Field theoretical modelling of the
QCD phase diagram
NJL
MODEL
QCD Phase Diagram
QCD Phase Diagram
•
The most important point
Revealed
that cosmological QCD phase transition was most probably a
crossover .The
universe evolves at small baryon chemical potential and
high temperature
spontaneous
symmetry breaking
thermal phase transitions
•
The 1/2 term
is divergent and represents the infinite contribution from the
zero
temperature We will
disregard this
term and
substituting
the occupation number
np
with the
Bose

Einstein
distribution
function.
Mean field
plays the role of the
order parameter in this transition
Presence
of the
term
in
Veff signals a first order
phase transition
.
How can we build an Effective field
theory?
The symmetry
structure is what mainly determines the behavior of a field
theory
1

chiral symmetry breaking
2

confinement
What is the chiral symmetry?
Why chiral symmetry is broken in QCD?
•
Chirality (“handedness”) is a conserved property of elementary particles with zero
mass.
If m=0 left handed and right handed fields decoupled
Mass term mixed left handed and right handed fields
, if m=0 we dont have mixing
term.if mix field exist, that means chirality flip ,that mean mass term exist and
chiral symmetry is broken.
Chiral symmetry in QCD
In this discussion , we will ignore all bu the lightest quarks
u
and
d
The fermionic part of lagrangian is
Symmetry group SU(2)
L
×
SU(2)
R
×
U(1)
L
×
U(1)
R
Currents associated with these symmetries
Q
denotes quark duoblet
Why we might expect the Chiral symmetry
spontaneosly broken in QCD?
Symmetry group SU(2)
SU(2) = SU(2)
L
×
SU(2)
R
= SU(2)
V
×
SU(2)
A
For massless fermions contain no coupling between left and right handed quarks this
lagrangian actually is symmetric under the seprate unitary transformation.
A quark
–
antiquark pair with zero total momentum and angular momentum
Thus we expect that the vacume of QCD contain a condensate of quark anti quark pairs.
Why we might expect the Chiral symmetry
spontaneosly broken in QCD?
•
The vacuum state with a quark pair condensate is characterized by a nonezero
expectation value for scalar opretor
The expectation value signal spontanous breaking of the fully symmetric group
SU(2)
V
×
SU(2)
A
SU(2)
V
Chiral symmetry Chiral symmetry broken
If Chiral symmetry broken then axial current is not conserved.
GOLDSTONE THEOREM
Why Pions are not massless particles?
This means that, in QCD with massless
u
and
d
quarks we should
find massless particle.
The real strong interaction do not contain any massless particle
,but they do contain an isospin triplet of relatively light
mesons,
THE PIONS
Axial Current matrix element
Matrix element connecting the vacuum to the pion, via the axial current
.
If we have the Chiral symmetry and =0 for an on shell pion it
must be massless ,as required by Goldston theorem.
pions are massive particles
BUT
quarks have small masses, and The Axial currents are no longer an exactly conserved quantity. In
this case Chiral symmetry is
explicitly
broken , and no spontaneously broken..massless goldston
bosons related to the spontaneously symmetry broken of a continuous symmetry in this case
≠ 0
and pions are massive particles
.
Goldstone bosons corresponding A
small quark mass tilts the effective
potential
to
flat directions in the effective
potential s electing
one direction for the true vacuum
and
giving the Goldstone bosons a mass
Chiral
Effective
Lagrangian
An
effective
low energy theory of QCD must be constructed such that chiral
symmetry and its spontaneous and explicit breaking are
implemente.
The effective
degrees of freedom in QCD at low energies are no longer
the
elementary
quarks and gluons
,
and the degrees of freedom at low energy were identified with mesons and
nucleons.
The Lagrangian of NJL can
in principle be obtained from QCD by “integrating
out” the
gluonic degrees
of freedom
, replacing them by local four

point
color

current
interactions.
NJL Model
Strength of the NJL
mode
:
1

Chiral symmetry and its spontaneous and explicit breaking
2

the dynamic generation of fermion
masses brought
about by the
breaking of chiral
symmetry
Weakness of the NJL
model
:
1

is not
renormalizable
2

it does not incorporate
confinement
*(
this is a consequence of replacing
the local
color symmetry by a
global
one
)*
NJL Model is a phenomenological Model ,
What is the phenomenological Model?
NJL Lagrangian
•
As the model we want to construct is a model dealing with
bulk properties
of
strong interacting matter, we integrate out effects on small length
scales 0.2 fm
Doing
this we replace the
local
SU(3)
C
gauge
symmetry by a global symmetry
.
The
effects
of gluons
are limited to the pressure the gluon gas generates and to the
strength of the
quark

quark coupling.
It
is usually implied that
NJL models
have only point like interaction
terms
The
standard interaction term used by Nambu and
Jona

Lasinio
NJL
models are designed to describe the dynamics
below
the Nambu and Jona

Lasinio model with its quark quasiparticle degrees of freedom
can reproduce the meson spectrum.
Here only the two flavor case is considered for simplicity
.
only
the pion
has to be
reproduced in its mass. In fact pion properties will be used to fix key
parameters
of
the model
.
The
spontaneous
symmetry breaking
mechanism is implemented into the model via a
so

called gap

equation
.
One way
to derive the
gap

equation
is to calculate the
vacuum energy of the quarks with a
Dyson equation..
Dyson
equation for the calculation of the self
energy
Hartree

Fock
approximation.
Derivation of
Feynman rules
for
NJL models
after explicitly breaking the chiral symmetry
As
the NJL model should model the
chiral symmetry
break down at low temperatures, it also
should
incorporate information about
theGoldstone bosons, or what is left of them after
explicitly breaking the chiral symmetry .
We build a
meson from a quark and an antiquark demanding that this pair may propagate
together.
A
Bethe

Salpeter
equation can be used to evaluate the mass of such a compound
state.
The thermodynamic potential is proportional to the effective action
.
In thermal equilibrium of a system the thermodynamic
potential is
minimal.
The
field equations are then
just the
gradients of the thermodynamic
potential with respect to the
fields
, set to
zero
Bosonization refers to the transformation which eliminates fermionic
degrees of freedom
and replaces
them by bosonic degrees
freedom.
The generating
functional is
defined as
In order to replace the
four

fermion terms
we use the path integral of a gaussian of a
new
(bosonic)
field
φ
r
doing some simple algebra this is transformed
In the case of
the NJL Lagrangian
substituted into the formula for the generating
functional
So far only an auxiliary field
φ
was introduced. This allowed to eliminate the four

fermion
terms, and
to absorb them into the bosonic
fields
In exchange additional interaction terms
appeared.One
couples the fermion field to
the auxiliary field
φ
.
Now
can be evaluated further by integrating out the fermions
.
Before
this
can be
done the fermion
sources
have to be uncoupled from the
fermion and anti

fermion
fields.
Additionally a shifted inverse propagator is defined
by
The generating functional
becomes
bosonized Lagrangian is read off
as
In
a strict sense this bosonic Lagrangian is an effective Lagrangian with respect to the fermion
fields that are integrated out
it is often possible to relate the integral along the real time axis with the integral
along the imaginary
time
axis
This also means going from Minkowski to Euclidean space.
more important application of the Wick rotation concerns quantum field
theories at non

zero
temperatures.
In quantum field theories the partition
function is
defined
as
Calculating thermal expectation values it is useful
to introduce
the
density
We define propagating oprator U
The similarity of U
to
Z
ρ
is quite
obvious:
Transforming U
into Z
ρ
is to change the integrational limits
such that
The
operator U(t
b
, ta) can be written in terms of a path
integral
the generating functional can be Wick rotated to obtain
the generating
functional of
the Euclidean quantum field
theory.
One
finds
All that needs to be done to evaluate thermal expectation values is to take the trace
.
As
the trace
sums matrix elements of equal final and initial states the path integral representation
is constrained
at the upper and the lower integration
limit.
the
path integrals are subject to the constraint that bosonic field need
to be
β
periodic, while
fermionic fields are
β
antiperiodic.
for NJL Model
One can easily confirm that the
condition
leads to the gap
equation.
NJL Phase Diagram
MIT BAG MODEL?!
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