Chiral Critical Surface in the NJL model
Hiroaki Kohyama
(幸山 浩章)
Academia Sinica / NCTS National Taiwan University
2010.04.30 Fri 14:20
Contents
1. Introduction
1.1 QCD phase diagram 1.2 Chiral vs Confinement 1.3 Chiral Critical Surface
2. NJL model
3. Our recent work 4. future works
1. Introduction
QCD phase diagram
Quark Chemical Potential
Temperature
Quark-Gluon Plasma
Hadron Color Super
CFL
Quark Chemical Potential
Temperature
QGP
Hadron CS
QCD phase diagram
Heavy Ion C.
Lattice QCD
Model Studies Neutron Star
Quark Chemical Potential
Temperature
Asymptotic free
Quark Confined
QCD phase diagram
Exp.
Lattice
Quark Chemical Potential
Temperature
Asymptotic free
Quark Confined
QCD phase diagram
Exp.
Lattice
Nothing concrete!
We have to rely on some effective model
Chiral vs Confinement?
Chiral vs Confinement?
: for deconfinement phase transition : for chiral phase transition
(’t Hooft & Casher)
A. Casher, PLB 83 (1979) 395.
G. ’t Hooft, et al, some book, NY (1980).
Critical temperature at Critical Point
Although it is not proven, but
Chiral phase diagram
O.K., let’s study chiral phase transition.
Quark Chemical Potential
Temperature
Chiral restored
Chiral broken
Chiral Critical Surface
Critical Surface?
We treat current quark masses as parameters.
Idea:
Quark Chemical Potential
Temperature
Chiral Critical Surface
P. de Forcrand & O. Philipsen, arXiv:0811.3858.
T
CP
1st Crossover
µ µ
Crossover
T
Text
Columbia Plot
airXiv:hep-ph/0303042 (review)
2. Nambu Jona-Lasinio
NJL Lagrangian
Lagrangian:
What the relation between QCD?
Natural question:
QCD & NJL
4-point interaction may come from
Contact interaction
NJL model
Parameters are fixed by
Gap equation
Effective potential:
where
Solutions of Gap eq.
0 100 200 300 400 500 600
0 50 100 150 200 250 300 350
Constituent Quark Masses[MeV]
Temperature[MeV]
Mu Ms
0 100 200 300 400 500 600
0 100 200 300 400 500 600
Constituent quark masses[MeV]
Chemical potential µ[MeV]
Mu Ms
Crossover 1st
Phase diagram in NJL
0 50 100 150 200 250
0 100 200 300 400 500
Temperature [MeV]
Quark Chemical Potential [MeV]
Critical Point
Chiral broken
Chiral restored
Crossover
1st
Critical behavior
0 2 4 6 8 10 12 14
0 0.5 1 1.5 2 2.5 3
Strange Quark Mass[MeV]
Light Quark Mass[MeV]
Columbia Plot Critical Surface
1st
Crossover
0 1
2 3 4
5 0
5 10 15 20 25 0
50 100 150 200 250 300
Chemical potential[MeV]
Light Quark Mass [MeV]
Strange Quark Mass [MeV]
3. Our recent work
based on
J.-W. Chen, K. Fukushima,† H. K., K. Ohnishi, U. Raha, PRD 81, 071501(R) (2010)
Motivation
0 20 40 60 80 100 120 140
0 5 10 15 20 25 30 35
Strange Quark Mass[MeV]
Light Quark Mass[MeV]
LQCD phys. point
NJL
Huge difference! WHY?
Lattice, too coarse?
Phillipsen, arXiv:0910.0785 Endrodi, et al, arXiv:0710.0998
Lattice discretization Nt may be crucial.
Lattice discretization
Time d. : Nt = 4 Space d. : Ns = 32
seemingly too coarse
Ns Nt
Check in NJL
Space direction is OK, let’s leave it intact.
Time d. : Nt = 4 problematically small Space d. : Ns = 32 decent
Time direction may be trouble, so we will try
Strategy:
the finite frequency summation.
We shall modify the model cut-off.
Modified NJL
Finite temperature field theory:
Let’s change summation,
leading smaller cut-off (larger ).
This mimics Lattice situation.
Parameter fit
Fit the parameters by
Larger N means smaller
Running coupling constant
Results
Comparison
Nt=4
Conclusion
Lattice discretization is too coarse!
4. future works
Generalize to finite mu
0 50 100 150 200 250
0 100 200 300 400
Temperature [MeV]
Quark Chemical Potential [MeV]
Critical Point
N=!
N=50 N=15
Critical Surface
0 1
2 3
4 5 0
5 10 15
20 25 0
50 100 150 200 250 300
Chemical potential[MeV]
Light Quark Mass [MeV]
Strange Quark Mass [MeV] 0 1
2 3 4 5 0
5 10
15 20
25 0
50 100 150 200 250 300
Chemical potential[MeV]
Light Quark Mass [MeV]
Strange Quark Mass [MeV]
N=15 N=∞
Summary
Tested the temporal UV-cutoff effect
explains the difference LQCD & NJL
Ask for ...
Confinement & Chiral transition
Maybe from some correspondence
Sign Problem in Lattice (maybe Dr.Takimi can)
Finite chemical potential region
Something new from condensed matter p. ? Effective model from first principle?
Mathematical aspects
Confining order parameter by string?