Thermodynamics of the 2 1D Gross-Neveu model

1
Thermodynamics of the 21D 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 21d)
  with discrete
• chiral symmetry undergoes PT (no continuous PT
  in 2 space dim)

guide to thermodynamics of chiral
symmetry restoration in QCD
GN model
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 !)
5
(No Transcript)
6
(No Transcript)
7
Diagrams to be evaluated up to O(?²) when going
BEYOND large N
?
O(?)
O(?²)

DVeff/N
1/N
1/N
1/N ²
8
1st 1/N contribution appears at order-?
where
GENERAL PMS SOLUTION
9
Applying the optimization procedure ( ??/? )
(order d and next-to-leading order in 1/N)
10
(N3)
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)

11
(No Transcript)
12
Landau expansion for the potential
For a0 and bgt0, cgt0 ? 2nd order PT (Tc)
for b lt0, cgt0 ? 1st order PT (?c)
For a0, b0 and cgt0 ? 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
In the OPT ? Perturbative Evaluation and
Renormalization of IR regulated contributions.
NON perturbative results generated by
variational criterion.
Analytical results for lt?gt, ?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 0705.0675 (In press PRD 2007) arXiv
0705.0673 Phys. Rev. D74, 125020 (2006) Braz.
Jour. Phys. 37, 258 (2007)
Partially supported by