Elina Seel In collaboration with S. Strüber, F. Giacosa, D. H. Rischke

1. Elina Seel In collaboration with S. Strüber, F. Giacosa, D. H. Rischke CJT study of the O(N) linear and nonlinear sigma-model at nonzeroTwithin the auxiliary field method

2. 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 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

3. 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 infinitelysteep in the σ-direction: • The dynamics is confined to: • “ chiral circle“

4. The O(N) Model • 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

5. 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 thedeltafunction:

6. The Model The lagrangian for the O(N) nonlinear σ-model • The effective potential is computed using the CJT-Formalism • Shifting σand α around their vacuum expectation values generates a mixing term in the tree-level potential: • This mixing term renders σand α non diagonal

7. 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

8. 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 σ-mass becomes infinitely heavy for ε → 0 the σ-field is not dynamical in the nonlinear limit

9. 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

10. 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

11. 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:

12. The gap equations • The corresponding gap equations read: • In the large-N limit the gap equations simplify to

13. 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

14. Results • 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

15. Results Explicitly broken symmetry in the linear case Crossover phase transition for Second order phase transition for First order phase transition for

16. Results Results Explicitly broken symmetry in the linear case Crossover phase transition in TR and in LN-CTR First order phase transition in CTR

17. Results • Explicitly broken symmetry in the nonlinear case: • The phase transition is cross-over • The σ-field becomes frozen due its infinitely heavy mass there are only pionic excitations left

18. Results Chiral limit in the linear case: • The phase transition is second order with in LN-CTR • The Goldstone's Theorem is fulfilled

19. Results • Chiral limit in the nonlinear case: • Second order phase transition, in TR and in LN-CTR • The Goldstone's Theorem is fulfilled • the σ-field is excluded from the thermodynamics

20. 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

21. Results The effective potential in the chiral limit in the nonlinear case:

22. Summary • 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

23. 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

24. Thank you for your attention

25. 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

26. The Lagrangian • The resulting Lagrangian is given by: • The inverse tree-level propagators and the tree-level masses: