Toward the understanding of QCD phase structure at finite temperature and density

Shinji Ejiri Niigata University

WHOT-QCD collaboration

S. Ejiri1, S. Aoki2, T. Hatsuda3, K. Kanaya2, Y. Nakagawa1, H. Ohno4, H. Saito2, and T. Umeda5

1Niigata Univ., 2Univ. of Tsukuba, 3RIKEN, 4Bielefeld Univ., 5Hiroshima Univ.

Osaka Univ., 2012/10/2

QCD phase structure at high temperature and density

Lattice QCD Simulations

• Phase transition lines • Equation of state

• Direct simulation: Impossible at µ≠0.

quark-gluon plasma phase

hadron phase

RH

IC

SPS

AG

S

color super conductor? nuclear

matter µq

T

RHIC E-scan J-PARC

LH

C

quarkyonic?

deconfinement?

chiral SB?

Histogram method Problem of Complex Determinant at µ≠0

Boltzmann weight: complex at µ≠0 Monte-Carlo method is not applicable.

Distribution function in Density of state method (Histogram method)

X: order parameters, total quark number, average plaquette etc.

Expectation values

( ) ( ) ,,, ,, µ=µ ∫ TmXWdXTmZ

( ) ( ) ( )( ) gSN emMXX-DUTmXW −µδ≡µ ∫ ,det ˆ ,,; f

[ ] ( ) [ ] ( ) ,,, 1,,

µ= ∫µTmXWXOdX

ZXO

Tm

( ) ( ) gSNTm

emMODUZ

O −

µµ= ∫ f),(det 1

,,complex

( ) ( )

( ) ( )( )

( )( ) ( )

( )'

0

'6

)0,(

)0,(0

'6f

0site

0

0

f

0site

0,det,det

ˆ

0,det,det ˆ

P

NPN

N

PN

mMmMe

P-P

mMmMP-P

ePR

µ≡

′δ

µ′δ

=′ β−β

=µβ

=µββ−β

(β, m, µ)-dependence of the Distribution function • Distributions of plaquette P (1x1 Wilson loop for the standard action)

( ) ( ) ( )( ) PNNmMP-PDUmPW ˆ6 sitef e ,det ˆ,,, βµ′δ≡µβ′ ∫

Effective potential:

( ) ( )[ ] ( ) ( )µββ−β=µβ−=µβ ,,,,ln0,,,,,ln,,, 0000effeff mmPRmPVmPWmPV

(Reweight factor) ( ) ( ) ( )0,,,,,,,,,, 0000 mPWmPWmmPR βµβ≡µββ

( ) ( ) ( )( )

P

N

mMmMPNPR

f

0,det,det ln6ln0

0site

µ+β−β=

independent of β0

PNSgˆ6 siteβ−=

• Sign problem: If changes its sign frequently,

Sign problem

( )( ) error) al(statistic

0,det,det~)(

fixed 0

<<µθ

X

i

mMmMeXW

θie

θ: complex phase of (detM)Nf

( )( )

( )( )

fixed 0

fixed 0

ff

0,det,det

0,det,det

X

Ni

X

N

mMmM e

mMmM µ

=

µ θ

Overlap problem

• W is computed from the histogram. • Distribution function around X where is minimized: important. • Veff must be computed in a wide range.

( ) ( ) ( )( ) lnexp1 1eff∫∫ +−== dXORXV

ZdXXORW

ZOR

)(ln)(eff XWXV −=

( )ORXV ln)(eff −

+ =

( )ORXV ln)(eff −( )ORln−)(eff XV

reliable range

Out of the reliable range

If X-dependence is large. reliable range

Distribution function in quenched simulations Effective potential in a wide range of P: required.

Plaquette histogram at K=1/mq=0. Derivative of Veff at β=5.69

dVeff/dP is adjusted to β=5.69, using

These data are combined by taking the average.

( ) ( ) ( )12site1eff

2eff 6 β−β−β=β N

dPdV

dPdV,4243

site ×=N 5 β points, quenched.

Distribution functions for ΩR and P

• If W(P,Ω) is a Gaussian distribution, – The peak position of W(P,Ω) (<P>,<Ω>) – The width of W(P,Ω) susceptibilities χP, χΩ

• If W(P,Ω) have two peaks, first order transition

4κ 4κ4

f* 48 KN+β=β

Expectation value of Polyakov loop and its susceptibility by the reweightuing method at µ=0.

4f

* 48 KN+β=β

lattice 4243 ×Ω

Ωχ

Ω

Ωχ~

Ω

µ-dependence of the effective potential

1st order phase transition

Critical point ( )µ,,eff TXV

Crossover

Correlation length: short V(X): Quadratic function

Correlation length: long Curvature: Zero

Two phases coexist Double well potential

)(ln)(eff XWXV −=( ) ( ),,, , µ=µ ∫ TXWdXTZX: order parameters, total quark number, average plaquette, quark determinant etc.

T

µ

hadron

QGP

CSC?

Quark mass dependence of the critical point

• Where is the physical point? • Extrapolation to finite density

– investigating the quark mass dependence near µ=0 • Critical point at finite density?

mud 0

0 ∞

∞

Physical point

1storder

2ndorder

Crossover

1storder

ms

Quenched Nf=2

0=µ

µ

Distribution function in the heavy quark region

• We study the properties of W(X) in the heavy quark region.

• Performing quenched simulations + Reweighting.

• We find the critical surface. • Standard Wilson quark action +

plaquette gauge action, • lattice size: • 5 simulation points; β=5.68-5.70. (WHOT-QCD, Phys.Rev.D84, 054502(2011))

( )( ) ( ) ( )( )( )+Ωµ+Ωµ⋅+=

µIR

Ns

N TiTKNPKNNM

KMN tt sinhcosh2122880,0det,detln 34

siteff

phase

4243 ×

Hopping parameter expansion

P: plaquette, Ω=ΩR+iΩI : Polyakov loop ( ) 10,0det =M

PNSg β−= site6

Effective potential near the quenched limit(µ=0) WHOT-QCD, Phys.Rev.D84, 054502(2011)

• detM: Hopping parameter expansion, • First order transition at K = 0 changes to crossover at K > 0.

Quenched Simulation (mq=∞, K=0)

K~1/mq for large mq

Critical point for Nf=2: Kcp=0.0658(3)(8)

02.0≈πm

Tc

( ) ( )[ ] ( ) ( )µββ−β=µβ−=µβ ,,,,ln0,,,,,ln,,, 0000effeff mmPRmPVmPWmPV

( ) ( ) ( )( )

P

N

mMmMPNPR

f

0,det,det ln6ln0

0site

µ+β−β=

at phase transition point

Order of the phase transition Polyakov loop distribution

54 100.2 :point Critical −×≈κ

Effective potential of |Ω| on the pseudo-critical line at µ=0

Ωχ

• The pseudo-critical line is determined by χΩ peak.

• Double-well at small K – First order transition

• Single-well at large K – Crossover

Polyakov loop distribution in the complex plane

• on βpc measured by the Polyakov loop susceptibility.

0.04 =κ 64 100.5 −×=κ 54 100.1 −×=κ

54 105.1 −×=κ 54 100.2 −×=κ 54 105.2 −×=κ

critical point

RΩ RΩ RΩ

RΩRΩ RΩ

IΩ IΩ IΩ

IΩ IΩ IΩ

(µ=0)

Distribution function of ΩR at finite density

• Hopping parameter expansion

• Adopting • Effective potential:

• V0 is Veff (µ=0) when we replace (at µ=0, )

( ) ( ) ( )( ) PNNRRR eMDUKW ˆ6 sitef det ˆ ,,, −κΩ−Ωδ=µβΩ ∫

( ) ( ) ( )0,;

3f0effeff

0lncosh2120,0,,,

=µ=βΩ

θ−Ωµ×−β=µβK

iR

Ns

N

R

tt eTKNNVKV

( )( )IN

sN TKNN tt Ωµ⋅=θ ˆsinh212 f

3

( ) ( )µβΩ−=µβΩ ,,;ln,,;eff KWKV RR

,48 4f0 KN+β≡β

( )0,;0

0ln ,,

=µ=βΩ

θ−µβ≡K

i

ReKV

( )TKK tt NN µ⇒ cosh

Phase-quenched part

( )( ) ( )( ) ( )[ ]

0,;

3fsite

4f0

0 0

ˆcosh212ˆ2886exp0,0,

,,=µ=βΩ

θ+Ωµ×−+β−β=β

µβKR

Ns

N

R

tt iTKNNPNKNW

KW

Phase average

( ) ( ) RN

sN tt KNNVKV Ω×−β=β 3

f0effeff 2120,0,0,,

• Sign problem: If changes its sign,

• Cumulant expansion

– Odd terms vanish from a symmetry under µ ↔ −µ (θ ↔ −θ) Source of the complex phase

If the cumulant expansion converges, No sign problem.

+θ+θ−θ−θ=

Ω

θ CCCCP

i iieR

432

, !41

!321exp

Avoiding the sign problem (SE, Phys.Rev.D77,014508(2008), WHOT-QCD, Phys.Rev.D82, 014508(2010))

error) al(statistic fixed ,

<<Ω

θ

RP

ie

=θθ+θθ−θ=θθ−θ=θθ=θΩΩΩΩΩΩΩ CPPPPCPPCPC RRRRRRR

43

,,,

2

,

332

,,

22,

,23 , ,

cumulants →0 →0

<..>P,ΩR: expectation values fixed P and ΩR.

θie

θ: complex phase MdetlnIm≡θ ( ) IN

sN TKNN tt Ωµ⋅≈ sinh212 f

3

Convergence in the large volume (V) limit The cumulant expansion is good in the following situations. • If the phase is given by

– No correlation between θx.

– Ratios of cumulants do not change in the large V limit. – Convergence property is independent of V, although the phase fluctuation becomes larger as V increases. – The application range of µ can be measured on a small lattice.

• When the distribution function of θ is perfectly Gaussian, the average of the phase is give by the second order,

∑θ=θx

x

θ= ∑Ω

θ

nC

nn

P

i

nie

R !exp

,

θ=≈

∑= ∑∑∏ Ω

θ

Ω

θ

Ω

θ

x nC

nx

n

xP

i

P

i

P

i

nieee

R

x

R

xx

R !exp

,,

,

)(~ VOx

C

nxC

n ∑ θ≈θ

θ−=

Ω

θ

CP

i

Re 2

, 21exp

Cumulant expansion

• The effect from higher order terms is small near the critical point of phase-quenched part.

69.5* =β +θ−θ+θ−=Ω

θ

CCC

i

Re 642

!61

!41

21 ln

( ) ( ) ( ) ( ) ( )TKTKK ttt NNN μsinhμμcoshμ0 cpcpcp >=

( ) ( ) 00002.0μsinhμ4 =TK

( ) ( ) 0001.0μsinhμ4 =TK

( ) ( ) 00005.0μsinhμ4 =TK

~0.00002

Effect from the complex phase factor • Polyakov loop effective potential for each

at the pseudo-critical (β, K). – Solid lines: complex phase omitted, i.e.,

– Dashed lines: complex phase is estimated with

( ) 0μsinh =T

( )TK tN µcosh

( ) ( )TT µ=µ coshsinh

The effect from the complex phase factor is very small except near ΩR=0.

( ) ( )

( )fixed:

20

fixed:0eff

21

ln

R

R

R

iRR

V

eVV

Ω

Ω

θ

θ+Ω≈

−Ω=Ω

( ) )sinh212( f3

IN

sN TKNN tt Ωµ⋅≈θ

Critical line in 2+1-flavor finite density QCD • The effect from the complex phase is very small for the determination of Kcp.

( ) ( ) ( )TKK tt NN µµ= cosh0 cpcp

Critical line for µu=µd=µ, µs=0 Critical line for µu=µd=µs=µ

The critical line is described by

( )( ) ( )+Ω×+=

R

Ns

N tt KNPKNM

KM 34site 2122882

0detdetln2

( )( ) ( )( )( )

( ) +Ω

+

×++=

R

sNs

udNuds

Nsud

sud

TK

TKNPKKN

MKMKM

tttμcoshμcosh22122288

0detdetdetln 344

site3

2

t2)fcp(2μcoshμcosh2 N

NsN

sudN

ud KT

KT

K tt==

+

Nf=2 at µ=0: Kcp=0.0658(3)(8)

Nf=2+1

(WHOT-QCD, Phys.Rev.D84, 054502(2011))

Distribution function in the light quark region WHOT-QCD Collaboration, in preparation,

(Nakagawa et al., arXiv:1111.2116)

• Perform phase quenched simulations • Add the effect of the complex phase by the reweighting. • Calculate the probability distribution function.

• Goal

– The critical point – The equation of state

Pressure, Energy density, Quark number density, Quark number susceptibility, Speed of sound, etc.

Probability distribution function by phase quenched simulation

• We perform phase quenched simulations with the weight:

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )µβ×=

µ−δ−δ=

µ−δ−δ=µβ

θ

−θ

−

∫∫

,,,','

,det 'ˆ'ˆ

,det'ˆ'ˆ ,,,','

0','

f

f

mFPWe

emMeFFPPDU

emMFFPPDUmFPW

FP

i

SNi

SN

g

g

( ) gSN emM −µ ,det f

( ) ( )( )0det

detlnsite

f

MM

NNF µ

=µ

( ) ( ) ( ) PNN eMFFPPDUFPW ˆ60

sitef det'ˆ'ˆ ',' β−δ−δ= ∫

MN detlnImf≡θP: plaquette

Distribution function of the phase quenched.

expectation value with fixed P,F histogram

µ-dependence of the effective potential Curvature of the effective potential

+ =

1st order phase transition

( )[ ] ,ln 0 β− PW

Critical point

[ ] ln θ− ie( )[ ] ,ln 0 β− PW( )[ ] ,ln β− PW

+ =

Crossover

phase effect

phase effect

Curvature: Zero

Curvature: Negative

[ ] ln θ− ie

T

µ

hadron

QGP

CSC

Curvature of the effective potential • If the distribution is Gaussian,

( ) ( )

−

χ−

πχ×

−

χ−

πχ≈ 2sitesite2sitesite

0 2exp

226exp

26),( FFNNPPNNFPW

FFPP

( )2site6 PPNP −=χ

( ) ( )P

NFPP

Wχ

=∂−∂ site

20

2 6,ln ( ) ( )F

NFPF

Wχ

≈∂−∂ site

20

2

,ln

( )2site FFNF −=χ

at the peak of the distribution

Complex phase distribution • We should not define the complex phase in the range from -π to π. • When the distribution of q is perfectly Gaussian, the average of the

complex phase is give by the second order (variance),

• Gaussian distribution → The cumulant expansion is good. • We define the phase

– The range of θ is from -∞ to ∞.

( ) ( )( ) ( )

µ

µ∂

∂=

µ=µθ

µ

µ

∫ Td

TMN

MMN

T

0ffdetlnIm

0detdetlnIm

π −π π −π θ θ

W(θ) W(θ)

C

2θNo information

θ−=θ

CFP

ie 2

, 21exp

Distribution of the complex phase

• Well approximated by a Gaussian function. • Convergence of the cumulant expansion: good.

θ−≈θ 2

, 21exp

FP

ie

at the peak of W0 in each simulation

00 ,

2

,

2

21

21

µβθ≈θ

FP

Simulations

• Simulation point in the (β, µ0/T) • Peak of W0(P,F) for each µ

lattice 483 × 8.0≈ρπ mm

2-flavor QCD Iwasaki gauge + clover Wilson quark action Random noise method is used.

Curvature of the effective potential -lnW0

• The curvature for F decreases as µ increases.

Effect from the complex phase

• Rapidly changes around the pseudo-critical point.

Critical point at finite µ

• zero curvature: expected at a large µ.

Curvature of the effective potential • Without the complex phase effect

( ) ( )F

NFPF

Wχ

≈∂−∂ site

20

2

,ln ( )2site FFNF −=χ

Phase average • 2nd order cumulant

21ln

,

2

, FPFP

ie θ−≈θ00 ,

2

,

2

21

21

µβθ≈θ

FP

Curvature of the effective potential • The effect of the phase incruded.

zero curvature Critical point

Peak position of W(P,F) • The slopes are zero at

the peak of W(P,F).

( ) ( )

( ) ( ) ( )P

eFP

PRNFP

PW

P

eFP

PWFP

PW

FP

i

site

FP

i

∂

∂+µµ

∂∂

+β−β+µβ∂

∂=

∂

∂+µβ

∂∂

=µβ∂

∂

θ

θ

,0000

0

,0

ln,,,ln6,,,ln

ln,,,ln,,,ln

0ln ,0ln=

∂∂

=∂

∂FW

PW

( ) ( )( )

µβµβ

=µµ00

00 ,,,

,,,,,,FPWFPWFPR

( ) ( )

( ) ( )F

eFP

FRFP

FW

F

eFP

FWFP

FW

FP

i

FP

i

∂

∂+µµ

∂∂

+µβ∂

∂=

∂

∂+µβ

∂∂

=µβ∂

∂

θ

θ

,000

0

,0

ln,,,ln,,,ln

ln,,,ln,,,ln If these terms

are canceled,

( ) ( ) )const.(,,,,,, 000 ×µβ≈µβ FPWFPW

• W(β, µ) can be computed by simulations around (β0,µ0).

=0

=0

),(ln FPW−

QCD phase diagram

phase-quenched QCD finite-density QCD

pion condensed phase

mπ/2 µ

Τ

0=θiecolor super-conductor phase?

µ

Τ

( ) ( )µβ=×µβ θ ,,,,,,,,,0 mFPWemFPWFP

i

Summary • We studied the quark mass and chemical potential dependence

of the nature of QCD phase transition.

• The shape of the probability distribution function changes as a function of the quark mass and chemical potential.

• To avoid the sign problem, the method based on the cumulant expansion of θ is useful.

• Our results by phase quenched simulations suggest the existence of the critical point at high density.

• To find the critical point at finite density, further studies in light quark region are important applying this method.

Backup

Complex phase • Gaussian distribution → The cumulant expansion is good. • We define the phase

– The range of θ is from -∞ to ∞.

• At the same time, we calculate F as a function of µ,

• The reweighting factor is also computed,

( ) ( )( ) ( )

µ

µ∂

∂=

µ=µ

µ

µ

∫ Td

TMN

MMNF

T

0ffdetlnRe

0detdetln

( ) ( )( ) ( )

µ

µ∂

∂=

µ=µθ

µ

µ

∫ Td

TMN

MMN

T

0ffdetlnIm

0detdetlnIm

( ) ( )( ) ( )

µ

µ∂

∂=

µµ

=µµ

µ

µ∫ Td

TMN

MMNC

T

T0

detlnRedetdetln f

0f

Distribution function for P and ΩR

• Effective potential • Hopping parameter expansion

• 2 parameters in V0: – V0 is the same as Veff (µ=0) when

• 1 parameter in θ:

( ) ( ) ( ) ( )( ) PNNRRR eMPPDUPW ˆ6 sitef det ˆ ˆ ,,, −κΩ−Ωδ−δ=κβΩ ∫

( ) ( ) ( )( ) ( )R

tt

P

iR

Ns

N eTKNNPNKNVVΩ

θ−Ωµ×−+β−β−=β−κβ,

3fsite

4f00effeff lncosh21228860,,

( )( )IN

sN TKNN tt Ωµ⋅=θ ˆsinh212 f

3

( ) ( )κβΩ−=κβΩ ,;,ln,;,eff RR PWPV

,48 *4f β≡+β KN ( )TK tN µcosh

( )RP

ieVΩ

θ−κβ≡,0 ln ,

( )TKK tt NN µ⇒ cosh

( ) ( ) ( ) ( )TKTTKTK ttt NNN µ<µµ=µ coshtanhcoshsinh

Phase-quenched part

( )( )

( ) ( ) ( ) ( )( )

( ) ( )( ) ( )

( )RP

NPN

KRR

K

NPN

RR

MKMe

-PP-

MKMe-PP-

WKW

Ω

β−β

=µ=β

=µ=β

β−β

µ≡

ΩΩδδ

µΩΩδδ

=β

µβ

,

ˆ6

)0,(

)0,(

6

0

f

0site

0

0

f

0site

0,0det,det

ˆˆ

0,0det,det ˆˆ

0,0,,,

Order of phase transitions and Distribution function

• Peak position of W:

0effeff =Ω

=Rd

dVdP

dV

crossover 1 intersection

first order transition 3 intersections

Lines of zero derivatives for first order

( ) ( )κβΩ−=κβΩ ,;,ln,;,eff RR PWPV

( ) ( ) ( ) ( )( ) PNNRRR eMPPDUPW sitef 6 det ˆ ˆ ,,, −κΩ−Ωδ−δ=κβΩ ∫

Derivatives of Veff in terms of P and ΩR

• Contour lines of and at (β,κ) = (β0,0) correspond to

the lines of the zero derivatives at (β,κ).

( ) ( ) ( )( ) site4

f00effeff 28860,, NKN

dPdV

dPKdV

+β−β−=β

−β ( ) ( )

µ

×−=Ω

β−

Ωβ

TKNN

ddV

dKdV

tt Ns

N

RR

cosh2120,, 3f

0effeff

Phase-quenched part: when is neglected,

constant shift constant shift

dPdVeff

RddV

Ωeff

( ) ( ) ( )( ) ( ) RN

sN TKNNPNKNVV tt Ωµ×−+β−β−=β−κβ cosh21228860,, 3

fsite4

f00effeff

θieln

P PRΩRΩ

measured at κ = 0

dPdVeff

RddV

Ωeff

lines of and in the (P,Ω) plane

• Small K: lines of : S-shape first order • Large K: lines of : straight line crossover

0eff =dP

dV 0eff =ΩRd

dV

( ) ( ) ( )0*

site4

f0site0eff 64860,

β−β=+β−β=β NKNN

dPdV ( )

µ

×=Ω

βT

KNNd

dVtt N

sN

R

cosh2120, 3f

0eff

RΩRΩ

P P

0eff =dP

dV 0eff =ΩRd

dV

5.70 5.69 b*= 5.68

K4=0.0

K4=1.0E-5 K4=2.0E-5

K4=3.0E-5

0eff =ΩRd

dV

0eff =ΩRd

dV