Transcript An Introduction to Relativistic Heavy

Introduction to QCD
adopted from Peter G. Jones
THE UNIVERSITY
OF BIRMINGHAM
Layout
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
1.2
Essential ingredients
Peter G. Jones
• The structure of matter
Proton (uud)
Neutron (udd)
• Fundamental constituents of the Standard Model
Quarks
Leptons
u 5 MeV c 1500 MeV t 180000 MeV
d 10 MeV s 150 MeV b 5000 MeV
m
e
t
n
n
n
e
m
Gauge
bosons
t
Gluon
W±,Z0
Photon
Graviton ?
Table of “bare” quark masses, leptons and gauge bosons
1.3
A brief history ...
Peter G. Jones
1.4
Energy scales
Peter G. Jones
• 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
3 mins
Primordial nucleosynthesis
3105 yrs Radiation and matter decouple - atoms form
200 MeV
5 MeV
1 eV
• Large scale structure
1 bill yrs
3 bill yrs
5 bill yrs
Today
Proto-galaxies and the first stars
Quasars and galaxy spheroids
Galaxy disks
Life !
1.5
Quantum Chromodynamics
Peter G. Jones
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
1.6
Confinement
Peter G. Jones
• The strong interaction potential
– Compare the potential of the strong and electromagnetic interaction
Vem  
q1q2
c

4 0 r
r
Vs  
c
 kr
r
c, c, k constants
– Confining term arises due to the self-interaction property of the colour field
q1
q2
Charges
Gauge boson
Charged
Strength
a) QED or QCD (r < 1 fm)
r
q1
b) QCD (r > 1 fm)
q2
 em 
e2
4 0 c
QED
QCD
electric (2)
g(1)
no
e2
1
em 

4 137
colour (3)
g (8)
yes
 s  0.1  0.2
in MKS units
= c =1 natural units
 0  m0  1 Heaviside- Lorentz units
1.7
Asymptotic freedom - effective charge
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.
dielectric
q
q
d
~ molecular spacing
• Effect in QED
– The “vacuum” is also a polarisable medium.
– Charges are surrounded by virtual e+e- pairs.
Et ~
– Observed charge increases when r < d.
t ~ / mc 2
– Where d is given by the electron Compton wavelength.
C
 ct  / mc
1.8
Asymptotic freedom - the coupling “constant”
Peter G. Jones
• It is more usual to think of coupling strength rather than charge
– and the momentum transfer squared rather than distance.
2Mn  Q2  W 2  M 2
M  initial state mass
n  energy transfer
W  final state mass
Q  momentum transfer
• In both QED and QCD the coupling strength depends on distance.
– In QED the coupling strength is given by:
 
em Q2 

1  3  lnQ
2
m2

em  em
Q2»m2
where  = (Q2  0) = e2/4 = 1/137
– In QCD the coupling strength is given by:


 s Q 
 

2 33  2n f 
2
1   m 
ln Q
s m2
2

 
s
12


m 

which decreases at large Q2 provided nf < 16.
2

Q2 = -q2
1.9
Asymptotic freedom - summary
Peter G. Jones
• 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.
1.10
Quark deconfinement - medium effects
Peter G. Jones
• 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:
r 
1 1
1
V(r)   exp
where rD  3

r
r
n
rD 

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.
1.11
Debye screening
Peter G. Jones
• Modification of Vem - the Mott Transition
a) d > rD
V(r)
V(r)
d
r
V(r)  
b) d < rD
V(r)
d
V(r)
1
r
Unbound electron(s)
r
r 
1
V(r)   exp

r
rD 

1.12
Debye screening in nuclear matter
Peter G. Jones
• 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.
1.13
Chiral symmetry
Peter G. Jones
• 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.
Below the Curie
temperature the
underlying rotational
symmetry is hidden.
Above the Curie
temperature the
rotational symmetry
is restored.
– In the sense that any direction is possible the symmetry is still present.
1.14
Chiral symmetry explained ?
Peter G. Jones
• Chiral symmetry and quark masses ?
Red’s rest frame
Lab frame
a) blue’s velocity > red’s
Blue’s handedness
changes depending
on red’s velocity
Red’s rest frame
Lab frame
b) red’s velocity > blue’s
1.15
Estimating the critical parameters, Tc and c
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.
Temperature
– Two solutions:
Quark-Gluon
1. Phenomenological models
Tc
Plasma
2. Lattice QCD calculations
Hadronic
matter
• Modelling confinement - MIT bag model
Nuclear
matter
rc
Density
– 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)
ig m m  M   M  mq V  0
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.
1.16
The MIT bag model
Peter G. Jones
• Solutions
– Inside the bag, we are left with the free Dirac equation.
– For m = 0 and spherical bag radius R, find solutions:
Ei   i
c
with  i  2.04, 5.40,
R
– 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.
Mn 
c
4 3


 i R B
R i
3
e.g. nucleon ground state is
3 quarks in 1s1/2 level
Mn
c
  2   i  4R2 B  0
R
R i
1/ 4
 c
1 
R

 i 

4

B 
 i
B
1.17
Bag model results
Peter G. Jones
• 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
1.18
Phase transition phenomenology
Peter G. Jones
• The quark-gluon and hadron equations of state
– The energy density of (massless) quarks and gluons is derived from FermiDirac statistics and Bose-Einstein statistics.
p 3dp
 g  2  bp
2 e  1
1
q 
1

p3 dp
2 2 e b  p m   1
 g 
 2T 4
30
natural units
7 2T 4 m 2 T 2 m 4
 q   q 

 2
120
4
8
where m is the quark chemical potential, mq = - mq and b = 1/T.
– Taking into account the number of degrees of freedom
 TOT  16 g  12 q   q 
– 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).
mB = 3 mq
1.19
Estimates of the critical parameters
Peter G. Jones
• High temperature, low density limit - the early universe
– Two terms contribute to the total energy density
 qg  37
– For a relativistic gas:
– For stability:
2
30
T4
1
Pqg   qg
3
Pnet  Pqg  B  0
14
90
Tc   2 B
37 
Pqg
B
MIT bag model
 100  170 MeV
• Low temperature, high density limit - neutron stars
– Only one term contributes to the total energy density
3
 q  2 mq4
2

2
– By a similar argument: mc  2 B

14
 300  500 MeV
~ 2-8 times normal nuclear matter density
given pFermi ~ 250 MeV and r ~ 2m3/32
1.20
The quark-hadron phase diagram
Peter G. Jones
• 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 MeV
Tc (m=0) ≈ 170 MeV
 qg  37
2
T 4  1.3 GeV/fm 3
30
 c  197 MeV fm 
 nucl  0.16 GeV/fm 3
1.21
Lattice QCD
Peter G. Jones
• 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)
• There are two order parameters
1. The Polyakov Loop
2. The Chiral Condensate
L ~ Fq
 ~ mq
• The lattice provides a natural
momentum cut-off
pure gauge = gluons only
pmax  ,
pmin 
a
Ns  a
• Recover the continuum limit by
letting a  0
b  1  2s
1.22
Summary of lecture 1
Peter G. Jones
• 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.
1.23