The QCD phase diagram at non-zero baryon density...The QCD phase diagram at non-zero baryon density Owe Philipsen Durham, 2009 Introduction Lattice techniques for ﬁnite temperature

The QCD phase diagram at non-zero baryon density

Owe Philipsen

Durham, 2009

Introduction

Lattice techniques for finite temperature and density

The phase diagram: the confusion before clarity?

Original work with Ph. de Forcrand (ETH/CERN)

Annual Theory Meeting

QCD at high temperature/density: change of dynamics

chiral condensate , Cooper pairs

0.1

0.2

0.3

0.4

0.5!s

1 10 100

Measured

QCD

Energy in GeV

Chiral symmetry: broken (nearly) restored

QCD at high temperature/density: change of dynamics

chiral condensate , Cooper pairs

0.1

0.2

0.3

0.4

0.5!s

1 10 100

Measured

QCD

Energy in GeV

Chiral symmetry: broken (nearly) restored

Phase transitions?

The QCD phase diagram established by experiment:

B

Nuclear liquid gas transition, Z(2) 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 i

on coll

isions

compact stars ?

B

Model predictions for critical end point (CEP)

How to get funding for heavy ion programs:

Thermal QCD in experimentThis is how experiment probes the phase transition & QGP....

heavy ion collision experiments at RHIC, LHC, GSI.... have finite baryon density!

?

Phase boundary from hadron freeze-out?

Theory: The Monte Carlo method

QCD partition fcn:

links=gauge fields

lattice spacing a

How to measure p.t., critical temperature

Lee,Yang:

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

chiral p.t.restoration of global

µ = 0

deconfinement p.t.: breaking of global

SU(2)L × SU(2)R × U(1)A

Z(3)

anomalous

How to identify the critical surface: Binder cumulant

B4(ψ̄ψ) ≡〈(δψ̄ψ)4〉

〈(δψ̄ψ)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/ν(m − mc0), ν = 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−amc0)+ . . . , c(L) % L1/&

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

How to identify the order of the phase transition

x− xcparameter 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

How to identify the critical surface: Binder cumulant

B4(ψ̄ψ) ≡〈(δψ̄ψ)4〉

〈(δψ̄ψ)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/ν(m − mc0), ν = 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−amc0)+ . . . , c(L) % L1/&


How to identify the order of the phase transition

x− xcparameter 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 γ5-hermiticity: D/ (µ)† = γ5D/ (−µ∗)γ5

⇒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)† = γ5D/ (−µ∗)γ5

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 γ5-hermiticity: D/ (µ)† = γ5D/ (−µ∗)γ5

⇒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!

1dim. illustration

Finite density: methods to evade the sign problem

Reweighting:

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

coeffs. one by one, convergence?

requires convergencefor anal. continuation

µ/T < 1

〈O〉(µi) =N∑

k=0

ck( µi

πT

)2k, µi → −iµ

〈O〉(µ) = 〈O〉(0) +∑

k=1

ck( µ

πT

)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 ~ 0.85(1)

~ 0.45(5)

~ 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

The good news: comparing Tc(µ) de Forcrand, Kratochvila 05

The (pseudo-) critical temperature

very flat, but not yet physical masses, coarse lattices

indications that curvature does not grow towards continuum de Forcrand, O.P. 07

extrapolation to physical masses and continuum is feasible! Budapest-Wuppertal 08

Comparison with freeze-out curve so far

freeze-out

The calculable region of the phase diagram

T

µ

confined

QGP


Tc!

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

Here: phase diagram itself, most difficult!

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

(entire curve generated from one point!)

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 convergence

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

µETE

= limn→∞

√∣∣∣∣c2n

c2n+2

∣∣∣∣, limn→∞

∣∣∣∣c0c2n

∣∣∣∣

12n

n→∞

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

Approach 2: follow chiral critical line surface

mc(µ)mc(0)

= 1 +∑

k=1

ck( µ

πT

)2k

chir

al p

.t.

chir

al p

.t.

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

Approach 2: imaginary rather than imagined (?) CEP

mc(µ)mc(0)

= 1 +∑

k=1

ck( µ

πT

)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

πT

)2k

T

µ

confined

QGP


m > mc(0)

Tc

T

!

confined

QGP


m > > mc(0)

Tc!

Curvature of the chiral critical surface

de Forcrand, O.P. 08,09

The chiral critical surface on a coarse lattice:

Higher order terms? Convergence?

Cut-off effects?

In any case: picture for QCD phase diagram not as clear as anticipated.....

08

Un-discovering a critical point feels like...

Scenario unusual? ...the same happens for heavy quark masses!

effective heavy quark theory, same universality class: 3-state Potts model

-3 -2 -1 0 1 2 3 4 5

(µ/!)^2

0

2

4

6

8

10

12

M/T

from µ_"from µM_infinity limit

critical M in Potts, 72^3

first order transition

cross-over

QCD critical point DISAPPEARED

crossover 1rst0

#

Real world

X

Heavy quarks

mu,dms

µ

Real µ: first order region shrinking! de Forcrand, Kim, Takaishi

also for finite isospin chemical potential Kogut, Sinclair

N.B.: non-chiral critical point still possible!

Recent model studies with similar results

0 0.5 1 1.5 2 2.5 0 2

4 6

8 10

0

50

100

150

200

250

300

Light Quark Mass [MeV] Strang

e Quar

k Mass

[MeV

]

Qua

rk C

hem

ical

Pot

entia

l [M

eV] GV = 0.8GS

K. Fukushima 08

NJL-Polyakov loop model with vector-vector interaction

Bowman, Kapusta 08

Linear sigma model with quarks

Qualitative behaviour as in exotic lattice scenario!

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

Nt=6

mcπ(Nt = 4)mcπ(Nt = 6)

≈ 1.77

Physical point deeper in crossover region as

Cut-off effects stronger than finite density effects!

Curvature of crit. surface (so far) consistent with 0; strong quark mass sensitivity!

a→ 0

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

The interplay of Nf=2 and Nf=2+1

Xphys.

O(4)

The interplay of Nf=2 and Nf=2+1

Xphys.

which critical surface do we need?

O(4)

More (and non-chiral) critical points?

Good time for models: NJL with vector interactions Zhang, Kunihiro, Fukushima 09 Ginzburg-Landau approach Baym et al. 06 for quark condensates ...

Conclusions

Working lattice methods available for

, EoS under control at small density

On coarse lattices a~0.3 fm no chiral critical point for

Large cut-off and quark mass effects

Uncharted territory: do QCD critical points exist?

Tc(µ)

µ < T

µ < T

The equation of state

Continuum extrapolation with physical quarks feasible in the near future

Equation of state at finite baryon density

Taylor expansion, second order Bielefeld-Swansea Reweighting Wuppertal

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

Documents

The QCD phase diagram at non-zero baryon density...The QCD phase diagram at non-zero baryon density Owe Philipsen Durham, 2009 Introduction Lattice techniques for ﬁnite temperature