# Field theoretical modelling of the QCD phase diagram

Mechanics

Oct 27, 2013 (4 years and 6 months ago)

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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
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
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
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

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

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