1. 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

2. 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

3. 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

4. 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 !)

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

8. 1st 1/N contribution appears at order-:
where:
GENERAL PMS SOLUTION:

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

10. 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)

12. 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

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

14. 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

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