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Introduction to QCD. adopted from Peter G. Jones. THE UNIVERSITY OF BIRMINGHAM. 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

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introduction to qcd

Introduction to QCD

adopted from Peter G. Jones



  • 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
essential ingredients
Essential ingredients
  • The structure of matter
  • Fundamental constituents of the Standard Model

Proton (uud)

Neutron (udd)




Graviton ?


5 MeV


1500 MeV


180000 MeV





10 MeV


150 MeV


5000 MeV











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

energy scales
Energy scales
  • The beginning

The 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-structure

10-5 s QCD scale - protons and neutrons form 200 MeV

3 mins Primordial nucleosynthesis 5 MeV

3105 yrs Radiation and matter decouple - atoms form 1 eV

  • Large scale structure

1 bill yrs Proto-galaxies and the first stars

3 bill yrs Quasars and galaxy spheroids

5 bill yrs Galaxy disks

Today Life !

quantum chromodynamics
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


Charges electric (2) colour (3)

Gauge boson g (1) g (8)

Charged no yes




  • 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

a) QED or QCD (r < 1 fm)




b) QCD (r > 1 fm)

asymptotic freedom effective charge
Asymptotic freedom - effective charge
  • 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.


















d ~ molecular spacing

asymptotic freedom the coupling constant
Asymptotic freedom - the coupling “constant”
  • 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 a = a(Q2  0) = e2/4p= 1/137

    • In QCD the coupling strength is given by:

which decreases at large Q2 provided nf < 16.

em  em


Q2 = -q2

asymptotic freedom summary
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 aem 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 as is strongly varying with distance - can’t be ignored.
quark deconfinement medium effects
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.

debye screening
Debye screening
  • Modification of Vem - the Mott Transition

a) d > rD

b) d < rD








Unbound electron(s)


debye screening in nuclear matter
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.

chiral symmetry
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.

Below the Curie temperature the underlying rotational symmetry is hidden.

Above the Curie temperature the rotational symmetry is restored.

chiral symmetry explained
Chiral symmetry explained ?
  • Chiral symmetry and quark masses ?

a) blue’s velocity > red’s

b) red’s velocity > blue’s

Red’s rest frame

Lab frame

Blue’s handedness changes depending on red’s velocity

Lab frame

Red’s rest frame

estimating the critical parameters t c and e c
Estimating the critical parameters, Tc and ec
  • 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 qV= 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.







Nuclear matter



the mit bag model


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.

e.g. nucleon ground state is

3 quarks in 1s1/2 level

bag model results
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 as = 2.2 and ms = 279 MeV

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

phase transition phenomenology
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 m is the quark chemical potential, mq = - mq and b = 1/T.

    • Taking into account the number of degrees of freedom
    • Consider two extremes:

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

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

natural units

mB = 3 mq

estimates of the critical parameters



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:

MIT bag model

~ 2-8 times normal nuclear matter density

given pFermi ~ 250 MeV and r ~ 2m3/3p2

the quark hadron phase diagram
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 (p-, p0, p+)
    • 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 MeV

Tc (m=0) ≈ 170 MeV

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

Lattice QCD
  • There are two order parameters

pure gauge = gluons only

summary of lecture 1
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.