The QCD phase diagram at non-zero baryon density Owe Philipsen - - PowerPoint PPT Presentation

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!

early universe h e a v y i

• n

c

• l

l i s i

• n

s compact stars ?

B

Model predictions for critical end point (CEP)

How to get funding for heavy ion programs:

How to get funding for heavy ion programs:

Thermal QCD in experiment

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<< hadron << L ! thermodynamic behaviour, large V ! typically dim. integral Monte Carlo, importance sampling

> 108 − 1010 U det M e−Sgauge T = 1 aNt Nt → ∞, a → 0

Continuum limit:

Nt = 4, 6

Here:

a ∼ 0.3, 0.2 fm Z =

• DU
• f

det M(µf, mf; U) e−Sgauge(β;U) Light fermions expensive, , here staggered fermions

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( ¯ ψψ) ≡ (δ ¯ ψψ)4 (δ ¯ ψψ)22

V →∞

− →    1.604 3d Ising 1 first − order 3 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 − mc

0),

ν = 0.63

µ = 0 :

• 0.5

1 1.5 2 2.5 3 3.5 4

• 1
• 0.5

0.5 1 First order Crossover V1 V2 > V1 V3 > V2 > V1 V="

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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.01 0.02 0.03 0.04 ams amu,d Nf=2+1 physical point ms

tric - C mud 2/5 phys. point

N = 2 N = 3 N = 1

f f f

m

s s

m Gauge m , m

u

1st

2nd order O(4) ? 2nd order Z(2) 2nd order Z(2)

crossover 1st

d tric

! !

Pure

• deconf. p.t.

chiral 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, mtric

s

∼ 2.8T Nf = 2 UA(1) chiral critical line

µ = 0

How to identify the critical surface: Binder cumulant

B4( ¯ ψψ) ≡ (δ ¯ ψψ)4 (δ ¯ ψψ)22

V →∞

− →    1.604 3d Ising 1 first − order 3 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 − mc

0),

ν = 0.63

µ = 0 :

• 0.5

1 1.5 2 2.5 3 3.5 4

• 1
• 0.5

0.5 1 First order Crossover V1 V2 > V1 V3 > V2 > V1 V="

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 requires positive weights Dirac operator:

⇒det(M) complex for SU(3), µ = 0 ⇒real positive for SU(2), µ = iµi

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

D / (µ)† = γ5D / (−µ∗)γ5 Z =

• DU [det M(µ)]fe−Sg[U]

⇒real positive for

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 S

µ=0 finite µ

~exp(V) statistics needed,

• verlap problem
• coeffs. one by one,

convergence? requires convergence for 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

integrand

Comparing approaches: the critical line

uni

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 0.5 1 1.5 2 1.0 0.95 0.90 0.85 0.80 0.75 0.70 0.1 0.2 0.3 0.4 0.5

• T/Tc

µ/T a µ

confined QGP

<sign> ~ 0.85(1) <sign> ~ 0.45(5) <sign> ~ 0.1(1) D’Elia, Lombardo 16

3

Azcoiti et al., 8

3

Fodor, Katz, 6

3

Our reweighting, 6

3

deForcrand, Kratochvila, 6

3

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 mass

Nt = 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 Color superconductor 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?

Much harder: is there a QCD critical point?

1

Much harder: is there a QCD critical point?

1 2

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 fermions

Nt = 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?

µE TE = lim

n→∞

• c2n

c2n+2

• ,

lim

n→∞

• c0

c2n

• 1

2n

n → ∞

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1 2 3 4 5 6 µB / Tc(0) T / Tc(0) !2 !4 nf=2+1, m"=220 MeV nf=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 chiral p.t. chiral p.t.

hard/easy de Forcrand, O.P. 08,09

Approach 2: imaginary rather than imagined (?) CEP

mc(µ) mc(0) = 1 +

• k=1

ck µ πT 2k chiral p.t. chiral p.t.

hard/easy de Forcrand, O.P. 08,09

Finite density: chiral critical line critical surface

phys. point

N = 2 N = 3 N = 1

f f f

m

s s

m Gauge m , m

u

1st

2nd order O(4) ? 2nd order Z(2) 2nd order Z(2)

crossover 1st

d tric

! !

Pure

* QCD critical point crossover 1rst ! Real world X Heavy quarks mu,d ms ! QCD critical point DISAPPEARED crossover 1rst ! Real world X Heavy quarks mu,d ms !

mc(µ) mc(0) = 1 +

• k=1

ck µ πT 2k

T µ confined QGP Color superconductor m > mc(0) Tc T ! confined QGP Color superconductor m > > mc(0) Tc

!

Finite density: chiral critical line critical surface

phys. point

N = 2 N = 3 N = 1

f f f

m

s s

m Gauge m , m

u

1st

2nd order O(4) ? 2nd order Z(2) 2nd order Z(2)

crossover 1st

d tric

! !

Pure

* QCD critical point crossover 1rst ! Real world X Heavy quarks mu,d ms ! QCD critical point DISAPPEARED crossover 1rst ! Real world X Heavy quarks mu,d ms !

mc(µ) mc(0) = 1 +

• k=1

ck µ πT 2k

T µ confined QGP Color superconductor m > mc(0) Tc T ! confined QGP Color superconductor 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...

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

1 2 3 4 5 (µ/)^2 2 4 6 8 10 12 M/T from µ_ from µ M_infinity limit

critical M in Potts, 72^3

first order transition cross-over

QCD critical point DISAPPEARED crossover 1rst

• Real world

X Heavy quarks mu,d ms 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

0.5 1 1.5 2 2.5 0 2 4 6 8 10 50 100 150 200 250 300 Light Quark Mass [MeV] S t r a n g e Q u a r k M a s s [ M e V ] Quark Chemical Potential [MeV]

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

N = 2 N = 3 N = 1

f f f

m

s s

m Gauge m , m

u

1st

2nd order O(4) ? 2nd order Z(2) 2nd order Z(2)

crossover 1st

d tric

! !

Pure

Nt=4 Nt=6

mc

π(Nt = 4)

mc

π(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. 07 Endrodi et al 07

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

X phys.

O(4)

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

X phys.

O(4)

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

X phys.

O(4)

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

X phys.

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

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