## 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?