status of the qcd phase diagram from the...

Status of the QCD phase diagram from the lattice

Owe Philipsen

QGHXM St. Goar, 15.03.2011

Is there a critical end point in the QCD phase diagram?

Is it connected to a chiral phase transition?

Imaginary chemical potential: rich phase structure, benchmark for models!

The conference location...

The conference location...

...a critical point for Rhine navigation

The QCD phase diagram established by experiment:

B

Nuclear liquid gas transition with critical end point

QCD phase diagram: theorist’s view

T

µ

confined

QGP

Color superconductor

Tc!

~170 MeV

~1 GeV?

Expectation based on models: NJL, NJL+Polyakov loop, linear sigma models, random matrix models, ...

Until 2001: no finite density lattice calculations, sign problem!

earl

y un

iver

se

heavy ion collisio

ns

compact stars ?

B

QGP and colour SC at asymptotic T and densities by asymptotic freedom!

The Monte Carlo method, zero density

QCD partition fcn:

links=gauge fields

lattice spacing a<< hadron << L ! thermodynamic behaviour, large V !

Monte Carlo by importance sampling U

det M e!Sgauge

T =1

aNtNt !", a! 0Continuum limit:

Here:

Z =!

DU"

f

det M(µf , mf ;U) e!Sgauge(!;U)

Nt = 4! 10 a ! 0.1" 0.3 fm staggered fermions

Theory: how to calculate p.t., critical temperature

= crit. exponent

The order of the p.t., arbitrary quark masses

chiral p.t.restoration of global symmetry in flavour space

µ = 0

deconfinement p.t.: breaking of global symmetry

SU(2)L ! SU(2)R ! U(1)A

Z(3)

anomalous

chiral critical line

deconfinement critical line

How to identify the critical surface: Binder cumulant

B4(!̄!) !"("!̄!)4#

"("!̄!)2#2V !"

$%

!

"

#

1.604 3d Ising1 first $ order3 crossover

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.018 0.021 0.024 0.027 0.03 0.033 0.036

B4

am

L=8

L=12

L=16

Ising

B4(m, L) = 1.604 + bL1/!(m ! mc0), ! = 0.63µ = 0 :

university-logo

Intro Tc CEP Results Discussion Concl. Others Strategy

Observable: Binder cumulant

• Probability distribution of order parameter

- distinguishes crossover (Gaussian) vs 1rst order (2 peaks)

- 2nd order: scale-invariant distribution with known Ising exponents

- encoded in Binder cumulant

• Measure B4(!̄!) ! "("!̄!)4#"("!̄!)2#2

!!!"("!̄!)3#=0

=

"

#

$

3 crossover

1 first$order1.604 3d Ising

for V % #

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1

First order

Crossover

V1V2 > V1

V3 > V2 > V1V=$$"

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.018 0.021 0.024 0.027 0.03 0.033 0.036

B4

am

L=8

L=12

L=16

Ising

• Finite volume, µ= 0: B4(am) = 1.604+ c(L)(am$amc

0)+ . . . , c(L) % L1/&

Ph. de Forcrand INT, Aug. 2008 Controlled crit. pt.

How to identify the order of the phase transition

x! xc

parameter along phase boundary, T = Tc(x)

Hard part: order of p.t., arbitrary quark masses

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.01 0.02 0.03 0.04

ams

amu,d

Nf=2+1

physical point

mstric - C mud

2/5

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

!

!

Pure

deconf. p.t.

chir

al p

.t.

physical point: crossover in the continuum Aoki et al 06

chiral critical line on de Forcrand, O.P. 07

consistent with tri-critical point at

But: chiral O(4) vs. 1st still open Di Giacomo et al 05, Kogut, Sinclair 07 anomaly! Chandrasekharan, Mehta 07

Nt = 4, a ! 0.3 fm

mu,d = 0, mtrics ! 2.8T

Nf = 2UA(1)

chiral critical line

µ = 0

The nature of the phase transition at the physical point Fodor et al. 06

...in the staggered approximation...in the continuum...is a crossover!

The nature of the transition for phys. masses Aoki et al. 06

How to identify the critical surface: Binder cumulant

B4(!̄!) !"("!̄!)4#

"("!̄!)2#2V !"

$%

!

"

#

1.604 3d Ising1 first $ order3 crossover

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.018 0.021 0.024 0.027 0.03 0.033 0.036

B4

am

L=8

L=12

L=16

Ising

B4(m, L) = 1.604 + bL1/!(m ! mc0), ! = 0.63µ = 0 :

university-logo

Intro Tc CEP Results Discussion Concl. Others Strategy

Observable: Binder cumulant

• Probability distribution of order parameter

- distinguishes crossover (Gaussian) vs 1rst order (2 peaks)

- 2nd order: scale-invariant distribution with known Ising exponents

- encoded in Binder cumulant

• Measure B4(!̄!) ! "("!̄!)4#"("!̄!)2#2

!!!"("!̄!)3#=0

=

"

#

$

3 crossover

1 first$order1.604 3d Ising

for V % #

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1

First order

Crossover

V1V2 > V1

V3 > V2 > V1V=$$"

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.018 0.021 0.024 0.027 0.03 0.033 0.036

B4

am

L=8

L=12

L=16

Ising

• Finite volume, µ= 0: B4(am) = 1.604+ c(L)(am$amc

0)+ . . . , c(L) % L1/&


How to identify the order of the phase transition

x! xc

parameter along phase boundary, T = Tc(x)

The ‘sign problem’ is a phase problem

importance sampling requirespositive weights

Dirac operator:

Lattice QCD at finite temperature and density

Difficult (impossible?): sign problem of lattice QCD

Z =

!

DU [detM(µ)]fe!Sg[U ]

positivity governed by !5-hermiticity: D/ (µ)† = !5D/ (!µ")!5

"det(M) complex for SU(3), µ #= 0

"real positive for SU(2), µ = iµi

N.B.: all expectation values are real, but MC importance sampling impossible

The following methods evade the sign problem, they don’t cure it!

N.B.: all expectation values real, imaginary parts cancel, but importance sampling config. by config. impossible!

D/ (µ)† = !5D/ (!µ!)!5

Z =

!DU [detM(µ)]fe!Sg[U ]

Lattice QCD at finite temperature and density

Difficult (impossible?): sign problem of lattice QCD

Z =

!

DU [detM(µ)]fe!Sg[U ]

positivity governed by !5-hermiticity: D/ (µ)† = !5D/ (!µ")!5

"det(M) complex for SU(3), µ #= 0

"real positive for SU(2), µ = iµi

N.B.: all expectation values are real, but MC importance sampling impossible

The following methods evade the sign problem, they don’t cure it!

µu = !µd

Same problem in many condensed matter systems!

Finite density: methods to evade the sign problem

Reweighting:

Optimal: use |det| in measure, reweight in phase

Taylor expansion:

Imaginary : no sign problem, fit by polynomial, then analytically continue

All require !

U

Sµ=0 finite µ

~exp(V) statistics needed,overlap problem

coefficients one by one, convergence?

requires convergencefor analytic continuation

µ/T < 1

!O"(µi) =N!

k=0

ck

" µi

!T

#2k, µi # $iµ

!O"(µ) = !O"(0) +!

k=1

ck

" µ

!T

#2k

Z =!

DU det M(0)det M(µ)det M(0)

e!Sg

use for MC calculate

µ = iµi

inte

gran

d

Comparing approaches: the critical line

university-logo

Intro Tc CEP Results Discussion Concl.

The good news: curvature of the pseudo-critical line

All with Nf = 4 staggered fermions, amq = 0.05,Nt = 4 (a! 0.3 fm)

PdF & Kratochvila

4.8

4.82

4.84

4.86

4.88

4.9

4.92

4.94

4.96

4.98

5

5.02

5.04

5.06

0 0.5 1 1.5 2

1.0

0.95

0.90

0.85

0.80

0.75

0.70

0 0.1 0.2 0.3 0.4 0.5!

T/T

c

µ/T

a µ

confined

QGP<sign> ~ 0.85(1)

<sign> ~ 0.45(5)

<sign> ~ 0.1(1)

D’Elia, Lombardo 163

Azcoiti et al., 83

Fodor, Katz, 63

Our reweighting, 63

deForcrand, Kratochvila, 63

imaginary µ

2 param. imag. µ

dble reweighting, LY zeros

Same, susceptibilities

canonical

Agreement for µ/T ! 1


de Forcrand, Kratochvila LAT 05

; same actions (unimproved staggered), same massNt = 4, Nf = 4

Test of methods: comparing Tc(µ) de Forcrand, Kratochvila 05

The calculable region of the phase diagram

T

µ

confined

QGP


Tc!

need

Upper region: equation of state, screening masses, quark number susceptibilities etc.under control

µ/T <! 1 (µ = µB/3)

The (pseudo-) critical temperature

Tc(µ)Tc(0)

= 1! !(Nf , mq)! µ

T

"2+ . . .

de Forcrand, O.P. 03D’Elia, Lombardo 03

! ! Nf

Nc

Curvature rather small

Toublan 05

Tc(µ)Tc(0)

= 1! 0.059(2)(4)! µ

T

"2+ O

#! µ

T

"4$

Curvature of crit. line from Taylor expansion2+1 flavours, Nt=4, 8 improved staggered

Extrapolation to chiral limit assuming O(4),O(2) scaling of magn. EoS

Consistent with determinations at finite mass with imag. chem. pot.

Phase boundary in the chiral limit hotQCD 11

scaling form of magnetic EoS

=2!T

t0msh!(1!!)/!" dfG(z)

dz

z = h1/!"/t

Phase boundary for physical masses Endrödi et al. 11

Curvature of crit. line from Taylor expansion2+1 flavours, Nt=6,8,10 improved staggered

Observables

Continuum extrapolation:

!̄!r, "s

t =T ! Tc

Tc

Comparison with freeze-out curve

freeze-out

Much harder: is there a QCD critical point?


1


12

Fodor,Katz JHEP 04

abrupt change: physics or problem of the method? Splittorff 05; Han, Stephanov 08

Lee-Yang zero:

Critical point from reweighting

physical quark masses, unimproved staggered fermionsNt = 4, Nf = 2 + 1

Approach 1a: CEP from reweighting Fodor, Katz 04

Splittorf 05, Stephanov 08

Approach 1b: CEP from Taylor expansion

p

T 4=!!

n=0

c2n(T )" µ

T

#2n

Nearest singularity=radius of convergenceµE

TE= lim

n!"

!""""c2n

c2n+2

"""", limn!"

""""c0

c2n

""""

12n

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 1 2 3 4 5 6

µB / Tc(0)

T / Tc(0)

!2 !4

nf=2+1, m"=220 MeVnf=2, m"=770 MeV

CEP from [5]CEP from [6]

Bielefeld-Swansea-RBC

FK

Gavai, Gupta

Nf = 2

improved staggered Nt = 4

Predictivity ?

0

1

2

3

4

5

6

7

8

0.85 0.9 0.95 1 1.05 1.1

T/Tc(0)

! !2(p/T4)!4(p/T4)!2("B)!4("B)

Radius of convergence necessary condition, but can it proof the existence of a CEP?

Hadron resonance gas

C.Schmidt, hotQCD 09

Different definitions agree only for not n=1,2,3,... CEP may not be nearest singularity, control of systematics?

n!"

Approach 2: follow chiral critical line surface

mc(µ)mc(0)

= 1 +!

k=1

ck

" µ

!T

#2k

chir

al p

.t.

chir

al p

.t.

hard/easyde Forcrand, O.P. 08,09

Finite density: chiral critical line critical surface

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

!

!

Pure

* QCD critical point

crossover 1rst0

!

Real world

X

Heavy quarks

mu,dms

!

QCD critical point DISAPPEARED

crossover 1rst0

!

Real world

X

Heavy quarks

mu,dms

!

mc(µ)

mc(0)= 1 +

!

k=1

ck

" µ

!T

#2k

T

µ

confined

QGP


m > mc(0)

Tc

T

!

confined

QGP


m > > mc(0)

Tc

!

transition strengthens transition weakens

Curvature of the chiral critical surface

de Forcrand, O.P. 08,09

On coarse lattice exotic scenario: no chiral critical point at small density

Weakening of p.t. with chemical potential also for:

-Heavy quarks de Forcrand, Kim, Takaishi 05

-Light quarks with finite isospin density Kogut, Sinclair 07

-Electroweak phase transition with finite lepton density Gynther 03

Towards the continuum: Nt = 6, a ! 0.2 fm

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

!

!

Pure

Nt=4

mc!(Nt = 4)

mc!(Nt = 6)

! 1.77

Physical point deeper in crossover region as

Cut-off effects stronger than finite density effects!

Preliminary: curvature of chiral crit. surface remains negative de Forcrand, O.P. 10

No chiral critical point at small density, other crit. points possible

a! 0

Nf = 3 de Forcrand, Kim, O.P. 07Endrodi et al 07

Nt=6

Same statement with different methods

Endrödi et al., 11 findweakening of crossover

Study suitably defined width of crossover region

strengthening of transition

Chiral/deconf. and Z(3) transitions at imaginary

Nf=4: D’Elia, Di Renzo, Lombardo 07 Nf=2: D’Elia, Sanfilippo 09 Nf=3:

Strategy: fix , measure Im(L), order parameter at

determine order of Z(3) branch/end point as function of m

µi

T= !µi

T=

!

3, !

µ

de Forcrand, O.P. 10

0 0.5 1 1.5 2 2.5 3 3.5 4

T

µi

T /(!3 )

ordered

disordered

ordered

disordered

Results:

1

1.5

2

2.5

3

5.216 5.217 5.218 5.219

B4(Im

(L))

!

m=0.040

L=8L=12L=16

1

1.5

2

2.5

3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

B4(Im

(L))

(!-!c)L1/"

m=0.040

L=8L=12L=16

! = 0.33

B4 at intersection has large finite size corrections (well known), more stable!

B4(!, L) = B4(!c,!) + C1(! " !c)L1/! + C2(! " !c)2L2/! . . .

! = 0.33, 0.5, 0.63

for 1st order, tri-critical, 3d Ising scaling

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.01 0.1 1

!

quark mass

Second order

Tricritical

First order

On infinite volume, this becomes a step function, smoothness due to finite L

T

m

Phase diagram at fixedµi

T=

!

3, !

triple line

triple line

3d Ising

ordered phase

disordered phase

Critical lines at imaginary µ

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

!

!

Pure

µ = i!T

3µ = 0

-Connection computable with standard Monte Carlo! -Here: heavy quarks in eff. theory

triple

tricritical1st ordertriple

1st order

2nd order 3d Ising

mu,d

ms

?

Critical surfaces

shape, sign of curvature determined by tric. scaling!

Similar chiral crit. surface: tric. line renders curvature negative!

de Forcrand, O.P. 10

(µ/T)2

ms!

mu,d!

0crossover

1.O.chiral 1.O.

deconf.

tric.

x phys. point

(µ/T)2

ms!

mu,d!

0crossover

1.O.chiral 1.O.

deconf.

tric.

x phys. point

-("/3)2-("/3)2-("/3)2-("/3)2

0.5

1

1.5

2

2.5

3

3.5

4

-1.5 -1 -0.5 0 0.5 1

mc

(µ/T)2

Potts: QCD, Nt=1, strong coupling series:Langelage, O.P. 09

mc

T(µ2) =

mtric

T+ K

!"!

3

#2+

" µ

T

#2$2/5

tri-critical scaling: exponent universal

6.5

7

7.5

8

8.5

9

9.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

mc /

T

(µ/T)2

Imaginary µ Real µ

6.6+1.6((!/3)2+(µ/T)2)2/5

Conclusions

Reweighting, Taylor: indications for critical point, systematics ?

Chiral crit. surface, deconfinement crit. surface: Transitions weaken with chemical potential

For lattices a~0.3, 0.2 fm no chiral critical point for

Still possible: chiral critical point at large chemical potential non-chiral critical point(s)

µ/T <! 1

?

status of the qcd phase diagram from the...

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