Nuclear and Particle Physics Franz Muheim 1
QCD QCD ------ Quantum Quantum ChromodynamicsChromodynamics
QCD PrinciplesQuantum field theory, analogy with QEDVertex, coupling constantColour “red”, “green” or “blue”Gluons
Quark and Gluon InteractionsConfinementAsymptotic FreedomQCD potential
QCD ExperimentsExperimental evidence for quarks, colour and gluons e+e- annihilationsCharmoniumScattering, DIS
QuestionsWhy is strong interaction short range?Why are “free quarks” not observed?How do quarks and gluons fragment into hadronic jets?
OutlineOutline
Nuclear and Particle Physics Franz Muheim 2
QCD QCD vsvs QEDQED
QEDQuantum theory of electromagnetic interactionsmediated by exchange of photonsPhoton couples to electric charge eCoupling strength ∝ e ∝ √α
QCDQuantum theory of strong interactionsmediated by exchange of gluons between quarksGluon couples to colour charge of quarkCoupling strength ∝ √αS
Fundamental verticesQED QCD
α = e2/4π ≈ 1/137 αS = gS2/4π ~ 1
Coupling constantStrong interaction probability ∝ αS > αCoupling strength of QCD much larger than QED
Nuclear and Particle Physics Franz Muheim 3
ColourColour
What is Colour ?Charge of QCD Conserved quantum number“Red”, “green” or “blue”
QuarksCome in three colours r g bAnti-quarks have anti-colours
Leptons, other Gauge Bosons - γ, W±, Z0
Don’t carry colour, “zero colour charge”Don’t participate in strong interaction
Caveat“… colour is not to be taken literally.”
Interaction QED QCDConserved charge
electric charge e
colour charges r, g ,b
Coupling constant
α = e2/4π αS = gS2/4π
Gauge boson Photon 8 gluonsCharge carriers
fermions (q ≠ 0)
quarks gluons
r g b
Nuclear and Particle Physics Franz Muheim 4
GluonsGluons
Gluon PropertiesGluons are massless spin-1 bosons
QCD propagator 1/q2
Emission or absorption of gluons by quarks changes colour of quarks - Colour is conserved
Gluons carry colour charge themselvese.g. rg gluon changes red quark into green QCD very different from QED, q(photon) = 0
Number of gluonsNaively expect 9 gluons rb, rg, gb, gr, br, bg, rr, gg, bbSymmetry -> 8 octet and 1 singlet states
8 gluons realised by Nature (colour octet)
Nuclear and Particle Physics Franz Muheim 5
Quark & Gluon Quark & Gluon InteractionsInteractions
Quark-Antiquark Scatteringdescribes a mesone.g. π+ = (u-dbar)Single gluon exchangeat short distance ≤ 0.1 fm
QCD Potentialat short distance ~ 0.1 fmattractive - negative sign
QED-like apart from colour factor 4/3 More than one gluon -> colour factor
Gluon Self-Interactions QED versus QCD - So far pretty similarPhotons and gluons – massless spin-1 bosonsBig difference - gluons carry colour charge
Gluons interact with each other3-gluon vertex 4-gluon vertex
Origin of huge differencesbetween QCD and QED
rrV S
QCD 34)( α
−=
Nuclear and Particle Physics Franz Muheim 6
ConfinementConfinement
Experimental EvidenceDo not observe free quarksQuarks confined within hadrons
Strong Interaction DynamicsGluons attract each other - self-interactions
Colour force lines pulled together in QCD
Colour Forcebetween 2 quarks at “long” distances O(1 fm)String with tension k -> Potential V(r) = krStored energy/unit length is constantSeparation of quarksrequires infinite amount of energy
ConfinementDirect consequence of gluon self-interactionsParticles with colour - quarks and gluons -confined inside QCD potential, must combineinto hadrons with zero net colour charge
Nuclear and Particle Physics Franz Muheim 7
QCD PotentialQCD Potential
Mesonsquark-antiquark pair colour wave fct.with zero net colour chargeSingle gluon exchange -> colour of individual q or anti-q can change
QCD potential
QED-like at short distance r ≤ 0.1 fmQuarks are tightly bound αS ≈ 0.2 .. 0.3String tension -> Potential increases linearly at large distance r ≥ 1 fm
Potential similarfor quarks in baryons
ForceBetween two quarks at large distanceF = |dV/dr| = k = 1.6 10-10 J/ 10-15 m = 16000 NEquivalent to weight of large car
αS = 0.2k = 1 GeV/fm
krr
rV +−= SQCD 3
4)( α
bbrrrrrr →→
Nuclear and Particle Physics Franz Muheim 8
Coupling Constant Coupling Constant ααSS
PropertiesαS --- coupling strength of strong interactionRecall QED - coupling constant varies
with distance - running αIn QED – bare electron charge is screened by cloud of virtual e-e+ pairsIn QCD – similar effects
QCD Quantum FluctuationsCloud of virtual q-anti-q pairs around a quark
Screening of colour charge Colour charge decreases with distance
Cloud of virtual gluons --- no equivalent in QEDdue to gluon self-interactionsColour charge of gluons contributes to effective colour charge of quark
Anti-screening of colour chargeColour charge increases with distance
Nuclear and Particle Physics Franz Muheim 9
Running of Running of ααSS
Screening and Anti-screeningAnti-screening dominatesEffective colour charge increases with distanceAt large distances / low energies αS ~ 1 - largeHigher order diagrams -> αS increasingly largerSummation of diagrams divergesPerturbation theory fails
Asymptotic Freedom
Coupling constantαS = 0.12 at q2 = (100 GeV)2
small at high energiesRunning of αS
depends on q2 and # of colours and flavoursEnergetic quarks are (almost) free particlesSummation of all diagrams convergesQCD Perturbation theory works
πβ
µµαβ
µαα
12211
ln)(1
)()(
2
22
S
2S2
S
fn
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
Nobel prize 2004Gross, Politzer, Wilczek
n = 3 coloursf = 3 … 6 flavours
Asymptotic FreedomAsymptotic Freedom
Nuclear and Particle Physics Franz Muheim 10
HadronisationHadronisation & Jets& Jets
What happens when quarks separate?Example: annihilation
Quarks separateEstring increases - when Estring > 2 mqString breaks up into pairs - fragmentation
HadronisationAs energy decreasesFormation of hadrons(mesons and baryons)Hadrons follow directionof original
qqee →−+
Jets
Observe collimated jetsback-to-back in CoM frame
hadronsionhadronisat
→
→−+
−+
eeqqee
LEP√s = 91 GeV
Nuclear and Particle Physics Franz Muheim 11
e+ee+e-- AnnihilationAnnihilation
Feynman Diagrams
Quark and muon masses are neglected Only difference in coupling of virtual photon to final state fermion pair is charge Qf
muons Qµ = ±1 quarks Qq = ±2/3 or ±1/3
Cross sectionFor a single quark flavour --- without colourexpect cross section ratio
With colour – each quarks has NC = 3 final statesRule is to sum over all available final states
HadronisationMeasure
-pairs fragment and form hadronic jetsJets from different -pairs are similarat high energies compared to quark masses
−−−+−+ →→ µµeeqqee and
( )( )
22
2
q QQQ
eeqqeeR ==
→→
= −+−+
−+
µµµσσ
qqeeee →→ −+−+ not hadrons
( )( )
22 3 qqCq QQNee
qqeeR ==→→
= −+−+
−+
µµσσ
qq−+µµ
qqqq
Nuclear and Particle Physics Franz Muheim 12
e+ee+e-- AnnihilationAnnihilation
Ratio R
Sum is over all quark flavours (u, d, s, c, b, t) kinematically accessible at CoM energy, √s, of collider, and 3 colours (r,g,b) for each flavour
Measurements
R increases in steps with √sR ≈ 3.85 ≈ 11/3 at √s ≥ 10 GeV
Overwhelming evidence for colour√s < 10 GeV -- resonances (c-cbar and b-bar)
( )
( )
( ) bcsdumsR
csdumsR
sdumsR
b
c
s
,,,,3
1131
32
31
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323GeV 10~2
,,,3
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31
31
323GeV 3~2
,,231
31
323GeV 1~2
22222
2222
222
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −+⎟
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=⎟⎟⎠
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=⎟⎟⎠
⎞⎜⎜⎝
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⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
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( )( ) ∑=
→→
= −+−+
−+
q qQee
eeR 23hadronsµµσ
σ
u,d,s,c,bu,d,s,c
u,d,sNo colour
√s
R
Nuclear and Particle Physics Franz Muheim 13
CharmoniumCharmonium
qqee →−+
−+−+ → µµee
−+−+ → eeee
XeeBep −+→Discovery of Charm Quark1974 Brookhaven and SLACNarrow resonance at 3.1 GeVdecays into e+e-, µ+µ-, hadronsdid not fit in existing schemes
J/ψ MesonMass mJ/ψ = 3.1 GeV/c2
Narrow width, smaller thanexperimental resolution Total width Γ = 0.087 MeVLifetime τ = ħ/Γ = 7.6 ·10-21 s= 1000 x expected forstrong interaction process
Branching FractionJ/ψ decays many final states withpartial decay width ΓiTotal decay width
Branching fraction
. )%10.093.5()/()%10.088.5()/(
)%5.07.87()/(
±=→
±=→
±=→
−+
−+
eeJBJB
qqJB
ψ
µµψ
ψ
∑ Γ=Γi i
ΓΓ
= iiB
Nuclear and Particle Physics Franz Muheim 14
CharmoniumCharmonium
Quark Model ExplanationJ/ψ is new quark (c-cbar) bound state
Strong decay for J/ψ (diagram b) is forbiddenby energy conservation at √s =mJ/ψ < 2mDAllowed transition (diagram a) has three gluonsDecay rate suppressed ∝ αS
6
J/ψ =ψ(1S) resonance established quarksas real particles
Excited Charmonium statesFound more statesψ(2S), ψ(3S)e.g.
ψ states - spin J = 1 (like γ)Observe also ηc (J = 0)and P states χc (L = 1)In agreement with QCD potential calculationsSimilar to positronium (e+e-)
−+
−+
→
→
eeJJSψ
πψπψ
//)2(
Nuclear and Particle Physics Franz Muheim 15
Evidence for GluonsEvidence for Gluons
Quarks radiate Gluons2nd order diagram
Experimental SignatureGluons confined, fragmentshadronises into jet
3-jet events
Measurement of αSWhen including gluon radiationadditional factor √αS in matrix elementadds term with factor αS in cross section
e.g R(q2 = (25 GeV)2) ≈ 3.85 > 11/3 → αS = 0.15
gqqee →−+
JADE √s = 35 GeV
( )( ) ⎟
⎠⎞
⎜⎝⎛ +=
→→
= ∑−+−+
−+
πα
µµσσ S2 13hadrons
q qQee
eeR
LEP √s = 91 GeV
Nuclear and Particle Physics Franz Muheim 16
Running of Running of ααSS
Measurementsat many energies √s = 1.5 GeV to 200 GeVe+e- Annihilations
Ratio R ∝ (1+αS/π)Ratio of 3 jet versus 2 jet events ∝ αSEvent shapes - angular distributions
Hadronic collisionsDeep Inelastic scattering Charmonium and UpsilonTau decaysLattice QCD calculations
αS is running αS (MZ) = 0.1187 ± 0.002
e+e- annihilationMany methods
Nuclear and Particle Physics Franz Muheim 17
Evidence for ColourEvidence for Colour
Ratio RDiscussed in previous slides
∆++ BaryonStrong interaction resonance - spin 3/2Quark model explains ∆++ as (uuu)Wave function for (u↑u↑u↑) is symmetricunder interchange of identical quarksAppears to violate Pauli Principle
Led to introduction of colour1964 Greenberg
Antisymmetric colour wave function for baryons
Same arguments for ∆- (ddd) and Ω- (sss)Decay rate π0 → γγ
Γ(π0 → γγ) ∝ N2colour
Measurement: Ncolour = 2.99 ± 0.12
Nuclear and Particle Physics Franz Muheim 18
Elastic Elastic ee--pp ScatteringScattering
e-p → e-pProbe structure of proton with electron beam
KinematicsLaboratory frame, proton at restEnergy and momentum transfer
q2 and ν not independent, E3 and scattering angle θ related, only need to measure E1 and θ
Cross SectionForm factor F(q2)describes deviation from a point chargeF(q2) isFourier transform of charge distributioninside proton,see Nuclear Physics
( )( ) ( ) ( )
022
,0,
,
222
222
24
22
31
<−=⇒=⋅++
=⋅=⋅=+
=−=
ν
νν
νν µ
MqMqpMq
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qqEErr
rqr,ν
( ) 22
point
qFdd
dd
Ω=
Ωσσ
022 <−= νMq
-q2
F(q2)
Nuclear and Particle Physics Franz Muheim 19
Deep Inelastic ScatteringDeep Inelastic Scattering
e-p → e-X ScatteringAt high |q2| proton breaks up into hadrons
q2 and ν independent, hadronic mass Wdefine dimensionless variable x with 0 < x < 1Form factor F(q2) → Structure function F2(ν, q2)
Experimental ResultsInelastic cross section independent of q2
dependent on x → F2 (x)Evidence for point-likeparticles inside proton
PartonsPoint-like constituents inside nucleons Feynman
electron scatters off “free” parton with mass mx is fraction of proton 4-momentum
νMqx
MqpMqW
2
22
22
222
−=
≠⋅++=
2222
222
22
22
20
2MxpxExm
Mm
Mqx
mq
=−=
=−
=⇒=+r
νν
νMqx
2
2−=
-q2
F2
Nuclear and Particle Physics Franz Muheim 20
PartonsPartons
Parton Distribution Functions fi(x)Probability that parton i carries fraction x of particle momentum i = u,d,s (valence quarks), sea quarks, gluonsRequire Quarks carry only 54% of proton momentumGluons carry remaining 46%
Partons arequarks and gluons
Quark- Quark Scattering
2 jet events at p-pbar collider √s = 315 GeV|q2| ≈ 2000 GeV2 see QCD points
QED points are Geiger & Marsden (1911)Rutherford scattering
( )2/sin4
2S
2SS
θασ
αα
∝Ω
⇒
∝
dd
qM
( ) 1=∑∫ dxxfxi
i