Latti

ce A

ppro

ach

to

finite

den

sity

QC

D

QC

D a

nd D

ense

Mat

ter:

From

Lat

tices

to S

tars

June

3, 2

004

INT,

Sea

ttle

Ats

ushi

Nak

amur

a, R

IISE

, Hiro

shim

a U

niv.

Pla

n of

the

Talk

•B

rief O

verv

iew

of L

attic

e Q

CD

with

Che

mic

al

Pot

entia

l–

Form

ulat

ion

–W

hat p

eopl

e ar

e do

ing

•M

eson

(scr

eeni

ng) m

asse

s at

sm

all µ

Ban

dµ Ι

–Q

CD

-TA

RO

Col

labo

ratio

n•

Pha

se o

f Fer

mio

nD

eter

min

ant a

nd E

igen

valu

es–

Sas

aki,

Taka

ishi

and

AN

•(T

wo-

Col

or Q

CD

with

Wis

onFe

rmio

ns)

–M

uroy

a, N

onak

a, Ii

da a

nd A

N

Intro

duct

ion

Form

ulat

ion

and

Pro

blem

QC

D a

s a

func

tion

of T

and μ

.

Crit

ical

end

poi

nt ?

μ

Crit

ical

end

poi

nt

2SC

CFL

TR

HIC

GS

I, JH

FJ-

PA

RC

Col

or S

uper

Con

duct

ivity

Orig

inal

Col

or S

uper

Con

duct

ivity

B.C

. Bar

rois

Nuc

l.Phy

s.B

129

(197

7) 3

90D

. Bai

lin a

nd A

. Lov

e,

Phy

s.R

ep. 1

07 (1

984)

325

. M

. Iw

asak

i and

T. I

wad

o, P

hys.

Lett.

B35

0 (

1995

) 163

(gap

ene

rgy)～

μ/1

000

Rev

ival

M. A

lford

, K. R

ajag

opal

and

F. W

ilcze

k, P

hys.

Let

t. B

422(

1998

) 24

7.

R. R

app,

T. S

chae

fer,

E. V

. Shu

ryak

, M. V

elko

vsky

,Phy

s. L

ett.

81 (1

998)

53.

(gap

ene

rgy)～

μ

Col

or-F

lavo

r-Lo

ckin

g, M

. Alfo

rd, K

. Raj

agop

al a

nd F

. W

ilcze

kN

uclP

hys

B53

7(1

999)

443

D. T

. Son

, Phy

s. R

ev. D

59 (1

999)

, 094

019

(gap

ene

rgy)～

exp(-

c/g),

not ～

exp(-

c/g^

2)

Rev

iew

K.R

ajag

opal

and

F.W

ilcze

k, ``

At t

he fr

ontie

r of P

artic

le P

hysi

cs-

Han

dboo

k of

QC

D'',

ed. b

y M

. Shi

fman

, Vol

. 4, C

hap.

35, W

orld

S

cien

tific

, 200

2. h

ep-p

h/00

1133

3.

M. A

lford

, Ann

. Rev

. Nuc

l. P

art.

Sci

. 51

(200

1) 1

31he

p-ph

/010

2047

“Col

or s

uper

cond

uctin

g qu

ark

mat

ter”

飯田

圭「カラー超

伝導

」日

本物

理学

会誌

2002

年12

月号

S. M

uroy

a, A

. Nak

amur

a, C

. Non

aka

and

T.Ta

kais

hihe

p-la

t/030

6031

, P

rog.

The

or. P

hys.

110

(200

3) 6

15“L

attic

e Q

CD

at F

inite

Den

sity

–A

n in

trodu

ctor

y R

evie

w”

Fini

te d

ensi

ty e

ven

in n

orm

al N

ucle

ar M

atte

r ?

ρ0=

0.16

～0.

17/fm

^3

ρ 0

T

ρ

Usi

ng

latt

ice Q

CD

, w

e w

an

t to

stu

dy

h

ere

!

Com

pres

sed

Bar

yoni

c M

atte

r Wor

ksho

p, M

ay 1

3-16

, 200

2, G

SI D

arm

stad

t:H

. App

elsh

aeus

er, D

ilept

ons

from

Pb-

Au

Col

lisio

ns a

t 40

AG

eVht

tp://

ww

w.g

si.d

e/cb

m20

02/tr

ansp

aren

cies

/hap

pels

haeu

ser1

/inde

x.ht

ml

Larg

er e

nhan

cem

ent a

t 40

AG

eV c

ompa

red

to 1

58 A

GeV

KE

K-P

S E

325

Col

labo

ratio

n(ta

ken

from

Yak

kaic

hi’s

pres

enta

tion

atN

ucle

arC

hira

l

take

n fro

m P

rof.A

kais

hi’s

talk

QC

D a

s a

func

tion

of T

and μ

・In

tere

stin

g an

d so

und

phys

ics

from

th

eore

tical

and

exp

erim

enta

l poi

nt

of v

iew

s.

・La

ttice

QC

D s

houl

d pr

ovid

e fu

ndam

enta

l inf

orm

atio

n as

a fi

rst

prin

cipl

e ca

lcul

atio

n.

μ

Crit

ical

end

poi

nt ?

2SC

CFL

TR

HIC

GS

I, JH

FJ-

PA

RC

Map

of W

onde

r Wor

ld o

f Hig

h D

ensi

ty Yes

, I w

ill ex

plor

e th

is w

onde

rful

wor

ld !

Sig

n P

robl

emTw

o-C

olor

µ I

µ B<ψψ

>

∆

Tri-

Crit

ical

P

oint

CS

C

Latt

ice

QC

DG

GS

SDU

DDUD

−∆

+−

∫∫

∆=

=Ζ

ede

te

)(

ψψ

ψψ

)(

e)

(x

iAx

Uµ

µ=

mD

+=

∆ν

νγ :)

(xA µ

)(x

Uµ

)(x

ψ

(SU

(N)

Colo

r G

roup

)

Glu

on F

ield

s:)

(),

(x

xψ

ψQ

uark

Fie

lds

{}

∑ =−

++

++

−−

=∆

4 1ˆ

,'ˆ

,')'

()

1()

()

1(1

µµ

µµ

µµ

µδ

γδ

γκ

xx

xx

xU

xU

On

the

latt

ice

(Wils

on f

erm

ions

)

κ：ho

ppin

g pa

ram

eter

qua

rk m

ass

Latt

ice

QC

D w

ith

Che

mic

al P

oten

tial

In f

ree

case

,i.e.

,I

xU

xiA

==

)(

e)

(µ

µ {}

∑ =

−+

+−

−=

∆4 1

)1(

)1(

1)

(µ

µµ

γγ

κipx

ipx

ee

p

A n

atu

ral

way t

o i

ntr

od

uce

th

e c

hem

ical

po

ten

tial

µiP

P−

⇒4

4

{} 4̂

,'4

44̂

,'4

4)'

()

1(e

)(

)1(

e

−

+−

++

+−

−x

xx

xx

Ux

Uδ

γδ

γκ

µµ

{}

∑ =−

++

++

−−

=∆

3 1ˆ

,'ˆ

,')

()

1()

()

1(1

ii

xx

ii

ix

xi

ix

Ux

Uδ

γδ

γκ

GSe

det

1−

∆=

∫DU

Z)

(e

)(

xiA

xU

µµ

=

)(

)(

eTr

eψ

ψµ

βψ

ψ∆

+−

−−

∫=

=Ζ

GSN

HD

DUD

0µγ

γ νν

++

=∆

mD

55

0γ

γµγ

γ νν

∆≠

++

−=

∆+m

Dco

mpl

ex

:∆

det

0=

µ†

∆=

∆=

∆=

∆de

tde

tde

t)

(det

55

*γ

γ

real

:∆

det

),(

)(

xU

ex

Ut

tµ

→

)(

)(

xU

ex

Ut

t†

†µ

−→

0≠

µ

At At

ZGS/

ede

t−

∆

In M

onte

Car

lo s

imul

atio

n, c

onfig

urat

ions

ar

e ge

nera

ted

acco

rdin

g to

the

Pr

obab

ility

:

GSO

DU

ZO

−∆

>=<

∫e

det

.1

det

:!

Complex

∆M

onte

Car

lo S

imul

atio

nsve

ry d

iffic

ult

!

G

G

G

G

Si

S

S

Si

eDUDU

DU

eO

DU

−

−

−

−

∆

∆×

∆∆

∫∫∫

∫e

det

ede

t

ede

t

ede

t.

θ

θ

>=<O

00

detde

t≈

><

><

=θθ

ii

eOe

if th

e ph

ase

θflu

ctua

te r

apid

ly.

Rec

ent

prog

ress

es in

Lat

tice

Had

ron

mas

s re

spon

se t

o ch

emic

al p

oten

tial

S.Ch

oeet

al.

(QC

D-T

aro

colla

bora

tion

Phys

. Rev

. D65

(20

02)

0545

01

T

µ

∂ ∂µ∂ ∂µ

µµ

MM

==

0

2

20

,|

:M

PS m

eson

scr

eeni

ng m

ass

11

(0,

)(0

,

()

())de

tGS

t

Gt DUT

et

r−

−−

∆Γ∆

=∆

∫co

sh(

(/2

))tL

Cm

=−

(1)

(2)

1 ,det

,,

:Cm

−∆

∆fu

nctio

n of

µ

i)D

eter

min

e C

and

m b

y fit

ting

Eq.

(1)

ii)Ta

ke th

e de

rivat

ive

of (1

) and

(2) w

ith re

spec

t to

µ

iii)

By

fittin

g th

ese

Eqs

. , d

eter

min

/,

/C

Mµ

µ∂

∂∂

∂

Rew

eigh

ting

)(

e)(

det

1β

µgS

DUO

ZO

−

∫∆

=

∫∆∆

∆=

−−

)0(de

t)

(de

te)0(

det

e1

)(

)(

)(

00

µβ

ββ

gg

gS

SS

DUO

Z

Fodo

r and

Kat

zM

ulti-

para

met

er re

wei

ghtin

gte

chni

que

Allt

onet

al.

(Bie

lefe

ld-S

wan

sea)

Tayl

or e

xpan

sion

at h

igh

T an

d lo

w µ

n

n

n

n nµ

µµ

∂∆

∂= ⎟⎟ ⎠⎞

⎜⎜ ⎝⎛∆∆

∑∞ =

)0(de

tln

!)0(

det

)(

det

ln1

Fodo

r-Ka

tz, J

HEP

03(2

002)

014

T EE

=±

=±

160

3572

535

.,

MeV

MeV

µ

12

+=

FN

2.0

,

025

.0,

==

sd

um

m8 ,6 ,4

,43

=×

ss

NNSta

ndar

d ga

uge

+ S

tagg

ered

ferm

ion

162

2,

ET=

±M

eV

360

40Eµ

=±

MeV

Allto

net

al.

(Bie

lefe

ld-S

wan

sea) Im

prov

ed a

ctio

n +

Impr

oved

sta

gger

ed fe

rmio

n

416

3×

170

MeV

0.2

,1.0=

qm

µa=0

.29

µ

Imag

inar

y Ch

emic

al P

oten

tial

deFo

rcra

ndan

d P

hilip

sen

hep-

lat/0

2050

16

µµ

µµ

βIm

)(

)(

21

0

=+

=

I

II

Ca

cc

a

(D’E

liaan

d Lo

mba

rdo

hep-

lat/0

2050

22)

At s

mal

l µ)

()

(lo

g6

44

22

0µ

µµ

µO

aa

aZ

++

+=

)(

)(

log

64

42

20

II

II

Oa

aa

Zµ

µµ

µ+

+−

=µ

µIm

=I

com

plex

:de

t∆re

al:

detM

µµ

Re

Imi−

=

Sta

ndar

d ga

uge

+ S

tagg

ered

ferm

ion

,2=FN

250.0=

qm

46 ,4

83

3×

×

3πµ

≤I

Z(3)

sym

met

ry

µ=0.

0µ=

0.2

deFo

rcra

nd-P

hilip

sen

For

smal

l µ, w

e m

ay

have

a lo

ok o

f th

e ph

ase

tran

sitio

n lin

e.

•Th

ese

new

met

hods

wor

k on

ly a

t sm

all µ

Larg

e µ

and

smal

l TSt

ill v

ery

diff

icu

lt !

Anoth

er s

trate

gy:

Use c

olo

r S

U(2

)as t

he f

irst

ste

p !

Latt

ice

QC

D w

ith

two

Col

or

•Fo

r Col

or S

U(2

) cas

e,

22

*σ

σµ

µU

U=

)2(SU

U∈

µfo

r

),

(de

t))

,(

(det

**

*µ

µγ

γU

U∆

=∆

),

(de

t)

,(

det

2*

2µ

µγ

σγ

σU

U∆

=∆

=

:)(

det

µ∆

Rea

l !

Col

or S

U(2

) Q

CD

•P

oor p

erso

n’s

QC

D–

Asy

mpt

otic

free

Non

-Abe

lian

Gau

ge th

eory

–C

onfin

emen

t/Dec

onfin

emen

ttra

nsiti

on•

’t H

ooft’

sm

onop

ole

pict

ure:

S

U(2

) par

t is

esse

ntia

l.

•B

ut B

aryo

ns a

re q

qst

ates

, no

t qqq

!

SU

(3)

SU

(2)

Ana

lyse

s of

Tw

o-co

lor

QC

D•

SU

(2) l

attic

e ga

uge

theo

ry a

tN

akam

ura

(PLB

140(

1984

)391

)•

The

first

cal

cula

tion,

Pse

udo-

Ferm

ion

Met

hod

Han

ds,K

ogut

,Lom

bard

oan

d M

orris

on (N

PB

558(

’99)

327)

•S

tagg

ered

ferm

ion,

HM

C a

nd M

olec

ular

dyn

amic

sH

ands

,Mon

tvay

,Mor

rison

,Oev

ers,

Sco

rzat

oan

d S

kulle

rud

,Eur

.Phy

s.J.

C17

(200

0) 2

85 (h

ep-la

t/000

6018

)

•S

tagg

ered

ferm

ion,

HM

C a

nd T

wo-

Ste

p M

ulti-

Bos

on

algo

rithm

Kog

ut, T

oubl

anan

d S

incl

airP

LB51

4 (2

001)

77

(hep

-lat/0

1040

10)

Kog

ut, S

incl

air,

Han

ds a

nd M

orris

on,P

RD

64(2

001)

0945

05 (h

ep-

lat/0

1050

26)

Kog

ut, T

oubl

an, a

nd S

incl

airh

ep-la

t/020

5019

Mur

oya,

Nak

amur

a, N

onak

a(h

ep-la

t/001

007,

hep

-lat/0

1110

32, h

ep-

lat/0

2080

06, P

hys.

Let

t. B5

51 (2

003)

305

-310

)

•W

ilson

ferm

ion,

Lin

k-by

-Lin

k up

dateµ

≠0

Sta

ndar

d ga

uge

+ S

tagg

ered

ferm

ion

05.0 ,6

12 ,48

,8

33

4=

××

m 05.0 ,6

12 ,16

34

=×

m

Evid

ence

of

di-q

uark

cond

ensa

tion

Vec

tor m

eson

at F

inite

µ

Per

iodi

c bo

unda

ry c

ondi

tion

Vect

or m

eson

mas

s be

com

es s

mal

l !(T

his

rem

inds

us

of

CERES

exp

erim

ent.

)

Mur

oya,

AN

, N

onak

a

Pol

yako

vLi

ne C

orre

latio

nS

ingl

etTr

iple

t

04

80

48

V1

0.1 0

-0.1

µ=0.

0µ=

0.4

µ=0.

6µ=

0.7

V3

0.1 0

-0.1

Rµ=0.

0µ=

0.4

µ=0.

6µ=

0.7

prel

imin

ary

R

Glu

on P

ropa

gato

r

•8X

4x12

X4,

Nf=

3,

κ=0.

170

)z)A

Az

Ga

a

a0,

(,

()

,(

PP

P−

=∑

νµ

µν

⎟⎟ ⎠⎞⎜⎜ ⎝⎛

=0,0,

xN2π

P

Ezz

G−

∝e

),

(ttP

Scr

eeni

ng m

ass

µ=0.

0µ=

0.7

Con

finem

ent D

econ

finem

ent

Pha

se tr

ansi

tion?

Free

1

),

(tt

zGP

410

−

210

−

126

0Z

µ=0.

0µ=

0.4

µ=0.

6µ=

0.7

prel

imin

ary

Glu

on P

ropa

gato

r

•8X

4X12

X4,

Nf=

3,

κ=0.

170 Ez

zG

−∝

e)

,(

yyP

126

0Z

10 1

0.1

0.01

),

(yy

zG

P

µ=0.

0µ=

0.4

µ=0.

6µ=

0.7

prel

imin

ary

Had

ron

mas

s re

spon

se t

o ch

emic

al p

oten

tial

or T

aylo

r ex

pans

ion

of H

adro

nM

ass

S.Ch

oeet

al.

(QC

D-T

aro

colla

bora

tion

Phys

. Rev

. D65

(20

02)

0545

01

T

µ

∂ ∂µ∂ ∂µ

µµ

MM

==

0

2

20

,|

:M

PS m

eson

scr

eeni

ng m

ass

11

(0,

)(0

,

()

())de

tGS

t

Gt DUT

et

r−

−−

∆Γ∆

=∆

∫co

sh(

(/2

))tL

Cm

=−

(1)

(2)

1 ,det

,,

:Cm

−∆

∆fu

nctio

n of

µ

i)D

eter

min

e C

and

m b

y fit

ting

Eq.

(1)

ii)Ta

ke th

e de

rivat

ive

of (1

) and

(2) w

ith re

spec

t to

µ

iii)

By

fittin

g th

ese

Eqs

. , d

eter

min

/,

/C

Mµ

µ∂

∂∂

∂

Par

amet

ers

in th

e fo

llow

ing

•Q

uark

Mas

s m

a=0.

05, 0

.10

βTe

mpe

ratu

re•

Iso-

scal

ar C

hem

ical

Pot

entia

lµ 1

=µ2=

µ S•

Iso-

vect

or C

hem

ical

Pot

entia

lµ 1

=-µ 2

=µV

•W

e co

nsid

er P

seud

o-S

cala

r (pi

), V

ecto

r (rh

o)

Seco

nd

Der

ivat

ive

of

Pion

Prop

agat

ors

β=5.

35β=

5.45

β=5.

50β=

5.60

The

first

der

ivat

ive

is z

ero

•Li

ttle

diffe

renc

e is

obs

erve

d be

twee

n Q

CD

( µB) a

nd F

inite

Isos

pin

Mod

el (µ

I).•

But

(Scr

eeni

ng) M

ass

Beh

avio

rs a

re v

ery

diffe

rent

.•

We

are

now

cal

cula

ting

also

Bar

yon

Mas

s R

espo

nse.

D.T

oubl

an, T

alk

at IN

T

Fini

te Is

ospi

n M

odel

vs.

QC

D-F

inite

Isos

pin

mod

el =

Tw

o-fla

vor Q

CD

with

P

hase

Que

nchi

ng

†0

()

Dm

νν

µγ

µγ∆

=−

++

50

()

()

Dm

νν

γγ

µγµ

=+

−=

∆−

2†

det

()

det

()d

et(

)µ

µµ

∆=

∆∆

21

det

()d

et(

)µ

µ=

∆∆

The

diffe

renc

e is

due

to th

e P

hase

!

•S

incl

air,

Talk

at F

DQ

CD

at N

ara,

hep

-lat/0

3110

19

/2i eθ

Res

ults

with

and

with

out P

hase

•N

akam

ura,

Tak

aish

i and

Sas

ai, t

o be

pub

lishd

ed

Tous

sain

t 19

90

Tow

ards

larg

e de

nsity

QC

D:

Dre

am ?

Diff

icul

ty o

f lar

ge C

hem

ical

Pot

entia

l, or

S

ecre

t of F

odor

-Kat

z

mRe

Im

0E

igen

Val

ue D

istri

butio

n

(0)

Dm

νν

µγ

∆=

=+

D ννγ

: Ant

i-Her

mite

Whe

n μ

incr

ease

s,

max

min

λ λ⇒

∞Im

mμ

Con

juga

te G

radi

ent

to c

alcu

late

does

not

con

verg

eRe

1(

)µ

−∆

01

()

µ−

∆B

ut w

e ne

edfo

r Upd

ate

!E

igen

Val

ue D

istri

butio

n

All

full

QC

D u

pdat

e al

gorit

hms

requ

ire

Fodo

r-K

atz

algo

rithm

doe

s no

t ca

lcul

ate

,

but

1(

)µ

−∆

1(

)µ

−∆

det

()

det

(0)µ

∆ ∆

(The

y up

date

at

) 0µ

=

K. S

plitt

orff,

Tal

k at

INT

Fini

te D

ensi

ty L

attic

e Q

CD

is s

till i

n S

tone

-Age

. W

e sh

ould

inve

stig

ate

mor

e to

un

ders

tand

wha

t is

real

ly p

robl

em.

Latti

ce Q

CD

nee

ds h

elp

from

Mod

el

calc

ulat

ions

to u

nder

stan

d w

hat i

s go

ing

on.

We

need

sev

eral

new

idea

s to

get

Dat

a of

hi

gh q

ualit

y in

Lat

tice

sim

ulat

ions

.