Finite  size  scaling  for  3  and  4-­‐flavor  QCD  with  finite  chemical  poten>al  

arXiv:1307.7205

Shinji  Takeda  Kanazawa  University  

 in  collabora>on  with  

X-­‐Y.  Jin,    Y.  Kuramashi,  Y.  Nakamura  &  A.  Ukawa  

Why  4-­‐flavor  ? n  Good  tes>ng  ground  before  3-­‐flavor  n  Depending  on  the  size  of  mass,  phase  diagram  changes  n  Reasonable  cost  to  survey  transi>on  region  

�

T

�

T

massless large  mass

La[ce  study  so  far n Mul>-­‐parameter  reweigh>ng  Fodor  &  Katz  01  n  Imaginary  chemical  poten>al  D’Elia  &  Lombardo  02  n  Canonical  approach  de  Forcrand  06,  Kentucky  10  

It  is  not  well  inves>gated  by  finite  size  scaling!

What  we  do  here n  Careful  finite  size  scaling  and  high  sta>s>cs  ~  105  conf.  n  Grand  canonical  approach  with  Wilson  type  fermions    n  Phase  reweigh>ng  

 n  Reduc>on  technique  

n exact  phase  &  quark  number  n GPGPU  

ZQCD(T, µ) =�

[dU ]e�Sg[U ] detD(µ; U)

�O� =�OeiNf��||�eiNf��||

Z||(T, µ) =�

[dU ]e�Sg[U ]| detD(µ; U)|

Danzer  &  Gafringer  (2008)

Complex

ST,  Kuramashi  &  Ukawa  (2011)

Phase  can  be  controlled  for  larger  temporal  size

µ

TCross%over/weak%1st%PT�

µ

Tstrong'1st'PT�

Simula>on  parameters n  Clover  fermions  and  Iwasaki  gauge  n  Parameters:  

Kentucky  group  (2010)    canonical  approach

Light  mass

• β=1.58 •  κ=0.1385 • V=63-­‐103 • μ=0.02-­‐0.30

• mπ/mρ=0.822 •  T/mρ=0.154

Heavy  mass

• β=1.60 •  κ=0.1371 • V=63-­‐83 • μ=0.10-­‐0.35

• mπ/mρ=0.839 •  T/mρ=0.150

Nt=4

μ-­‐reweigh>ng

n  Taylor  expansion  

n  Taylor  expansion  of  logarithm  

n  Winding  expansion  (fugacity  expansion  of  logarithm)  

 

�O(µ�)�µ� =

�O(µ�)det D(µ�)Nf

det D(µ)NfeiNf�(µ)

�

||µ�det D(µ�)Nf

det D(µ)NfeiNf�(µ)

�

||µ

Expansion  schemes  to  evaluate  ra>o  of  determinant

= 1 +��

n=1

(�µ/T )n

n!Qn

= exp

� ��

n=1

(�µ/T )n

n!Wn

��µ = µ� � µ

= exp

��

n�ZVn

�enµ�/T � enµ/T

��

Which  expansion  is  befer?

1

10

102

103

104

105

106

107

108

0 1 2 3 4 5

n

!=1.58 "=0.1385 aµ=0.14

<Qn>|| on 63

<Qn>|| on 83

<Qn>|| on 103

<Wn>|| on 63

<Wn>|| on 83

<Wn>|| on 103

Taylor

Taylor  log

Light  mass

 aμ’

0

0.5

1

1.5

2

0.12 0.13 0.14 0.15 0.16 0.17

r P

aµ

103 `=1.58 g=0.1385 aµ=0.14

WindingTaylor

Taylor Log

Phase-­‐reweigh>ng  factor �ei4��|| =

ZQCD

Z||

aµc = am�/2 � 0.7

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

<co

s(4

!)>

||

aµ

"=1.60

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

<co

s(4

!)>

||

aµ

"=1.58

63

668688

83

103

Light  mass heavy  mass

Suscep>bility

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

!P

"=1.58

63

668688

83

103

"=1.60

0

10

20

30

40

50

60

0.1 0.12 0.14 0.16 0.18 0.2

!q

aµ

0.16 0.18 0.2 0.22 0.24

aµ

Plaquefe

Quark  Number

Light  mass heavy  mass

Volume  scaling  of  suscep>bility  peak

1st  order  phase  transi>on

0.2

0.25

0.3

0.35

0.4

0.45

0.5

100 200 300 400 500 600

63 668 688 83

pe

ak

of

!P

V

"=1.60

!-1V+!0!0+!1/V

!-1V+!0+!1/V 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

200 400 600 800 1000

63 668 688 83 103

pe

ak

of

!P

V

"=1.58

!-1V+!0!0+!1/V

!-1V+!0+!1/V

Cross  over/weak  1st  order  PT

Plaquefe

Challa  Landau  Binder  cumulant

limV��

Umin =�

2/3 cross overothers 1st order

UX = 1 � 13�X4��X2�2

Challa,  Landau  &  Binder  86 Fukugita,  Okawa  &  Ukawa  89

0.6625 0.663

0.6635 0.664

0.6645 0.665

0.6655 0.666

0.6665 0.667

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24U

P

aµ

`=1.60 w/ phase

2/363

668688

83

103 0.6625

0.663 0.6635

0.664 0.6645

0.665 0.6655

0.666 0.6665

0.667

0.06 0.08 0.1 0.12 0.14 0.16 0.18

UP

aµ

`=1.58 w/ phase

Plaquefe

Scaling  for  the  min  of  CLB  cumulant lim

V��Umin =

�2/3 cross over

others 1st order

0.663

0.664

0.665

0.666

0.667

0.668

0 0.001 0.002 0.003 0.004 0.005

1/103

1/83

1/73

1/63

min

imum

of U

P

1/V

!=1.60

0.663

0.664

0.665

0.666

0.667

0.668

0 0.001 0.002 0.003 0.004 0.005

1/103 1/83 1/73 1/63

min

imu

m o

f U

P

1/V

!=1.58

2/3u0[1-u1/V]

2/3[1-u1/V+u2/V2]u0[1-u1/V+u2/V2]

1st    PT Cross  over/weak  1st  PT

Summary  for  Nf=4 n  μ-­‐reweigh>ng  works  very  well  n  Taylor  expansion  of  logarithm  of  determinant  is  a  

good  approxima>on  n Moments  analysis  shows  that  

n  Lee-­‐Yang  zero  analysis  will  be  presented  by  X-­‐Y.  Jin  awer  this  talk  

µ

TCross%over/weak%1st%PT�

µ

Tstrong'1st'PT�

 mπ/mρ=0.822  mπ/mρ=0.839

n  Purpose  :      Tracing  cri>cal  end  point    in  (m,μ)  plane    de  Forcrand  &  Philipsen  2006    n  Procedure  

n  Cri>cal  end  point  is  es>mated  by  Binder  cumulant  (kurtosis)  intersec>on  method                                                                        Karsch  et  al.  2001  

n  μ=0  is  discussed  by  Nakamura  on  Thu.  

Nf=3  finite  density  QCD µ

m

?�?�

-2

-1.5

-1

-0.5

0

1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78

gaug

e ac

tion

kurto

sis

inte

rsec

tion

`

aµ=0.10 w/ phase

L=8L=10L=12

Kurtosis  intersec>on

1st  PT Need  more    Sta>s>cs,  β  points  and  NL=14  

•  Iwasaki  gauge  &  clover  fermions  •  Grand  canonical  &  phase  reweigh>ng  •  NT=6              NL=8,  10,  12            aμ=0.1  (μ/T=0.6)

Cross  over

Gauge  ac>on  suscep>bility

0

2

4

6

8

10

0.138 0.1385 0.139 0.1395 0.14 0.1405 0.141 0.1415

gauge a

ctio

n s

usc

eptib

ility

g

aµ=0.10 w/ phase

`=1.70 NL=8`=1.70 NL=10`=1.73 NL=8

`=1.73 NL=10`=1.73 NL=12`=1.75 NL=8

`=1.75 NL=10`=1.75 NL=12`=1.77 NL=8

`=1.77 NL=10`=1.77 NL=12

1st  PT Cross  over

 β=1.77  β=1.73

 β=1.75

 β=1.70 Cri>cal  end  point?

BACK  UP  SLIDES

Applicable  range  (Taylor  log)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3

r P

aµ

63 `=1.58 g=0.1385

aµ=0.02aµ=0.14aµ=0.30

Comparison  between  QCD  and  phase  quenched  QCD

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.1 0.2 0.3

aµ

!=1.60 83

<P><P>||

Comparison  between  Grand  Canonical  and  Canonical  approach

1.6 1.8

2 2.2 2.4 2.6 2.8

3

3 4 5 6 7 8 9 10 11

0.140.160.180.200.220.24

µ B/T aµ

nB

GC phase reweighting

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3 4 5 6 7 8 9 10 11

µ!/T

nB

T=0.95Tc

maxwell construction

Our  transi>on  point

Kentucky  group  (2010) Our  results

Pressure  of  QCD  and  phase  quenched  QCD

0

0.002

0.004

0.006

0.008

0.01

0 0.05 0.1 0.15 0.2

aµ

!=1.58 103

pQCD(µ)-pQCD(0)pQCD||

(µ)-pQCD||(0)

-0.0009

-0.0008

-0.0007

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0

0 0.05 0.1 0.15 0.2

!p

aµ

"=1.58 103

T/V ln<cos(4#)>||

�cos(4�)�|| = exp�V

T

�pQCD(µ) � pQCD||(µ)

��= exp

�V

T�p(µ)

�

Pressure

0

0.002

0.004

0.006

0.008

0.01

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

pQ

CD

(µ)-

PQ

CD

(0)

aµ

!=1.60

0

0.002

0.004

0.006

0.008

0.01

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

pQ

CD

(µ)-

PQ

CD

(0)

aµ

!=1.58

63

83

103

Transi>on  point

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0 0.001 0.002 0.003 0.004 0.005

1/103 1/83 1/73 1/63

aµ

t

1/V

susceptibility !=1.58

plaquettegauge action

polyakov loopfuzzy polyakov loop

quark number 0.198

0.2

0.202

0.204

0.206

0.208

0.21

0.212

0.214

0.216

0 0.001 0.002 0.003 0.004 0.005

1/103 1/83 1/73 1/63

aµ

t

1/V

susceptibility !=1.60

-1

-0.5

0

0.5

1

SP

!=1.58

63

668688

83

103

!=1.60

-1

-0.5

0

0.5

1

0.13 0.14 0.15 0.16

Sq

aµ

0.17 0.18 0.19 0.2 0.21 0.22 0.23

aµ

Skewness

Plaquefe

Quark  Number

Kurtosis

-2

-1.5

-1

-0.5

0

0.5

KP

!=1.58 !=1.60

63

668688

83

103

-2

-1.5

-1

-0.5

0

0.5

0.13 0.14 0.15 0.16

Kq

aµ

0.17 0.18 0.19 0.2 0.21 0.22 0.23

aµ

Plaquefe

Quark  Number

B4 = K + 3

Scaling  of  min  of  kurtosis

-2

-1.5

-1

-0.5

0

0.5

0 0.001 0.002 0.003 0.004 0.005

1/103

1/83 1/688 1/668 1/6

3

min

imum

of K

1/V

!=1.60

-2

-1.5

-1

-0.5

0

0.5

0 0.001 0.002 0.003 0.004 0.005

1/103 1/83 1/688 1/668 1/63

min

imum

of K

1/V

!=1.58

plaquettegauge action

polyakov loopfuzzy polyakov loop

quark number

Zero  density  simula>on  1

0

0.02

0.04

0.06

0.08

0.1

1.4 1.45 1.5 1.55 1.6 1.65 1.7

G

!=1.600, "=0.1380

63

83

103

123

0.663

0.6635

0.664

0.6645

0.665

0.6655

0.666

0.6665

0 0.001 0.002 0.003 0.004 0.005

1/103 1/83 1/63

CL

B c

um

ula

nt

1/V

!=1.600, "=0.1380

2/3plaquette

gauge action 0.5

1

1.5

2

2.5

200 400 600 800 1000 1200 1400 1600 1800 2000

63 83 103 123

!

V

"=1.600, #=0.1380

plaquette x1gauge action /10polyakov loop /3

β=1.60 κ=0.1380 V=63-­‐123 mπ/mρ=0.825 T/mρ=0.155

Strong  1st  PT

Zero  density  simula>on  2 β=1.618 κ=0.1371 V=63-­‐123 mπ/mρ=0.834 T/mρ=0.155

Cross  over  Or  Weak  1st  PT

0.5

1

1.5

2

2.5

200 400 600 800 1000 1200 1400 1600 1800 2000

63 83 103 123

!

V

"=1.618, #=0.1371

plaquette x1gauge action /10polyakov loop /3

0.663

0.6635

0.664

0.6645

0.665

0.6655

0.666

0.6665

0 0.001 0.002 0.003 0.004 0.005

1/103 1/83 1/63

CL

B c

um

ula

nt

1/V

!=1.618, "=0.1371

2/3plaquette

gauge action

0

0.02

0.04

0.06

0.08

0.1

1.4 1.45 1.5 1.55 1.6 1.65 1.7

G

!=1.618, "=0.1371

63

83

103

123