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

qq

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

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 →−+

qq

qq

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

qq

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

31

323GeV 10~2

,,,3

1032

31

31

323GeV 3~2

,,231

31

323GeV 1~2

22222

2222

222

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛=>

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛=>

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛=>

( )( ) ∑=

→→

= −+−+

−+

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

MqMqpppq

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