scaling study of the chiral phase transition in two-flavor qcd for the improved wilson quarks at...

Scaling study of the chiral phase transitionin two-flavor QCD for the improved Wilson

quarks at finite density

H. Ohnofor WHOT-QCD Collaboration

The 3rd A01 group workshopCCS, University of Tsukuba, July 7, 2010


Plan of this talk

• Introduction– QCD phase diagram– Scaling behavior of QCD

• Scaling study with Wilson quarks– Scaling behavior at finite μ– Derivative of chiral order parameter with respect to μ/T

• Numerical results• Conclusion and future plan

2Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density


QCD phase diagram


Hadron gas

Quark-gluon-plasma

critical end pointCrossover

χSB

• Early universe• Heavy ion collisions• Neutron star

Important to understand


Critical behavior of QCD at zero chemical potential


mud

00

Physical point

1storder

2ndorder

Crossover

1storder

ms

QuenchedNf=2

N f=3

O(4)

Tri-criticalpoint mtc

Z(2)

Z(2)

Interesting subject:• O(4) scaling at ms >> mtc

• Tri-critical point• 1st order phase transition at ms << mtc


Scaling behavior of chiral order parameter in 2-flavor QCD


)4(~)2()2( OSUSU Chiral symmetry of 2-flavor QCD

3-d O(4) Heisenberg model

[Wilczek, Intn.J.Mod.Phys. A7('92)3911; Rajagopal, Wilczek, NP B399('93)395]

Recent lattice studies

Wilson quarks• Local for any Nf

• Conserved flavor symmetry• Explicit chiral symmetry breaking

• Higher computational cost

KS quarks• Non-local except for Nf = 4• Explicit flavor symmetry breaking

• Partly conserved chiral symmetry• Lower computational cost

Consistent with O(4) scaling

O(2)

Simulation at physical point

Subtraction and renormalization

[CP-PACS, PRD63, 034502(2000)]O(N) scaling?

[BNL-Bielefeld-GSI, arXiv:0909.5122]


Recent study with Wilson quarks

6/20Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density

[CP-PACS, PRD63, 034502(2000)]

• Iwasaki improved gauge• Clover improved Wilson quark• Nf = 2• 163×4 lattice

O(4) scaling function


Critical behavior of 2-flavor QCD at finite density


mud0

mud=0

O(4)

mud

Quenched

00

1storder

2ndorder

Crossover

1storder

mud

Interesting subject:• O(4) scaling at small μ• Curvature of critical or crossover line •Tri-critical point at finite μ

Nf = 2


Our study


We investigate scaling behavior

• of 2-flavor QCD• at finite chemical potential• with Wilson quarks

• using derivative of chiral order parameter• calculating curvature of 2nd order critical line in β-μ plane


Comparison with 3-d O(4) spin model


3-d O(4) spin model

magnetization: Mexternal field: hreduced temperature: t

2-flavor QCD

chiral order parameter:ud-quark mass: mq

critical value of β: βct

ψψM

amh q2

ctt

)( 11 htfhM Scaling function:


Chiral order parameter with Ward-Takahashi identity


[Bochichio et al., NPB262, 331 (1985)]

yxts

q yPxPNN

aZmψψ

,3

WI)()(

2

xyxyPxPamyPxA aaq

aa 2

,

0

02 4

PtP

PtAmamq

22KZ

aaaa TATP 55 ,

Ward-Takahashi identities in the continuum limit

Wilson quarks : explicit chiral symmetry breaking

Ward-Takahashi identities

: Tree level renormalization factor

A proper subtraction and renormalization are required.


Scaling behavior for finite density


The critical line runs on mq = 0 axis in (mq,μ) plane

t=0

μq/T

mq=0

β

2

2

T

c q

ctqct

c is the curvature of the critical ine in the (β, μq/T) plane.

To confirm this scaling property, we calculate the second derivative of

WI with respect to μq/T.

1

11htxdx

xdf

h

dtdM

,

00

2

2

q

q

dt

dMc

Td

Md

q

,WI

M

,2 amh q2

2

T

c

ct

qt


Derivative of chiral order parameter


M

NMMMM

Fdetlntr

2tr

f5

15

1

25

15

12

2

202332

WI2

03

WI 2 ,

2AFF

NN

maZ

TF

NN

maZ

tsqts

tNTa qq

,trtrdetln

,trdetln 11

2

21

2

21

M

MM

MM

MMM

MM

,tr 51

51

0 MMF

,detlndetln

2

f2

2

f2

M

NM

N

,2Tr4tr2tr

tr5

115

115

15

1115

15

12

21

25

15

12

M

MMM

MMMM

MM

MMMM

MM

MM

,tr2

tr5

15

1151

51

MM

MM

MM

,detln

trdetln

tr2

2f5

15

12

2

f51

51

M

NMMM

NMM

These operators can be calculated by random noise method


Simulation setup

• Action– Iwasaki improved gauge action– Clover improved Wilson fermion action– Nf =2

• Lattice size– 163×4

• # conf.– 500-600

• # noise vectors– 50 for each color and spin indices


β Κ T/Tpc

1.80 0.141139 0.93

1.85 0.140070 0.99

1.90 0.138817 1.08

1.95 0.137716 1.20

β Κ T/Tpc

1.80 0.145127 1.07

1.85 0.143502 1.18

1.90 0.141849 1.32

1.95 0.140472 1.48

mps/mv = 0.65 mps/mv = 0.80


Numerical result at zero μ (1)


Filled symbols: CP-PACS, PRD63, 034502(2000)Open symbols: New data in this study

mq : measured in T=0 simulations.

IW amq2as a function of


Numerical result at zero μ (2)


β,δ: critical exponents of O(4) spin model.

Filled symbols: CP-PACS, PRD63, 034502(2000)Open symbols: New data in this studyDashed line:O(4) scaling function [Toussaint,’97]

Comparison with O(4) scaling function

Scaling behavior is consistent with 3-d O(4) spin model.


Numerical result at finite density (1)


c =0.05

c =0.04

c =0.03

c =0.02

Second derivative ofIW

/1/

/1/1

0

2

2 )(

htxq dxxdf

chTd

Md

q

Solid line: O(4) scaling function with

Scaling behavior is roughly consistent with our expectation.


Numerical result at finite density (2)


350328.02

2

cTd

d

q

ct

/1/

/1/1

0

2

2 )(

htxq dxxdf

hTd

Mdc

q

dada

Td

d

Td

Td

T q

ct

q

c

c

2

2

2

21To calculate the curvature of Tc(μq),

we need a(dβ/da)in the chiral limit.

Curvature of βc(μq)

7df ,77.3df2 NN


Conclusion

• Scaling behavior of chiral order parameter is investigated in two-flavor QCD.

• At zero chemical potential, – it is consistent with 3-d O(4) spin model.

• At finite density,– chemical potential dependence for scaling variables is almost

consistent with our expectation– and the curvature of second order critical line in β-μ plane is

roughly calculated.


)35(0328.0c


Future plan

• More precise scaling study– increasing the statistics and the number of noise vectors

• Lighter mud

• 2+1-flavor simulation


scaling study of the chiral phase transition in two-flavor qcd for the improved wilson quarks at...

Documents

flavor qcd

chiral phase transition

order phase transition

whotqcd collaboration

wilson quarks scaling

o4 scaling

improved wilson quarks

finite density slide