CJT study of the O(N) linear and nonlinear O(N) linear and nonlinear -model at nonzero-model at nonzeroTTwithin the auxiliary within the auxiliary

field methodfield method

CJT study of the O(N) linear and nonlinear O(N) linear and nonlinear -model at nonzero-model at nonzeroTTwithin the auxiliary within the auxiliary

field methodfield method

Elina Seel

In collaboration with

S. Strüber, F. Giacosa, D. H. Rischke

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 σ-model in 3+1 dimensions.

For

Having the same symmetry breaking pattern the O(4) (non)linear σ-models are used to study the order and the critical temperature of the chiral phase transition

Introduction

Introduction

At very low temperatures only the pions are excited

In the nonlinear σ-model the σ-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) σ-model:

Sending the coupling to infinity,ε → 0, the potential becomes infinitely steep in the σ-direction:

The dynamics is confined to:

“ chiral circle“

The generating functional of the O(N) linear σ-model is given by

with the Lagrangian:

where and α is an unphysical auxiliary field

Integrating out α the generating functional reads

with

where 1/ε 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) σ-model is obtained by studying the limit ε → 0 :

where is identified with the precise formulation of the representation of thedelta function:

The O(N) Model

The lagrangian for the O(N) nonlinear σ-model The Model

The effective potential is computed using the CJT-Formalism

Shifting σ and α around their vacuum expectation values

generates a mixing term in the tree-level potential:

• This mixing term renders σ and α non diagonal

The lagrangian for the O(N) nonlinear σ-model The Lagrangian

Performing a further shift of α the mixing term can be eliminated

The resulting Lagrangian reads

The lagrangian for the O(N) nonlinear σ-model The Lagrangian

The inverse tree-level propagators

The tree-level masses

Advantages of the shift: 1) The Jacobian is unity 2) The σ-mass becomes infinitely heavy for ε → 0 the σ-field is not dynamical in the nonlinear limit

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 effective potential

The corresponding gap equations read:

In the large-N limit the gap equations simplify to

The gap equations

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

Counterterm Regularisation

1) N = 4:

2) The linear and nonlinear σ-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

Results

Crossover phase transition for

Second order phase transition for

First order phase transition for

Explicitly broken symmetry in the linear case

Results

Crossover phase transition in TR and in LN-CTR

First order phase transition in CTR

Results

Explicitly broken symmetry in the nonlinear case:

Results

The phase transition is cross-over The σ-field becomes frozen due its infinitely heavy mass

there are only pionic excitations left

Chiral limit in the linear case:

Results

The phase transition is second order with in LN-CTR

The Goldstone's Theorem is fulfilled

Chiral limit in the nonlinear case:

Results

Second order phase transition, in TR and in LN-CTR

The Goldstone's Theorem is fulfilled

the σ-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:

The study of the O(N) (non)linear σ-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 σ-model

The CJT-effective potential and the gap equations were derived

The regularization of divergent vacuum terms was done within the counter-term scheme

The numerical results for the temperature dependent masses and the condensate with and without explicitly broken chiral symmetry were presented

In the nonlinear version of the model the σ-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

Summary

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

Outlook

Thank you for your attention

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:

The Lagrangian