Introduction to QCD

adopted from Peter G. Jones

THE UNIVERSITYOF BIRMINGHAM

1.2

Peter G. Jones

Layout

• Phase transitions in the earlier universe– The sequence of events t = 10-43-10-5 s after the Big Bang

– Phase transitions in the early universe

– The QCD phase transition is the most recent of these

– It defines the moment when the strong interaction became STRONG

– Is it possible to study this phase transition in the laboratory ?

• Features of QCD– Confinement of quarks (r ~ 1 fm)

– Asymptotic freedom (r 0)

– Quark masses and chiral symmetry

• Phase transition phenomenology– MIT bag model

• Lattice QCD– Estimates of the critical parameters

1.3

Peter G. Jones

Essential ingredients

• The structure of matter

• Fundamental constituents of the Standard Model

electron

nucleus

nucleons

proton

neutron

gluons quarks

1500 MeV150 MeV

u

d s

c t

b

e

e

Quarks

Leptons

5 MeV10 MeV 5000 MeV

180000 MeV

Proton (uud)

Neutron (udd)

Table of “bare” quark masses, leptons and gauge bosons

Gaugebosons

GluonW±,Z0

Photon

Graviton ?

1.4

Peter G. Jones

A brief history ...Tem

pera

ture

(o K)

Time after the Big Bang (seconds)10-9 10-6 10-3 1 103 106 109 1012 1015 1018

1

103

106

109

1012

1015Quark-Gluon Plasma

Hadronization

Nucleosynthesis

Atoms Formed

Now

Acc

eler

ator

s

1.5

Peter G. Jones

Energy scales

• The beginningThe universe is a hot plasma of fundamental particles … quarks, leptons, force particles (and other particles ?)

10-43 s Planck scale (quantum gravity ?) 1019 GeV

10-35 s Grand unification scale (strong and electroweak) 1015 GeV

Inflationary period 10-35-10-33 s

10-11 s Electroweak unification scale 200 GeV

• Micro-structure10-5 s QCD scale - protons and neutrons form 200 MeV

3 mins Primordial nucleosynthesis 5 MeV

3105 yrs Radiation and matter decouple - atoms form 1 eV

• Large scale structure1 bill yrs Proto-galaxies and the first stars

3 bill yrs Quasars and galaxy spheroids

5 bill yrs Galaxy disks

Today Life !

1.6

Peter G. Jones

Quantum Chromodynamics

Important features of QCD

• Confinement– At large distances the effective coupling between quarks is large,

resulting in confinement.

– Free quarks are not observed in nature.

• Asymptotic freedom– At short distances the effective coupling between quarks decreases

logarithmically.

– Under such conditions quarks and gluons appear to be quasi-free.

• (Hidden) chiral symmetry– Connected with the quark masses

– When confined quarks have a large dynamical mass - constituent mass

– In the small coupling limit (some) quarks have small mass - current mass

1.7

Peter G. Jones

Confinement

• The strong interaction potential– Compare the potential of the strong and electromagnetic interaction

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

Vem q1q2

40r

cr Vs

c r

kr

c, c , k constants

em

e2

40c in MKS units

= c =1 natural units

0 0 1 Heaviside- Lorentz units

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

1.8

Peter G. Jones

• Influence of the “vacuum”– In relativistic quantum mechanics, vacuum fluctuations are possible.

– Need to consider interaction with virtual antiparticle-particle pairs.

– Analogy with electric charge in a dielectric medium.

– Introduces the concept of an effective charge.

• Effect in QED– The “vacuum” is also a polarisable medium.

– Charges are surrounded by virtual e+e- pairs.

– Observed charge increases when r < d.– Where d is given by the electron Compton wavelength.

Asymptotic freedom - effective charge

q+-

+-

+-

+ -

+-

+-

+-

q

d ~ molecular spacing

Et ~

t ~ / mc2

C ct / mc

dielectric

1.9

Peter G. Jones

• It is more usual to think of coupling strength rather than charge– and the momentum transfer squared rather than distance.

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

where = (Q2 0) = e2/4 = 1/137

– In QCD the coupling strength is given by:

which decreases at large Q2 provided nf < 16.

Asymptotic freedom - the coupling “constant”

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

W final state mass Q momentum 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

1.10

Peter G. Jones

Asymptotic freedom - summary

• Effect in QCD– Both q-qbar and gluon-gluon loops contribute.

– The quark loops produce a screening effect analogous to e+e- loops in QED

– But the gluon loops dominate and produce an anti-screening effect.

– The observed charge (coupling) decreases at very small distances.

– The theory is asymptotically free quark-gluon plasma !“Superdense Matter: Neutrons or Asymptotically Free Quarks”

J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 (1975) 1353

• Main points– Observed charge is dependent on the distance scale probed.

– Electric charge is conveniently defined in the long wavelength limit (r ).

– In practice em changes by less than 1% up to 1026 GeV !

– In QCD charges can not be separated.

– Therefore charge must be defined at some other length scale.

– In general s is strongly varying with distance - can’t be ignored.

1.11

Peter G. Jones

Quark deconfinement - medium effects

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

and rD is the Debye screening radius.

– Effectively, long range interactions (r > rD) are screened.

• The Mott transition– In condensed matter, when r < electron binding radius

an electric insulator becomes conducting.

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

nucleons (hadrons) cease to exist.

V(r) 1r

1r

exp rrD

where rD

1n3

1.12

Peter G. Jones

Debye screening

• Modification of Vem - the Mott Transition

a) d > rD

b) d < rD

V(r) V(r)

r

V(r) V(r)

r

V(r) 1r

V(r) 1r

exp rrD

d

d

Unbound electron(s)

1.13

Peter G. Jones

Debye screening in nuclear matter

• High (colour charge) densities are achieved by– Colliding heaving nuclei, resulting in:

1. Compression.

2. Heating = creation of pions.

– Under these conditions:1. Quarks and gluons become deconfined.

2. Chiral symmetry may be (partially) restored.

Note: a phase transition is not expected in binary nucleon-nucleon collisions.

Compression + Heating Quark-Gluon Plasma

1.14

Peter G. Jones

Chiral symmetry

• Chiral symmetry and the QCD Lagrangian– Chiral symmetry is a exact symmetry only for massless quarks.

– In a massless world, quarks are either left or right handed

– The QCD Lagrangian is symmetric with respect to left/right handed quarks.

– Confinement results in a large dynamical mass - constituent mass. chiral symmetry is broken (or hidden).

– When deconfined, quark current masses are small - current mass. chiral symmetry is (partially) restored

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

– In the sense that any direction is possible the symmetry is 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.

1.15

Peter G. Jones

Red

’s r

est

fram

e

• Chiral symmetry and quark masses ?a) blue’s velocity > red’s

b) red’s velocity > blue’s

Chiral symmetry explained ?

Lab

fram

e

Red

’s r

est

fram

e

Blue’s handedness changes depending on red’s velocity

Lab

fram

e

1.16

Peter G. Jones

• Mapping out the Nuclear Matter Phase Diagram– Perturbation theory highly successful in applications of QED.– In QCD, perturbation theory is only applicable for very hard processes.– Two solutions:

1. Phenomenological models

2. Lattice QCD calculations

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

of the quarks is small inside the bag (m) and very large outside (M)

where V = 1 inside the bag and 0 outside the bag.

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

Estimating the critical parameters, Tc and c

i M M m V 0

Tc

Hadronicmatter

Quark-GluonPlasma

Nuclear matter

Temperature

Densityc

1.17

Peter G. Jones

The MIT bag model

• Solutions– Inside the bag, we are left with the free Dirac equation.

– For m = 0 and spherical bag radius R, find solutions:

– The MIT group realised that Bogolioubov’s model violated E-p conservation.

– Require an external pressure to balance the internal pressure of the quarks.

– The QCD vacuum acquires a finite energy density, B ≈ 60 MeV/fm3.

– New boundary condition, total energy must be minimised wrt the bag radius.

Ei i

cR

with i 2.04, 5.40,

Mn cR

i 43i

R3B

Mn

R

c

R2 i 4i

R2B 0

R c4

i1Bi

1/ 4

e.g. nucleon ground state is 3 quarks in 1s1/2 level

B

1.18

Peter G. Jones

Bag model results

• Refinements– Several refinements are

needed to reproduce the spectrum of low-lying hadrons

e.g. allow quark interactions

– Fix B by fits to several hadrons

• Estimates for the bag constant– Values of the bag constant

range from B1/4 = 145-235 MeV

• Results– Shown for B1/4 = 145 MeV and s

= 2.2 and ms = 279 MeV

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

1.19

Peter G. Jones

Phase transition phenomenology

• The quark-gluon and hadron equations of state– The energy density of (massless) quarks and gluons is derived from Fermi-

Dirac statistics and Bose-Einstein statistics.

where is the quark chemical potential, q = - q and = 1/T.

– Taking into account the number of degrees of freedom

– Consider two extremes:

1. High temperature, low net baryon density (T > 0, B = 0).

2. Low temperature, high net baryon density (T = 0, B > 0).

g 1

2 2p3dp

ep 1 g

2T 4

30

q 1

2 2p3dp

e p 1 q q

7 2T 4

120

2T 2

4

4

8 2

TOT 16g 12 q q

natural units

B = 3 q

1.20

Peter G. Jones

Estimates of the critical parameters

• High temperature, low density limit - the early universe– Two terms contribute to the total energy density

– For a relativistic gas:

– For stability:

• Low temperature, high density limit - neutron stars– Only one term contributes to the total energy density

– By a similar argument:

qg 37 2

30T 4

Pqg 1

3qg

Pnet Pqg B 0

Tc 90

37 2 B

1 4

100 170 MeV

MIT bag model

B

Pqg

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

1.21

Peter G. Jones

The quark-hadron phase diagram

• Phase transition phenomenology– Compare and ideal of q+g (2 flavour +

3 colour) with an ideal hadron gas composed of pions (-, 0, +)

– State of higher pressure is stable against the state of lower pressure

– Phases co-exist when the pressure is the same in both phases

– Note: phase transition is first order by construction

Taking B1/4 ≈ 235 MeVTc (=0) ≈ 170 MeV

qg 37 2

30T4 1.3 GeV/fm3

c 197 MeVfm

nucl 0.16 GeV/fm3

1.22

Peter G. Jones

Lattice QCD

• Quarks and gluons are studied on a discrete space-time lattice

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

• The lattice provides a natural momentum cut-off

• Recover the continuum limit by letting a 0

• There are two order parameters

aa

Ns3 N

pmax

a

, pmin

Ns a

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

pure gauge = gluons only

1 s2

1.23

Peter G. Jones

Summary of lecture 1

• QCD is an asymptotically free theory.

• In addition, long range forces are screened in a dense medium.

• QCD possess a hidden (chiral) symmetry.

• Expect one or perhaps two phase transitions connected with deconfinement and partial chiral symmetry restoration.

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

• MIT bag model provides a phenomenological description of confinement.

• Thermodynamics of ideal gas of quarks and gluons plus the bag constant give an estimate of the critical parameters.

• More detailed estimates are obtained from lattice QCD calculations.

• The critical energy density should be in reach of modern-day particle accelerators.