CJT study of the O(N) linear and nonlinear
sigma-model at nonzeroT within the
auxiliary field method
Elina Seel
In collaboration with
S. Strüber, F. Giacosa, D. H. Rischke
Introduction
The global
symmetry of QCD with
massless
quark flavors is spontaneously broken at the ground state to
Lattice simulations indicate a restauration of chiral symmetry at
temperatures of 160 MeV

At nonzero temperature it is possible to investigate the
thermodynamics of QCD applying low-energy effective theories like
the O(4) linear and nonlinear  -model in 3+1 dimensions.
For

Having the same symmetry breaking pattern the O(4) (non)linear  models are used to study the order and the critical temperature of the
chiral phase transition
Introduction




At very low temperatures only the pions are excited
In the nonlinear  -model the  -field is eliminated as a dynamical
degree of freedom by sending its mass to infinity
This can be realised with an infinitely large coupling constant in the
linear O(N)  -model:
Sending the coupling to infinity, → 0, the potential becomes
infinitely steep in the  -direction:
The dynamics is confined to:
“ chiral circle“
The O(N) Model

The generating functional of the O(N) linear  -model is given by

with the Lagrangian:

where

Integrating out α the generating functional reads
and α is an unphysical auxiliary field
with
where 1/ is the coupling constant
The O(N) Model


In the nonlinear version of the model the fields are restrained by the
condition
The nonlinear O(N)  -model is obtained by studying the limit → 0 :
where
is identified with the precise formulation of the
representation of thedelta function:
The Model
The lagrangian for the O(N) nonlinear σ-model
The effective potential is computed using the CJT-Formalism

Shifting and α around their vacuum expectation values
generates a mixing term
•
in the tree-level potential:
This mixing term renders and α non diagonal
The Lagrangian
The lagrangian for the O(N) nonlinear σ-model

Performing a further shift of α the mixing term can be eliminated
The resulting Lagrangian reads
The Lagrangian
The lagrangian for the O(N) nonlinear σ-model
The inverse tree-level propagators
The tree-level masses

Advantages of the shift:
1) The Jacobian is unity
2) The  -mass becomes infinitely heavy for → 0
the  -field is not dynamical in the nonlinear limit
The effective potential
Restricting to the double bubble
approximation:
The CJT-effective potential coincides with its tree-level value
We use the imaginary-time formalism to compute quantities at finite
temperature, our notation is
The effective potential
From the stationary conditions for the effective potential
one derives two condensate equations
and the so-called Dyson-Schwinger equations for the full propagators
The effective potential


α is a Lagrange-multiplier and not an independent dynamical degree
of freedom
Substituting α by
we obtain the usual “Mexican hat” shape for the effective potential:
The gap equations
The corresponding gap equations read:
In the large-N limit the gap equations simplify to
Counterterm Regularisation
The lagrangian for the O(N) nonlinear σ-model
Matsubara summation of the thermal tadpole integral gives
Using the residue theorem the vacuum contribution can be rewritten
as
This term exhibits logarithmic and quadratic divergences and has to
be regularized accordingly
Results
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1) N = 4:
2) The linear and nonlinear  -model
3) Explicit symmetry breaking and the chiral limit
4) Counter-term regularization method (CTR) and trivial regularization
(TR)
5) Large-N limit in the counter-term regularization method (LN-CTR)

Common observation: The smaller
and/or the larger the difference
between
and
the more likely the pion
propagation becomes tachyonic at nonzero
temperature
Results
Explicitly broken symmetry in the linear case
Crossover phase transition for
Second order phase transition for
First order phase transition for
Results
Explicitly broken symmetry in the linear case
Crossover phase transition in TR and in LN-CTR
First order phase transition in CTR
Results
Explicitly broken symmetry in the nonlinear case:
The phase transition is cross-over

The  -field becomes frozen due its infinitely heavy mass
there are only pionic excitations left
Results
Chiral limit in the linear case:
The phase transition is second order with
The Goldstone's Theorem is fulfilled
in LN-CTR
Results
Chiral limit in the nonlinear case:

Second order phase transition,

The Goldstone's Theorem is fulfilled

in TR and
in LN-CTR
the  -field is excluded from the thermodynamics
Results
The pressure in the nonlinear case
The thermodynamic pressure is
determined by the minimum of
the effective potential:
where
are the
solutions of the gap equation
Results
The effective potential
in the chiral limit in the
nonlinear case:
Summary
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
The study of the O(N) (non)linear  -model at nonzero temperature
The auxiliary field method allows to properly incorporate the delta constraint
and to establish a well defined link between the linear and nonlinear 
-model
The CJT-effective potential and the gap equations were derived
The regularization of divergent vacuum terms was done within the counterterm scheme
The numerical results for the temperature dependent masses and the
condensate with and without explicitly broken chiral symmetry were presented
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In the nonlinear version of the model the  -field is infinitely heavy and
therefore excluded as a dynamical degree of freedom
As required, in the nonlinear limit the thermodynamics of the system is
completely generated by pionic excitations
The auxiliary field method results in a fulfillment of Goldstone's theorem and
renders the order of the phase transition to be in accordance with arguments
based on universality class reasoning
Outlook
Include sunset-type diagrams
in the 2PI effective action
which allows the computation of decay width
Nonzero chemical potential
Additional scalar singlet states
Addition of vector and axial vector mesonic degrees of freedom
Thank you
for your attention
The Lagrangian
The lagrangian for the O(N) nonlinear σ-model

The Lagrangian depends on Φ and α, which is a Lagrange-multiplier

Substituting α by its equation of motion
one recovers the Lagrangian for the O(N) linear σ-model

An infinitely large coupling in the linear σ-model, ε → 0, leads to the
δ-constraint in the nonlinear case
The Lagrangian

The resulting Lagrangian is given by:

The inverse tree-level propagators and the tree-level masses: