidly
createdandannihilated
07/03/16
F. Ould-Saada
3
¡
1stattempttoexplaininteractionbetweennucleonsinnuclei
§  quantummechanicalmodel–Yukawa1930s.
▪  InanalogywithQED–exchangeofmasslessγ,potentialV~1/r
▪  Strongforces–maximumrange~1fm,exchangebosonmustbeheavy~150MeV
▪  SochangeQEDpotentialsuchthatititquicklyvanisheswithdistanceduetoexchangeof
massiveparticles
!2 2
2
2 4
%
E =m c +p c
' * 1 ∂2 " 2 - m 2 c 2
"
⇒ , 2 2 22 −2 ∇ !/φ + 2 2 4 φ != 0
∂ " 2 ∂2ϕ"(r,& t)
¡  Klein-Gordonequation
c ∂t
c ∇ ϕ (r,.t) + m!c ϕ (r, t)
E →ih ; p −!
→− ih∇2' =+−!
(
∂t
∂t
§  E,p,mrelation
r
2 2
−
m
c
K
2
a
§  Correspondenceprinciple
∇ ϕ = 2 ϕ ⇒ ϕ = e !
r
§  φinterpretedaspotentialUincoordinate
space
§  staticpotentialU(r)
07/03/16
F. Ould-Saada
4
!
m2c2
1 ∂ % ∂U (
2
φ (r, t) → U(r) ⇒ ∇ U = 2 U = 2 ' r 2
*
&
"
r ∂r
∂r )
¡
g 1 −r/R
!
e ; R =
4π r
mc
!c
2
R ≈ 1.2fm ⇒ mc =
≈ 150MeV ≈ mπ
R
U(r) =
Huntingafterameson(me<mπ<mp)opened
§  3chargestatestoaccommodatepp,pn,nninteractions
¡
Wasbelievedthatthemuonfoundincosmicrayswasthepion
§  MuoncannotbeYukawa’smeson…doesnotinteractstrongly!
§  Pionsdiscoveredlater…
07/03/16
F. Ould-Saada
5
¡
Heisenberg,1932
§  Neutron&protonastwostatesofasingle
particle,thenucleonN,withrespectto
stronginteractions
§  Analogywithspin,isospinofN:I=1/2
▪  pandnareeigenstatesofI3
§  StronginteractiondependsonI,notonI3
▪  I3behavesas“Q”
Nucleon ≡ N ≡ (np) ; I =
p ≡
1 1
,+
2 2
▪  EMconservesI3
07/03/16
F. Ould-Saada
1 1
,−
2 2
≡(10)
Proton and neutron are eigenstates of I 3
I 3 p = + 12 p ; I 3 n = − 12 n
§  Isopinnotconservedindecaysinducedby
weakandEMinteractions
≡ (10) ; n ≡
1
2
Ii =
σi
2
! ⎛ I1 ⎞
; I = ⎜ I2 ⎟ ∈
⎝ I3 ⎠
Isospinspace
6
Isospinhassamemathematicalbehaviourasspin
¡
§  Spinisaphysicalquantity
▪  measuredinunitsofh
▪  withdimensionofEnergy.Time
§  Isospinisadimensionlessquantity
▪  helpsclassifyinghadronsintomultiplets
Ii =
σi
2
! ! I1 $
; I = # I2 & ∈
" I3 %
σ 1 = (10 01) ; σ 2 = (i0
Ladder operators: transformations p ↔ n
I + ≡ I1 + iI 2 = (00 10) ; I − ≡ I1 − iI 2 = (10 00)
Isospinspace
) ; σ = ( ) ; 1= ( )
−i
0
3
1 0
0 −1
1 0
0 1
I12 + I 22 + I 32 = I 2 → eigenvalues I(I +1)
I = 0,1 / 2,1, 3 / 2,...
−I ≤ I 3 ≤ I → multiplet (2I +1)
07/03/16
F. Ould-Saada
7
¡
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
§  ¡
Reactionprohibitedinstronginteractionsandallowedfor
EMinteraction
4
0
d + d → He + π
I 0 0 0 1 ← I not conserved
I 3 0 0 0 0 ← I 3 conserved
▪  ProcessmustproceedwithcrosssectionstypicalofEM
▪  Confirmedthatcrosssectionis100timessmallerthantypicalstrongcross
sections
07/03/16
F. Ould-Saada
9
¡
Mirrornuclei:samevalueofAbutvaluesof
NandZinterchanged
B ( A = 11, Z = 5, N = 6 ) !
#
10 pp,15nn pairs
#
" same np − pairs
11
6 C ( A = 11, Z = 6, N = 5)#
#
15pp,10 nn pairs
$
11
5
¡
¡
Chargesymmetryapproximatelyverifiedà
Ifdifferentnp-pairs,effectduetofactthat
nppairnotsubjecttoPauliprinciple
§  testofchargeindependence
¡
11
5
B , 116C , 114Be
Butevidencethatstrongforcestrongerfor
S=1thanS=0state
07/03/16
F. Ould-Saada
Lowlyingenergylevelsofmirrornuclei
10
¡
2-dimensionalspacerepresentation
¡
§  Nucleon:(p,n)orquarks(u,d):I=1/2
§  Generatorsare2x2Paulimatrices
¡
4-dimensionalspacerepresentation
Δ(1232):I=3/2àI3=3/2,1/2,-1/2,-3/2
§  Basis(Δ++,Δ-,Δ0,Δ-)
§
3-dimensionalspacerepresentation
§  Basis:(π+, π0, π-)withisospinI=1andI3=+1,0,-1
§  Generatorsare3x3matrices
¡
ConservationofIsospininSI
§  SIdependsonlyonIandisindependentonI3andQ
Q = I3 +
§  TotalIisconservedinSI
B
2
§  àselectionrulesandpreciserelationbetweenproductioncrosssectionsofrelatedprocesses
p + p → d + π +
I 1!"
/ 2 1#
/ 2 0 1
!
#
$
1
1
07/03/16
p + n → d + π 0
I 1!"
/ 2 1#
/ 2 0 1
!
#
$
1
0,1
F. Ould-Saada
σ ( pp → d π + )
σ ( pn → d π
0
)
=2
11
¡
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
¡
Flavourindependenceofcolourforcebetween2quarksat
samedistance–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
1
1(π ) × (N ) : 3 ⊗ 2 = 4 ⊕ 2
2
π ≡ (π +, π 0 , π − ) ; N ≡ ( p, n)
I = 3 / 2 : Δ ≡ (Δ ++, Δ +, Δ 0 , Δ − )
I =1/ 2 : N*
§  Next3slides:remindercouplingof2or3(iso)spins
07/03/16
F. Ould-Saada
13
1
2
1
× : 2 ⊗ 2 = 3⊕ 1
2
1
1× : 3 ⊗ 2 = 4 ⊕ 2
2
1 1
1
2
2
× × : 2 ⊗ 2 ⊗ 2 = 4S ⊕ 2 MS ⊕2 MA
2
07/03/16
F. Ould-Saada
14
1
1(π ) × (N ) : 3 ⊗ 2 = 4 ⊕ 2
2
π ≡ (π +, π 0 , π − ) ; N ≡ ( p, n)
π N States → π N; I, I 3
Δ
++
≡
3
3
,+
2
2
⎫
⎪
2
0p ⎪
π
⎪
3
⎬
2
0n
π
⎪
3
⎪
⎪⎭
=π p
+
Δ+ ≡
3
1
,+
2
2
=+
Δ0 ≡
3
1
,−
2
2
=
Δ− ≡
3
3
,−
2
2
= π −n
1
3
1
3
N *+ ≡
1
1
,+
2
2
=+
N *0 ≡
1
1
,−
2
2
=
π +n
π−p
2
3
2
3
+
+
π +n
π−p
−
−
1
3
1
3
Γ (Δ+ → π +n)
Γ ( Δ + → π 0 p)
=
1
2
π0p
π 0n
15
Δ0 ≡
N *0 ≡
3
1
,−
2
2
1
1
,−
2
2
=
=
1
3
π−p
2
3
π−p
2
3
+
−
π−p
π 0n ⎫⎪
⎬⇒
1
0n
π
π 0n
⎪⎭
3
=
1
1 3
,−
2
3 2
−
1
2 1
,−
2
3 2
=
1
2 3
,−
2
3 2
+
1
1 1
,−
2
3 2
⇒ M (π − p → π − p) = 13 M 3/2 + 23 M1/2 $&
%
−
0
2
2
M (π p → π n ) = 3 M 3/2 − 3 M1/2 &'
I(Δ) = 3 / 2 ⇒ M 3/2 >> M1/2 @ Δ(1232)
2
σ total (π − p) = σ (π − p → π − p) + σ (π − p → π 0 n ) ∝ 13 M 3/2 &
+
(σ
π
p)
(
(
total
3
3
++
+
Δ ≡ 2,+ 2 = π p
=3
'
−
σ total (π p)
(
2
σ total (π + p) ∝ M 3/2
()
07/03/16
F. Ould-Saada
16
σ total (π + p )
=3
−
σ total (π p )
07/03/16
F. Ould-Saada
17
¡
Crosssectionmeasurement
§  Informationontheinteractionpotential
§  Caseofshort-rangepotential,negligibleforr>Ro(Yukawa)
▪  ExpectpuregeometricCS:σ=πR02validforprojectilesizeλ<<targetsize
▪  P>>1GeVàpointlikeprojectilewrtnucleardimensions
¡
TotalCS
σ tot = σ el + σ inel
§  ElasticCS
▪  Projectile&targetunchanged&sameenergyincomsystem
§  InelasticCSdominantatHE
▪  àparticleexcitationsandnewparticleproduction
2π !c
pc
1.24
=
[ fm]
p [GeV ]
λ=
⎧ σ ≅ 40mb
⎧⎪ R ≅ 1.1 fm
⎪
pp
0
2
¡  Nuclearforceofshortrange p >> 1GeV ⇒ ⎨
; σ = π R0 ⇒ ⎨
lab
⎪⎩ σ π p ≅ 25mb
⎪⎩ R0 ≅ 0.9 fm
§  seeπpandppcrosssections
18
¡
σ=f(plab)
¡
Nuclearforcesareshortrange
¡
π-p,π-d
§  Athigh(plab),σ variesonly
slowlywithenergy
plab >> 1GeV
σ π p ≅ 25mb
R0 ≅ 0.9 fm
§  Atlowp,resonance
formations
07/03/16
F. Ould-Saada
19
σ=f(plab)
Nuclearforcesareshortrange
p-p,pbar-p
¡
¡
¡
§  Athigh(plab),σ variesonly
slowlywithenergy plab >> 1GeV
σ pp ≅ 40mb
R0 ≅ 1.1 fm
¡
Beforecontinuingwithlow
energyhadron-hadron
collisions,let’sintroduce
strangehadrons
20
¡
¡
B ≡ qqq ; B ≡ q q q
Hadrons
§  Baryonsandanti-baryons:half-integralspin
M ≡ qq
p = uud ; n = udd
§  Mesonsandanti-mesons:integralspin
π + = ud ;π 0 = uu , dd ; π − = u d
Strangeparticles
§  Firstdiscoveredin1947incloudchamberphotographsofcosmicrays.
§  K-mesonsorKaonsfirstcalledV-particles
§  Hyperonsinvolveprotonsintheirdecays
▪  ProducedinSIs(t~10-23s)butdecaythroughWIs(t~10-10s)
K 0 → π + + π −
π − + p →Λ 0 + K 0
Λ0 → π −+p
07/03/16
F. Ould-Saada
21
¡
Associatedproduction
§  Pais,1952:strangeparticlesproducedinpairs
§  Newquantumnumber,strangenessS,wasintroducedin1953byGell-
MannandNishijima.
S ≡ −N s = −( N(s) − N(s ))
§  Sisconservedinstrong&EMinteractionsbutnotinweakinteractions.
π − + p → Λ0 + K 0
€
mΛ = 1115.7MeV
mK 0 = 497.7MeV
S: 0
0
−1
+1 → ΔS = 0
Λ →π + p
0
S:
−1
−
0
0
→ ΔS =1
K − + p →Ω− + K + + K 0
S: −1
0
−3
+1 +1 → ΔS=0
K 0 →π + +π −
S:
07/03/16
+1
0
F. Ould-Saada
0
→ ΔS =1
22
¡
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
S = 1, I =
⎧ I 3 = + 12 K +
⎨
1
0
I
=
−
K
3
⎩
2
1
2
S = −1, I =
1
2
⎧ I 3 = + 12 K 0
⎨
1
−
I
=
−
K
3
⎩
2
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 −
23
¡
Featuresofhadronhadroncollisions
obtainedbyanalyzing
measurementsofσtotof
chargedhadronson
hydrogenanddeuterium,
§
¡
(seepreviousfiguresaswell)
Peaks&structures?
E ≤ 3GeV
Yes : π ± p, K − p, K − n
less : K + p, pp, pp
¡
¡
07/03/16
Tobeexplainedlater
Let’slookatK-p
F. Ould-Saada
24
¡
07/03/16
DirectproductionofΛ(1)
orthrougharesonance
(2,2’)?
F. Ould-Saada
(1) K − p →Λπ + π −, Λ → π − p
(2) K − p →Σ+*π −, Σ+* → Λπ +
(2')K − p →Σ−*π +, Σ−* → Λπ −
25
¡
Dalitzplot–2Dplot
§  K-(p=1.22GeV)p
interaction
§  Σ*±resonance
throughreactions(2),
2’)at1.385GeV
(1) K − p →Λπ + π −, Λ → π − p
(2) K − p →Σ+*π −, Σ+* → Λπ +
(2')K − p →Σ−*π +, Σ−* → Λπ −
¡
Effectiveorinvariant
mass:
2
2
2
mΛπ
+ = E
+ − p
Λπ
Λπ +
!
!
= (EΛ + Eπ + )2 − ( pΛ + pπ + )2
! !
= EΛ2 + Eπ2+ + 2EΛ Eπ + − pΛ2 − pπ2 + − 2 pΛ pπ +
Γ ~ 100MeV
!
⇒ τ = ~ 10 −23 s
Γ
= mΛ2 + mπ2 + + 2EΛ Eπ + − 2 pΛ pπ + cosθ Λπ
07/03/16
F. Ould-Saada
26
¡
Resonanceischaracterizedbydefined
AngularmomentumJ=l(forspinlessparticles)
ParityP
IsospinI
Mass=ER,totalenergyincenterofmassat
resonancemaximum.
§  Lifetime(τ),asdeterminedbywidthathalfmaximum(Γ)ofcurve.
§
§
§
§
¡
ShapeofBreit-Wignercurve
§  Fromformalismofamplitudesandphasesof
matterwaves
§  E-dependenceofamplitude=Fourier
transformofawavefunctiondescribinga
survivalprobabilitydecreasingexponentially
overtime,withlifetimeτ.
Breit–Wigner shape
Γ=width, ER=mass of resonnance
a + b → R → a'+ b'
§  ReadinbookthederivationoftheBreit-Wigner
equation(seenextslideaswell)
07/03/16
F. Ould-Saada
2J +1
Γ2 / 4
σ (E, J ) = 4πλ
(2sa +1) (2sb +1) ( ER − E )2 +Γ2 / 4
2
27
¡
¡
¡
UnstableresonanceRdescribedbythefree
particlewavefunctionmultipliedbyareal
functiondescribingitsdecayprobabilityasa
functionoftime
Probabilityoffindingtheparticleatatimet
*
t
2τ
−
t
τ
ψ (t) = ψ 0 e−iω Rt e
I(t) = ψ 02 e
χ (E) = ∫ ψ (t)eiEt dt =ψ 0 ∫ e
Fouriertransform
−
K2
⎡
Γ⎤
−⎢i(E R −E )+ ⎥t
⎣
2⎦
dt =
= ψ0e
= I0e
−i
−
ER
Γ
t − t
!
2!
e
t
τ
K
(ER − E) − i
Γ
2
2
; σ
=
π
(2
λ
)
0
Γ2
2
(ER − E) +
4
1 = χ * (ER ) χ (ER ) = 4K 2 / Γ2 ⇒ K 2 = Γ2 / 4
σ (E) = σ 0 χ (E) χ (E) = σ 0
¡
Addspinmultiplicities
2J +1
Γ2 / 4
σ (E, J ) = 4πλ
(2sa +1) (2sb +1) ( ER − E )2 +Γ2 / 4
2
28
¡
π+ptotalcross
section
§  Largepeakat
Tπlab=191MeV,
Ecm=1232MeV,
Γ=120MeV
§  Δ++:B=+1,I=3/2,
J=3/2
¡
Check
§  puttingthese
numbersintothe
BWequationà
σmax~188mb
compatiblewith
figure
§  J=1/2à94mb…
29
¡
(a)Resonanceformation
ins-channel
¡
§  π+pàΔ++àπ+p
§  π+pàΔ++àπ+p
§  (t-channel:pbarpàπ+π-)
¡
(c)Resonanceformationin
termsofquark
constitutents
(b)Resonanceproduction
§  π+pàΔ++π0àpπ+π0
07/03/16
F. Ould-Saada
30
§
π-pcrosssection
§
§
§
§
§
§
centreofmassenergy1.2-2.4
GeV
2(+2)enhancementsontop
ofnon-resonantcontributions
ResonancewidthsΓ~100MeV
Interactionstimesτ~10-23s
àStronginteraction
àConsistentwithtimetaken
forarelativisticpiontotransit
thedimensionofaproton
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
P
J =0
07/03/16
F. Ould-Saada
−
Q = I3 +
B+S
Y
= I3 +
2
2
P
J =
1+
2
34
¡
Stronginteraction
§  I,Sconserved
¡
EMinteraction
§  S,I3conserved,notI
¡
Weakinteraction
§  S,Iviolated
07/03/16
F. Ould-Saada
35
3 ⊗ 3 = 1⊕ 8
¡
¡
Meson=quark-antiquarkcombination 3 ⊗ 3 = 1 ⊕ 8
Totalspin:
§  S=0 (antisymmetric spin wave function) and S=1 (symmetric)
2 ⊗ 2 =1 ⊕ 3
P=(–1)l+1 C=(–1)l+S àPossible (allowed) states:
2S+1l
J
; JPC
1S
–+
0;0
3S
––
1;1
1P
¡
+–
1;1
3P
F. Ould-Saada
3P
++
1;1
3P
++
2;2
……..
Groundstatemesons
L=0,S=0,1
9 pseudoscalars
JPC = 0– +
07/03/16
++
0;0
9 vectors
(JPC = 1– –)
36
u,d,s
!
S = 1 2 + 1 2 + 1 2 →S = 1 2 , 3 2
! !
L = 0 →J = S
HyperonsJP=3/2+
Baryons JP=1/2+
*
*
*
07/03/16
F. Ould-Saada
*
Not
symmetric
enough
àsomething
ismissing!
*
37
The Ω- (S=-3) predicted to have a mass of 1680 MeV, discovered in 1964 at
mΩ = 1674 ±3 MeV, τΩ=82 ps
07/03/16
F. Ould-Saada
38
¡
u,d/charged,neutralhadron
massdifferenceverysmall
§  Atmost%level
§  DuetoEMinteraction
§  StrongIsospingoodsymmetry
¡
s-quarkmass>>mu,d
07/03/16
F. Ould-Saada
39
¡
¡
u,dquarksà22
combinations
u,d,sà32combinations
intotal
§  η1andη8mixtogiveη
andη’ mesonsobserved
innature
40
¡
u,dquarksà22
combinations
§  I=1:ρ+,ρ0,ρ-
§  I=0:ω
¡
u,d,sà32combinationsin
total
§  K*mesons
§  φ, ω mesonsobservedinnature
¡
Experimentally
§  φmostlytostrangeparticles
§  Althoughitisenergeticallypossibleto
decayto3pions,comparedto2kaons
§  Why?à“Zweigrule”vshigherorderQCD
41
¡
Zweigrule
§  Quarkflow–notreallyFeynmandiagrams
§  c)disconnectedlinesbetweeninitialandfinalstates
àSeveresuppression
07/03/16
F. Ould-Saada
42
¡
Fromthetotalangularmomentum
! ! ! 2 ! !
! !
J = L + S = L2 + S 2 + 2 L ⋅ S
(
)
§  Accountforspin-spacecoupling
§  With
¡
¡
¡
! ! J(J +1) − L(L +1) − S(S +1)
L⋅S =
2
L−S ≤ J ≤ L+S
Therearenoq-qbarmesonswithexotic
quantumnumberssuchas
JPC=0+-,1-+,…
Ontheotherhandexoticstates(non-qqbar)withnormalquantumnumbersare
notforbidden:gluballs,hybrids,
pentaquarks…
Itisthedutyofhadronsspectroscopyto
confirmtheexistinghadronsandtofind
themissingones.
07/03/16
F. Ould-Saada
43
I = 3 / 2; S = 0; Y = +1
Δ − ( I 3 = −3 / 2 ), Δ 0 (−1 / 2 ), Δ + (+1 / 2 ), Δ ++ (+3 / 2 )
I = 1; S = −1; Y = 0
Σ*− ( I 3 = −1), Σ*0 (0 ), Σ*+ (1)
I = 1 / 2; S = −2; Y = −1
Ξ*− ( I 3 = −1 / 2 ),Ξ*0 (+1 / 2 )
I = 0; S = −3; Y = −2
¡
Ω− ( I 3 = 0 )
SymmetrizedWFs
§  Explainingthe10completelysymmetricstates
mΣ − mΔ = 152MeV
1s − 0s
mΞ − mΣ = 149MeV
mΩ − mΞ = 139MeV
2s − 1s
3s − 2s
¡
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
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F. Ould-Saada
45
¡
p=uud
§  Spinwf2quarksà
(↑↓−↓↑) $
&
2 &
% ⇒ A = u↑d↓ − u↓d↑ − d↑u↓ + d↓u↑
antisymmetric
(ud − du) &
§  Flavourwfu,dàantisymmetric
2 &'
¡
N=ddu
§  Replaceuudbyddu!
¡
07/03/16
F. Ould-Saada
Exercise:usetheprotonandneutron
WFstoreproducethechargeand
magneticmomentofpandn.
46
¡
Diractheory
§  Point-likefermionwithelectric
chargeq,massm,spin½has
magneticdipolemoment
§  Magneticmoments
§  Useconstituentquarkmasses
! q! " q !
µ=
σ=
σ
2mc
2m
!
q
µf ≡ f µ f =
2m
!
q
q
µu = u µ u = u , µ d = d
2mu
2md
mu ≈ md ;qu = −2qd ⇒ µu ≈ −2µ d
!
8µu − 2µ d 3 "
µn
2
µp = p µ p =
= µu $
=−
6
2 $⇒
µp
3
#
4µ − µ u
$
µn = d
= −µ u
exp = −0.685
$%
3
"bare" : mu
07/03/16
Λ(1116) = uds"
# ⇒ µΛ = µs
s(ud) = 0
$
≈ md ≈ 5MeV; ms = md +150MeV
"constituent": mu
F. Ould-Saada
≈ md ≈ m p / 3; ms = md +150MeV
48
¡
¡
Prediction
agreeswith
experimentif
constituent
massesused
Quarkmodelis
notbadat
reproducing
magnetic
moments…
§
Theagreementis
howevernot
perfect
µN =
e!
⇒ µu = 2µ N ; µ d = −1µ N ; µ s = −0.67µ N
2m p c
µ p,n,Λ ⇒ µu = +1.852µ N ; µ d = −1.972µ N ; µu = −0.613µ N
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49
Supplement
¡
Hadronmasses
§  1/2+Baryonoctet
M Ξ − M Σ = M Ξ − M Λ = M Λ − M N = ms − mu,d
M Ω − M Ξ* = M Ξ* − M Σ* = M Σ* − M Δ = ms − mu,d
§  3/2+Baryondecuplet
§  àms-mu,d~120-200MeV/c2
§  L=0àdifferencesinmasses
duetospinstructureofstates
§  interactionenergybetween2
spin½particleswithmagnetic
moments
07/03/16
F. Ould-Saada
! !
µ ⋅µ
ΔE ∝ i 3 j
rij
! $ ei ' !
; if pointlike µi = & ) Si
% mi (
8π ei e j
2 ! !
ΔE =
ψ (0) Si ⋅ S j
3 mi m j
ψ (0) : wave function at the origin rij = 0
! !
Si ⋅ S j
Chromomagnetic interaction ⇒ ΔM ∝
mi m j
50
% 3
2
'
−
"
!2
! ! 2 !2 !2
! !
! ! ' 4
S ≡ S1 + S2 = S1 + S2 + 2 S1 ⋅ S2 ⇒ S1 ⋅ S2 = &
' + 1 "2
'( 4
(
)
M (meson ) = m1 + m2 + ΔM
! !
S ⋅S
ΔM ∝ i j
mi m j
M K = m + ms −
07/03/16
(S = 0)
(S = 1)
%
3a 1
' ΔM ( J P = 0 − ) = −
4 m1m2
'
&
' ΔM J P = 1− = a 1
(
) 4mm
'
1 2
(
Supplement
a : constant
3a
4mms
F. Ould-Saada
51
07/03/16
F. Ould-Saada
52
¡
¡
Problemwithspinstatistics
BaryonsarefermionsandtheirWFmustbe
globallyantisymmetricundertheexchangeofany
2ofthe3quarks
¡
DifficultiesinstaticQuarkModel
àintroductionofcolourquantumnumber
§  Non-observationoffreequarks
§  Disagreementbetweentheoryandexperiment
§  Wavefunctionofbaryonswithidenticalquarks
Δ+ + ≡ u ↑ u ↑ u ↑
Ω ≡s↑s↑s↑
−
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F. Ould-Saada
53
Symmetric WF
Pauli principle!
Way out: 3 colors
à Anti-symmetric WF
07/03/16
F. Ould-Saada
54
g1 = RG
g2 = RB
3 ⊗ 3 = 8a ⊕1s
g3 = GR
g4 = GB
g5 = BR
1
g0 =
RR + BB + GG )
(
3
07/03/16
g6 = BG
1
g7 =
RR − GG )
(
2
1
g8 =
(RR + GG − 2BB )
6
55
Experimentally:
¡
Colourcountingine+e-àsection9.2
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F. Ould-Saada
56
¡
Protonmass=sumofquarksmasses,plusworktobedoneonsystemtobring
constituentsintominimumenergyconfiguration
§  M(u+u+d)<2%mp:restcomesfromthekineticenergyofgluonsandvirtualparticles
¡
Energyscalerelatedtoquarkconfinedwithin1fm:200-300MeV
§  For3quarks:~1GeV
¡
Nucleonradius
§  Quarks:r<10-16cm
§  Hadrons:r~10-13cm–3quarks“choose”averagedistancestominimizeenergy
¡
¡
Quarkswereintroducedinordertoexplaintheregularityandthe
symmetrypropertiesofhadronspectroscopy
Dofreequarksexist?
§  Fractionalcharges…notobservedinnature…yet
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F. Ould-Saada
57
Top does not form
bound-states,
simply too heavy
¡
Charmedandbottomhadrons
c ≡ N c ≡ ( N(c) − N(c ))
b ≡ −N b ≡ − ( N(b) − N(b ))
¡
Hypercharge
B + S + c + b + (t)
Y
= I3 +
2
2
c: Charm; b: Bottom; (t :Top )
Q = I3 +
Y = B + S + c + b + (t)
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F. Ould-Saada
58
JPC = 1/2 +
JPC = 0– +
07/03/16
JPC = 3/2 +
JPC = 1– –
F. Ould-Saada
59
¡  Charmedmesons
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F. Ould-Saada
60
(I,I3)
07/03/16
S
S
CC
S
C
F. Ould-Saada
61
Symmetryagain
andagain
¡
¡
Afterthediscoveryoftheτlepton
in1975,nodoubtfora3rdquark
family
Anomaly-freetheory
Q(l − ) + Q(ν l ) + 3× [Q(qu ) + Q(qd )] = 0
N c = 3; l = e, µ, τ ; qu = u, c, t; qd = d, s, b
§  Sumofchargeswithinallfamilies
is0
§  Existenceof6quarkflavours
requiredwithintheCKMquark
mixingmatrixevenbeforethe
discoveryofcharmquarkin1974!
¡
b-quarkdiscoveredin1976
§  JPC=0-+B-mesons
§  2isospindoubletsand4isospin
singlets
07/03/16
F. Ould-Saada
! +
# B
# Bo
"
$ !
$ ! Bo
bu
&≡#
& #
& # bd & ; # B −
% "
% "
$ !
$
bd
& ≡ ##
&&
&
% " bu %
Bso ≡ ( bs ) ;
Bso ≡ ( bs ) Bc+ ≡ ( bc) ;
Bc− ≡ ( bc ) 62
¡
Hadronspectroscopy
§  Studyofhadronproperties:masses,lifetimes,decaymodes,spins,charges,
andotherQNs,ledtoQuarkModel(Gell-Mann,Zweig1964)
Baryons ≡ qqq
¡
Mesons ≡ qq
& p ≡ uud ; n ≡ udd
2
1
u(Q = + ) ; d(Q = − ) ⇒ ' +
−
3
3
(π ≡ ud ; π ≡ u d
Leptonscattering
§  ledtonucleonsubstructureandpoint-likeconstituents
€
€
¡
“Jet”production
e+ e− → q + q → jet + jet
07/03/16
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JADE@PETRA,DESY
63
¡
WeakInteractionsresponsibleforup-downtransitions
¡
InSIandEMIquarkscanonlybecreatedanddestroyedinpairs
¡
e+ e− → γ → cc
N c = +1 for c; N c = −1 for c; N c = 0 for others
e+e− → γ →
/ cu
p ≡ uud → N u = 2, N d = 1, N s,c,b,t = 0
Quarknumberconservation:
§  SI,EMI
¡
n → pe − νe
N f ≡ N( f ) − N( f )
( f = u, d, s, c, b, t)
Totalquarknumberconserved N q ≡ N(q) − N(q ) ; c(1) → s(1) + u(1) + d (−1)
§  InWI
¡
BaryonNumberconserved
§  Nucleonisbaryon,notpion
07/03/16
F. Ould-Saada
Nq
N ( q) − N ( q )
B≡
=
3
3
64