Transcript QCD – from the vacuum to high temperature

QCD – from the vacuum to
high temperature
an analytical approach
3 lectures
• Condensates and phase diagram
• Higgs picture of QCD
• Functional renormalization group
Condensates and
Phase Diagram
Action and functional integral
Generating functionals
Chiral symmetry
Condensates and symmetry breaking
Phase transition
Has Tc been measured?
Understanding the phase diagram
Cosmological phase transition…
…when the universe cools below 175 MeV
10-5 seconds after the big bang
QCD at high density
Nuclear matter
Heavy nuclei
Neutron stars
Quark stars …
Why analytical approach ?
Complementary to numerical investigations
(lattice – QCD )
Many questions cannot be addressed by
lattice - QCD
QCD
Quantum numbers
quark
quark
gluon
meson baryon
particle u
d
g
π+
p
electric 2/3
charge
isospin 1/2
-1/3
0
1
1
-1/2
0
1
1/2
color
3
8
1
1
3
Light particles
mesons
-0
+
0
+
0
• pseudoscalar octet: (π ,π ,π ,K ,K ,K ,K ,η)
• vector meson octet + …
baryons
• octet : (p,n,Λ,Σ+,Σ-,Σ0,Ξ0,Ξ-)
• decuplet + …
quarks and gluons
at high momenta
• Visible as jets at high virtual momenta
• Perturbation theory valid
Short and long distance
degrees of freedom
are different !
Short distances : quarks and gluons
Long distances : baryons and mesons
How to make the transition?
confinement
Action
+
QCD is microscopic theory of
strong interactions
Local gauge symmetry
Running coupling : QCD
effective gauge coupling
depends on
momentum scale μ
Generating functionals,
Effective action
Partition function and
functional integral
Integration over all configurations of gauge field A and quark fields ψ
Fermion fields are anticommuting Grassmann variables
This is the difficult part ! Needs regularisation !
Lattice QCD ; gauge fixing,ghosts,Slavnov Taylor identities …
Correlation functions
Only gauge invariant correlation functions can differ from zero.
Local gauge symmetries are not spontaneously broken
( exact proof )
• Scalar correlation functions
describes propagation of (pseudo)scalar mesons
in suitable channel
• Vector meson correlation function
Generating functionals
• Add to the action a source term
• Partition function becomes a functional of the
sources, Z[j]. Can be evaluated for arbitrary j(x)
Generating functional and correlation functions
Concentrate on scalars –
other channels can be treated in complete analogy
• Order parameter
• Correlation function
Generating functional for
connected Green’s functions
W[ j ]=ln Z[ j ]
Gc : propagator or
connected two point function
Effective action
Effective action is defined by Legendre transform
Matrix of second functional derivatives Γ(2) is the
inverse propagator
Physical sources
With this convention for physical sources the quark
mass term is not included in S anymore.
S: action for Nf massless quarks
Γ has the full chiral symmetry
“Solution” of QCD
Effective action ( for suitable fields ) contains all
the relevant information of the solution of QCD
Gauge singlet fields, low momenta:
Order parameters, meson-( baryon- ) propagators
Gluon and quark fields, high momenta:
Perturbative QCD
Aim: Computation of effective action
Symmetries
• If functional measure preserves the symmetries
•
of S :
Γ has the same symmetries as S
If not : anomaly
Examples for symmetries of Γ :
- gauge symmetry,
- global chiral flavor symmetry SU(Nf )xSU(Nf )
Example for anomaly: global axial U(1)
Chiral symmetry breaking
Chiral symmetry
For massless quarks the action of QCD is
invariant under independent phase rotations
acting on left handed and right handed
quarks.
SU(Nf )L x SU(Nf )R
Quark masses
• Quark mass terms mix left handed and right
handed quarks
• They break the axial symmetry
• Equal quark masses leave vectorlike “diagonal”
•
SU(N) unbroken.
Quark mass terms can be treated as sources for
scalar quark-antiquark bilinears.
Effective potential
Evaluate effective action
for homogenous field φ
Spontaneous symmetry breaking
SYM
<φ>=0
SSB
<φ>=φ ≠ 0
0
Pions
Two-flavor QCD , σ : 2x2 matrix
Effective potential
Spontaneous chiral symmetry breaking
For sufficiently small m² and positive λ1,2 :
( even for vanishing quark masses )
Goldstone bosons
Flat directions in potential,
dictated by symmetry breaking
massless (pseudo)scalar fields : pions
Presence of quark mass term shifts
minimum and gives mass to pions
Nonlinear σ - model
Nonlinear description
(neglects the scalar excitations beyond π,η)
Kinetic term
Chiral perturbation theory
Meson fluctuations
• Insert factor unity
Hubbard,Stratonovich
• New action and functional integral involves also σ - field
σ - interactions
• Four fermion interaction can cancel
•
corresponding piece induced by loops in QCD
Typical form
• Mass and kinetic term for σ
• Yukawa interaction between σ and quarks
• Source now for σ
Connection with quark bilinears
Effective action with scalar fluctuations
QCD phase transition
Phase diagram
<φ>=0
<φ>=φ0 ≠ 0
R.Pisarski
Chiral symmetry restoration
at high temperature
Low T
SSB
<φ>=φ ≠ 0
0
High T
SYM
<φ>=0
at high T :
less order
more symmetry
examples:
magnets, crystals
QCD at high temperature
• Quark – gluon plasma
• Chiral symmetry restored
• Deconfinement ( no linear heavy quark
potential at large distances )
• Lattice simulations : both effects happen
at the same temperature
Lattice results
e.g. Karsch,Laermann,Peikert
Critical temperature in chiral limit :
Nf = 3 : Tc = ( 154 ± 8 ) MeV
Nf = 2 : Tc = ( 173 ± 8 ) MeV
Chiral symmetry restoration and deconfinement at
same Tc
QCD – phase transition
Quark –gluon plasma
• Gluons : 8 x 2 = 16
• Quarks : 9 x 7/2 =12.5
• Dof :
28.5
Chiral symmetry
Hadron gas
• Light mesons : 8
• (pions :
3)
• Dof :
8
Chiral sym. broken
Large difference in number of degrees of freedom !
Strong increase of density and energy density at Tc !
Pressure
Analytical description of
phase transition
• Needs model that can account
simultaneously for the correct degrees of
freedom below and above the transition
temperature.
• Partial aspects can be described by more
limited models, e.g. chiral properties at
small momenta.
Quark descriptions ( NJL-model ) fail to describe
the high temperature and high density phase
transitions correctly
High T : chiral aspects could be ok , but glue …
(pion gas to quark gas )
High density transition : different Fermi surface for
quarks and baryons ( T=0)
– in mean field theory factor 27 for density at given
chemical potential –
Confinement is important : baryon enhancement
Berges,Jungnickel,…
Chiral perturbation theory even less complete
Universe cools below 170 MeV…
Both gluons and quarks disappear from
thermal equilibrium : mass generation
Chiral symmetry breaking
mass for fermions
Gluons ?
Analogous situation in electroweak phase
transition understood by Higgs mechanism
Higgs description of QCD vacuum ?
Order of the phase transition is
crucial ingredient for experiments
( heavy ion collisions )
and cosmological phase transition
Order of
the
phase
transition
Second order phase transition
First order phase transition
Matsubara formalism
• Only change for T ≠ 0 :
• Euclidean time on torus with
circumference β =1/T
Matsubara formalism for fermions
Thermodynamic potentials
and observables
Evaluate effective potential U at minimum
• free energy
• pressure
• energy density
vanishing chemical potential and sources
Thermodynamics
Effective action , when evaluated on torus
for euclidean time with circumference 1/T,
(and in presence of chemical potential )
contains full information about
thermodynamic behavior of QCD.
Has Tc been measured ?
• Observation : statistical distribution of hadron species with
“chemical freeze out temperature “
Tch=176 MeV
• Tch cannot be much smaller than Tc : hadronic rates for
T< Tc are too small to produce multistrange hadrons (Ω,..)
• Only near Tc multiparticle scattering becomes important
( collective excitations …) – proportional to high power of
density
Tch≈Tc
P.Braun-Munzinger,J.Stachel,CW
Heavy ion collision
Hadron abundancies
Very rapid density increase
…in vicinity of critical temperature
Extremely rapid increase of rate of
multiparticle scattering processes
( proporional to very high power of density )
Energy density
• Lattice
simulations
Karsch et al
Production time for Ω
multi-meson
scattering
π+π+π+K+K ->
Ω+p
strong
dependence on
pion density
P.Braun-Munzinger,J.Stachel,CW
Tch ≈ Tc
Phase diagram
quarks and gluons
hadrons
Running coupling : QED
Chiral perturbation theory
Matsubara formalism