Thermodynamics of the 2+1D Gross-Neveu model

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Presentation transcript:

Thermodynamics of the 2+1D Gross-Neveu model beyond the large N approximation Rudnei O. Ramos Department of Theoretical Physics, Rio de Janeiro State University Rio de Janeiro, Brazil 1- the Gross-Neveu model 2- optimized perturbation theory (or delta-expansion) 3- results: phase diagram and going beyond large N 4- final concluding remarks and future work

The Gross-Neveu (GN) model (Gross and Neveu 1974):  describes self-interacting fermions Y with N flavors  It is asymptotically free  Exactly soluble model of interest for QCD  In 3d it is renormalizable in the 1/N expansion Mass terms (which violate chiral symmetry explicitly) can be included as well without loss of solvability (at large N)  It can have either discrete or a continuous chiral symmetry At finite T only version of the model (in 2+1d) with discrete chiral symmetry undergoes PT (no continuous PT in 2 space dim) GN model guide to thermodynamics of chiral symmetry restoration in QCD

The original GN model: For studies in the large-N approximation (limit N  infinity) we redefine the coupling g to: Next rewrite the quartic interaction in terms of an auxiliary scalar field s : If: Mass term for y S Chiral SB

The phase diagram in the large-N approximation (3d) : (in units of the scalar field VEV ~ m0) Tc/m0= 1/(2 ln2) 1st order PT point  Only 2nd order PT line (no tricritical in 3d !)

 = Diagrams to be evaluated up to O(²) when going BEYOND large N : DVeff/N = + 1/N 1/N 1/N ²

1st 1/N contribution appears at order-: where: GENERAL PMS SOLUTION:

(order d and next-to-leading order in 1/N) Applying the optimization procedure ( =/|| ) (order d and next-to-leading order in 1/N)

We predict and are able to locate Line of 1st order PT Line of 2nd order PT We predict and are able to locate a tricritical point (result suggested by Hands, Kogut and collab. numerical MC simulations)

Landau expansion for the potential: For a=0 and b>0, c>0  2nd order PT (Tc) for b <0, c>0  1st order PT (c) For a=0, b=0 and c>0  tricritical points

From the PMS condition: 1st order iteration for the PMS solution 2nd order (in units of the large-N vev of the scalar field)

CONCLUSIONS Generalization to NJL model in 4d, etc In the OPT  Perturbative Evaluation and Renormalization of IR regulated contributions. NON perturbative results generated by variational criterion. Analytical results for <>, c, Tc with 1/N corrections Only available results for the Tricritical Points (they are predicted and located) and phase diagram beyond large N (second order corrections do not change our predictions) Generalization to NJL model in 4d, etc

In collaboration with: Jean-Loïc Kneur (UMII, Montpellier, France) Marcus Benghi Pinto and Ederson Staudt (UFSC, Florianopolis, Brazil) arXiv: 0705.0675 (In press PRD 2007) arXiv: 0705.0673 Phys. Rev. D74, 125020 (2006) Braz. Jour. Phys. 37, 258 (2007) Partially supported by: