Quantum Chromodynamics (QCD)

Main features of QCDMain features of QCD ConfinementConfinement

At large distances the effective coupling between quarks is large, resulting At large distances the effective coupling between quarks is large, resulting in confinement.in confinement.

Free quarks are not observed in nature.Free quarks are not observed in nature. Asymptotic freedomAsymptotic freedom

At short distances the effective coupling between quarks decreases At short distances the effective coupling between quarks decreases logarithmically.logarithmically.

Under such conditions quarks and gluons appear to be quasi-free.Under such conditions quarks and gluons appear to be quasi-free. (Hidden) chiral symmetry(Hidden) chiral symmetry

Connected with the quark massesConnected with the quark masses When confined quarks have a large dynamical mass - constituent massWhen confined quarks have a large dynamical mass - constituent mass In the small coupling limit (some) quarks have small mass - current massIn the small coupling limit (some) quarks have small mass - current mass

Confinement The strong interaction potentialThe strong interaction potential

Compare the potential of the strong & e.m. interactionCompare the potential of the strong & e.m. interaction

Confining term arises due to the self-interaction property of the Confining term arises due to the self-interaction property of the colour fieldcolour field

Vem q1q2

40r

cr

Vs c

r kr c, c , k constants

QED QCDCharges electric (2) colour (3)Gauge boson (1) g (8)Charged no yesStrength em

e2

4

1137

s 0.1 0.2

q1 q2

q1 q2

a) QED or QCD (r < 1 fm)

b) QCD (r > 1 fm)

r

It is more usual to think of coupling strength rather than chargeIt is more usual to think of coupling strength rather than chargeand the momentum transfer squared rather than distance.and the momentum transfer squared rather than distance.

In both QED and QCD the coupling strength depends on distance.In both QED and QCD the coupling strength depends on distance. In QED the coupling strength is given by:In QED the coupling strength is given by:

where where = = ((QQ2 2 0) 0) = = ee22/4/4 = 1/137= 1/137 In QCD the coupling strength is given by:In QCD the coupling strength is given by:

which decreases at large which decreases at large QQ22 provided provided nnff < 16 < 16..

Asymptotic freedom - the coupling “constant”

2M Q2 W 2 M2 M initial state mass energy transfer

W final state mass Qmomentum transfer

em Q2 1 3 ln Q2 m2

s Q2 s 2

1 s 2 33 2n f

12

ln Q2 2

Q2»m2

Q2 = -q2

e e

Asymptotic freedom - summary Effect in QCDEffect in QCD

Both q-qbar and gluon-gluon loops contribute.Both q-qbar and gluon-gluon loops contribute. The quark loops produce a screening effect analogous to eThe quark loops produce a screening effect analogous to e++ee-- loops in QED loops in QED But the gluon loops dominate and produce an anti-screening effect.But the gluon loops dominate and produce an anti-screening effect. The observed charge (coupling) decreases at very small distances.The observed charge (coupling) decreases at very small distances. The theory is asymptotically free The theory is asymptotically free quark-gluon plasma ! quark-gluon plasma !

““Superdense Matter: Neutrons or Asymptotically Free Quarks”Superdense Matter: Neutrons or Asymptotically Free Quarks”J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 (1975) 1353 J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 (1975) 1353

Main pointsMain points Observed charge is dependent on the distance scale probed.Observed charge is dependent on the distance scale probed. Electric charge is conveniently defined in the long wavelength limit (Electric charge is conveniently defined in the long wavelength limit (r r ).). In practice In practice emem changes by less than 1% up to 10 changes by less than 1% up to 102626 GeV ! GeV ! In QCD charges can not be separated.In QCD charges can not be separated. Therefore charge must be defined at some other length scale.Therefore charge must be defined at some other length scale. In general In general ss is strongly varying with distance - can’t be ignored. is strongly varying with distance - can’t be ignored.

Quark deconfinement - medium effects Debye screeningDebye screening

In bulk media, there is an additional charge screening effect.In bulk media, there is an additional charge screening effect. At high charge density, At high charge density, nn, the short range part of the potential becomes:, the short range part of the potential becomes:

and and rrDD is the Debye screening radius. is the Debye screening radius. Effectively, long range interactions (Effectively, long range interactions (r r >> r rDD) are screened.) are screened.

The Mott transitionThe Mott transition In condensed matter, when In condensed matter, when rr < < electron binding radiuselectron binding radius

an electric insulator becomes conducting.an electric insulator becomes conducting. Debye screening in QCDDebye screening in QCD

Analogously, think of the quark-gluon plasma as a colour conductor.Analogously, think of the quark-gluon plasma as a colour conductor. Nucleons (all hadrons) are colour singlets (qqq, or qqbar states).Nucleons (all hadrons) are colour singlets (qqq, or qqbar states). At high (charge) density quarks and gluons become unbound.At high (charge) density quarks and gluons become unbound.

nucleons (hadrons) cease to exist.nucleons (hadrons) cease to exist.

V(r)1r

1r

exp rrD

where rD

1n3

Debye screening in nuclear matter High (color charge) densities are achieved byHigh (color charge) densities are achieved by

Colliding heaving nuclei, resulting in:Colliding heaving nuclei, resulting in:1. Compression.1. Compression.2. Heating = creation of pions.2. Heating = creation of pions.

Under these conditions:Under these conditions:1. Quarks and gluons become deconfined.1. Quarks and gluons become deconfined.2. Chiral symmetry may be (partially) restored.2. Chiral symmetry may be (partially) restored.

Note: a phase transition is Note: a phase transition is notnot expected in binary nucleon-nucleon collisions. expected in binary nucleon-nucleon collisions.

The temperature inside a heavy ion collision at RHIC can exceed The temperature inside a heavy ion collision at RHIC can exceed

1000 billion degrees !! (about 10,000 times the temperature of the sun)1000 billion degrees !! (about 10,000 times the temperature of the sun)

Chiral symmetry Chiral symmetry and the QCD LagrangianChiral symmetry and the QCD Lagrangian

Chiral symmetry is a exact symmetry only for massless quarks.Chiral symmetry is a exact symmetry only for massless quarks. In a massless world, quarks are either left or right handed In a massless world, quarks are either left or right handed The QCD Lagrangian is symmetric with respect to left/right handed quarks.The QCD Lagrangian is symmetric with respect to left/right handed quarks. Confinement results in a large dynamical mass - constituent mass.Confinement results in a large dynamical mass - constituent mass.

chiral symmetry is broken (or hidden).chiral symmetry is broken (or hidden). When deconfined, quark current masses are small - current mass.When deconfined, quark current masses are small - current mass.

chiral symmetry is (partially) restoredchiral symmetry is (partially) restored

Example of a hidden symmetry restored at high temperatureExample of a hidden symmetry restored at high temperature Ferromagnetism - the spin-spin interaction is rotationally invariant.Ferromagnetism - the spin-spin interaction is rotationally invariant.

In the sense that any direction is possible the symmetry In the sense that any direction is possible the symmetry isis still present. still present.

T < Tc T > Tc

Below the Curie temperature the underlying rotational symmetry is hidden.

Above the Curie temperature the rotational symmetry is restored.

Chiral symmetry and quark masses ?Chiral symmetry and quark masses ?

a) blue’s velocity > red’sa) blue’s velocity > red’s

b) red’s velocity > blue’sb) red’s velocity > blue’s

Chiral symmetry explained ?

Blue’s handedness changes depending on red’s velocity

Red

’s r

est

fram

e

Lab

fram

e

Red

’s r

est

fram

e

Lab

fram

e

Modelling confinement: The MIT bag model

Modelling confinement - MIT bag modelModelling confinement - MIT bag model Based on the ideas of Bogolioubov (1967).Based on the ideas of Bogolioubov (1967). Neglecting short range interactions, write the Dirac equation so that the Neglecting short range interactions, write the Dirac equation so that the

mass of the quarks is small inside the bag (mass of the quarks is small inside the bag (mm) and ) and veryvery large outside large outside ((MM))

Wavefunction vanishes outside the bag if Wavefunction vanishes outside the bag if MM and satisfies a linear boundary condition at the bag surface.and satisfies a linear boundary condition at the bag surface.

SolutionsSolutions Inside the bag, we are left with the free Dirac equation.Inside the bag, we are left with the free Dirac equation. The MIT group realised that Bogolioubov’s model violated The MIT group realised that Bogolioubov’s model violated EE--pp

conservation.conservation. Require an external pressure to balance the internal pressure of the Require an external pressure to balance the internal pressure of the

quarks.quarks. The QCD vacuum acquires a finite energy density, The QCD vacuum acquires a finite energy density, BB ≈ 60 MeV/fm ≈ 60 MeV/fm33.. New boundary condition, total energy must be minimised wrt the bag New boundary condition, total energy must be minimised wrt the bag

radius.radius.B

Confinement Represented by Bag Model

Bag Model of Hadrons

Comments on Bag Model

Bag model results

RefinementsRefinements Several refinements are needed Several refinements are needed

to reproduce the spectrum of to reproduce the spectrum of low-lying hadronslow-lying hadrons

e.g. allow quark interactionse.g. allow quark interactions Fix Fix BB by fits to several hadrons by fits to several hadrons

Estimates for the bag constantEstimates for the bag constant Values of the bag constant Values of the bag constant

range from range from BB11/4/4 = 145-235 MeV = 145-235 MeV ResultsResults

Shown for Shown for BB11/4/4 = 145 MeV = 145 MeV and and ss = 2.2 = 2.2 and and mmss = 279 MeV = 279 MeV

T. deGrand et al, Phys. Rev. D 12 (1975) 2060

Summary of QCD input QCD is an asymptotically free theory.QCD is an asymptotically free theory. In addition, long range forces are screened in a dense In addition, long range forces are screened in a dense

medium.medium. QCD possess a hidden (chiral) symmetry.QCD possess a hidden (chiral) symmetry. Expect one or perhaps two phase transitions connected Expect one or perhaps two phase transitions connected

with deconfinement and partial chiral symmetry with deconfinement and partial chiral symmetry restoration.restoration.

pQCD calculations can not be used in the confinement pQCD calculations can not be used in the confinement limit.limit.

MIT bag model provides a phenomenological description MIT bag model provides a phenomenological description of confinement.of confinement.

Still open questions in the Standard Model

Chirality: Why Resonances ?

2212

21 ppEEminv

Bubble chamber, BerkeleyM. Alston (L.W. Alvarez) et al., Phys. Rev. Lett. 6 (1961) 300.

Invariant mass (K0+) [MeV/c2]

K*-(892)

640 680 720 760 800 840 880 920

Nu

mb

er

of

even

ts

0

2

4

6

8

10

Luis Walter Alvarez 1968 Nobel Prize for

“ resonance particles ” discovered 1960

K* from K-+p collision system Kp p

K

Resonances are:

• Excited state of a ground state particle.• With higher mass but same quark content.• Decay strongly short life time (~10-23 seconds = few fm/c ), width = natural spread in energy: = h/t. Breit-Wigner shape

• Broad states with finite and t, which can be formed by collisions between the particles into which they decay.

Why Resonances?:• Surrounding nuclear medium may change resonance properties• Chiral symmetry breaking: Dropping mass -> width, branching ratio

Strange resonances in medium

Short life time [fm/c] K* < *< (1520) < 4 < 6 < 13 < 40

Red: before chemical freeze outBlue: after chemical freeze out

Medium effects on resonance and their decay products before (inelastic) and after chemical freeze out (elastic).

Rescattering vs. Regeneration ?

dNll

dtd3xNcN f

2 e f

e

2 d3 p1d3 p2

2 6f 1

N f

f E1 f E2 M v12

The momentum distributions f(E1) and f(E2) depend on the thermodynamics of the plasma.

The cross-section for the sub-process (M) is calculable in pQCD.

Dilepton production in the QGPDilepton production in the QGP

The production rate (and invariant mass distribution) depends on the momentum distribution of q-qbar in the plasma.The production rate (and invariant mass distribution) depends on the momentum distribution of q-qbar in the plasma.

Reconstruct the invariant mass, Reconstruct the invariant mass, MM, of the dilepton pair’s hypothetical parent., of the dilepton pair’s hypothetical parent.

Dilepton production from hadronic mechanismsDilepton production from hadronic mechanisms

1. Drell-Yan1. Drell-Yan high Masshigh Mass

2. Annihilation and Dalitz decays2. Annihilation and Dalitz decays low Masslow Mass

3. Resonance decays3. Resonance decays discretediscrete

4. Charmed meson decays4. Charmed meson decays low Masslow Mass

Electromagnetic probes - dileptons

q

q

l+

l-

qq ll

ll ll , , and J/

D l X

CERES low-mass e+e– mass spectrumAlmost final results from the 2000 run Pb+Au at 158 GeV per nucleon

comparison to the hadron decay cocktail

Enhancement over hadron decay cocktail

for mee > 0.2 GeV:

2.430.21 (stat)

for 0.2 GeV<mee< 0.6 GeV:2.80.5 (stat)

• Absolutely normalized spectrum

• Overall systematic uncertainty of

normalization: 21%

NA60 Low-mass dimuons

, and even peaks clearly visible in dimuon channel

Net data sample: 360 000 events

Mass resolution:23 MeV at the position

Real

data !

Superb data!!!

Deconfinement at Initial Temperature

Thermometer for early stages: Tdis((2S)) < Tdis((3S))< Tdis(J/) Tdis((2S)) < Tdis((1S))

RHIC s = 200 GeV

Matsui & Satz (1986): (Phys. Lett. B178 (1986) 416)Color screening of heavy quarks in QGP leads to heavy resonance dissociation.

The suppression of heavy quark states signature of deconfinement at QGP.Jcc-bar) e+ +e,

(bb-bar) e+ +e, + +

Decay modes:c J/ + b +

Tota

l bott

om

/ c

harm

pro

duct

ion

Melting at SPS

Lattice QCD:SPS TI ~ 1.3 TcRHIC TI ~ 2 Tc

J/ suppression Charmonium productionCharmonium production

The The JJ// is a c-cbar bound state (analogous to positronium) is a c-cbar bound state (analogous to positronium) Produced only during the initial stages of the collisionProduced only during the initial stages of the collision

Thermal production is negligible due to the large c quark massThermal production is negligible due to the large c quark mass

Charmonium suppression (Debye screening)Charmonium suppression (Debye screening) Semi-classically (Semi-classically (EE = = T T + + VV))

Differentiate with respect to Differentiate with respect to rr to find minimum (bound state) to find minimum (bound state) Find there is no bound state ifFind there is no bound state if

For For s s = 0.52= 0.52 and and TT = 200 MeV = 200 MeV, , rrDD((pQCDpQCD) = 0.36 fm) = 0.36 fm

Compare with Compare with rrBohrBohr = 0.41 fm ( = 0.41 fm (setting setting rrDD above above))

Conclusion: the Conclusion: the JJ// is not bound in the plasma under these conditions is not bound in the plasma under these conditions

E(r)p2

2se

r / rD

r

p2 ~ 1 r2

mc 2

rD 1

0.84s

qq cc

mc 1500 MeV TQGP

rD pQCD 2

9s

1

T

Onium physics – the complete program

Melting of quarkonium states (Deconfinement TMelting of quarkonium states (Deconfinement TCC))

TTdissdiss((’) < T’) < Tdissdiss(((3S)) < T(3S)) < Tdissdiss(J/(J/) ) T Tdissdiss(((2S))(2S)) < <

TTdissdiss(((1S))(1S))

Future Measurements:

Resonance Response to Medium

Shuryak QM04

part

on

s

had

ron

s

Baryochemical potential (Density)

Temperature

Quark Gluon Plasma

Hadron Gas

Resonances below and above Tc:Resonances below and above Tc:

Gluonic bound statesGluonic bound states (e.g. Glueballs) Shuryak (e.g. Glueballs) Shuryak

hep-ph/0405066hep-ph/0405066

Deconfinement: Determine range of T Deconfinement: Determine range of T initial.initial.

J/J/andand state dissociationstate dissociation

Chiral symmetry restorationChiral symmetry restoration Mass and width of resonances Mass and width of resonances ( e.g. ( e.g. leptonic vs hadronic decay, leptonic vs hadronic decay,

chiral partners chiral partners and aand a11))

Hadronic time evolutionHadronic time evolution Hadronisation (chemical freeze-out)Hadronisation (chemical freeze-out) till kinetic freeze-out.till kinetic freeze-out.

Deconfinement: Melting of J/

J/

L

Projectile

Target

4.18 0.35mbJabs

J/ normal nuclear

absorption curve

Interaction length

SPS RHIC

J/ suppression at SPS and RHIC are the sameStrong signal for deconfinement in QGP phase RHIC has higher initial temperature Expect stronger J/ suppression Partonic recombination of J/

centcent NNparpar

ttNNcolcol

ll

339339 10410499

222222 590590

40-40-50%50%

6464 108108

2020 2222

2.82.8 2.22.2

Chiral Symmetry Restoration

Ralf Rapp (Texas A&M)Ralf Rapp (Texas A&M) J.Phys. G31 (2005) S217-S230 J.Phys. G31 (2005) S217-S230

Vacuum At Tc: Chiral Restoration

Measure chiral partnersNear critical temperature Tc (e.g. and a1)

Data: ALEPH Collaboration R. Barate et al. Eur. Phys. J. C4 409 (1998)

a1 +

Resonance Reconstruction in STAR TPC

Energy loss in TPC dE/dx

momentum [GeV/c]

dE/dx

p

K

e(1385)

-

-

p

(1520)

K- p

K(892) + K

(1020) K + K

(1520) p + K

(1385) + +

End view STAR TPC

• Identify decay candidates (p, dedx, E)• Calculate invariant mass

Invariant Mass Reconstruction in p+p

(1520)

STAR Preliminary

(1520)

— original invariant mass histogram from K- and p combinations in same event.— normalized mixed event histogram from K- and p combinations from different events. (rotating and like-sign background)

Extracting signal:After Subtraction of mixed event background from original event and fitting signal (Breit-Wigner).

2212

21 ppEEminv

Invariant mass:

Resonance Signal in p+p collisionsSTAR Preliminary

Statistical error only

K(892)

(1385)

STAR Preliminary

STAR Preliminary

ΦK+K-

p+p

STAR Preliminary

p+p

Δ++

Invariant Mass (GeV/c2)

Resonance Signal in Au+Au collisions

STAR Preliminary

Au+Au minimum biaspT 0.2

GeV/c

|y| 0.5

K*0 + K*0

(1520)

STAR Preliminary

(1020)

STAR Preliminary

*± +*±

K(892)

Mapping out the Nuclear Matter Phase DiagramMapping out the Nuclear Matter Phase DiagramPerturbation theory highly successful in Perturbation theory highly successful in

applications of QED.applications of QED. In QCD, perturbation theory is only applicable In QCD, perturbation theory is only applicable

for very hard processes.for very hard processes.Two solutions:Two solutions:

1. Phenomenological models1. Phenomenological models2. Lattice QCD calculations2. Lattice QCD calculations

Estimating the critical parameters, Tc and c

Lattice QCDQuarks and gluons are Quarks and gluons are

studied on a discrete studied on a discrete space-time lattice space-time lattice

Solves the problem of Solves the problem of divergences in pQCD divergences in pQCD calculations (which arise calculations (which arise due to loop diagrams)due to loop diagrams)

There are two order There are two order parametersparameters

aa

Ns3 N

1. The Polyakov Loop L ~ Fq2. The Chiral Condensate ~ mq

(F. Karsch, hep-lat/9909006)

/T4

T/Tc

Lattice Results Tc(Nf=2)=1738 MeVTc(Nf=3)=1548 MeV

0.5 4.5 15 35 GeV/fm375

T = 150-200 MeV ~ 0.6-1.8 GeV/fm3

Lattice QCD: the latest news(critical parameters at finite baryon density)

Phenomenology I: Phase transition The quark-gluon and hadron equations of stateThe quark-gluon and hadron equations of state

The energy density of (massless) quarks and gluons is derived from Fermi-Dirac The energy density of (massless) quarks and gluons is derived from Fermi-Dirac statistics and Bose-Einstein statistics.statistics and Bose-Einstein statistics.

where where is the quark chemical potential, is the quark chemical potential, qq = - = - qq and and = 1/ = 1/TT.. Taking into account the number of degrees of freedomTaking into account the number of degrees of freedom

Consider two extremes:Consider two extremes:

1. High temperature, low net baryon density (1. High temperature, low net baryon density (TT > 0, > 0, BB = 0 = 0).).

2. Low temperature, high net baryon density (2. Low temperature, high net baryon density (TT = 0, = 0, BB > 0 > 0).).

g 1

2 2p3dp

ep 1 g

2T 4

30

q 1

2 2p3dp

e p 1 q q

7 2T 4

1202T 2

44

8 2

TOT 16g 12 q q

B = 3 q

Phenomenology II: critical parameters High temperature, low density limit - the High temperature, low density limit - the

early universeearly universe Two terms contribute to the total energy Two terms contribute to the total energy

densitydensity

For a relativistic gas:For a relativistic gas:

For stability:For stability:

Low temperature, high density limit - neutron Low temperature, high density limit - neutron starsstars Only one term contributes to the total Only one term contributes to the total

energy densityenergy density

By a similar argument:By a similar argument:

qg 37 2

30T 4

Pqg 1

3qg

Pnet Pqg B0

Tc 90

37 2 B

1 4

100 170 MeV

q 3

2 2 q4

c 2 2B 1 4300 500 MeV

~ 2-8 times normal nuclear matter densitygiven pFermi ~ 250 MeV and ~ 23/32

How to create a QGP ?energy = temperature & density =

pressure