Exploring the QCD phase diagram

Statistical Mechanics and Field Theory INFN Bari, September 2006

Exploring the QCD phase diagram

Owe PhilipsenUniversitat Munster

With Ph. de Forcrand (ETH,CERN), hep-lat/0607017

• Introduction

• Lattice QCD at finite density, the imaginary µ approach

• Numerical results for Nf = 3

• Numerical results for Nf = 2 + 1

• Conclusions

QCD: conjectured phase structure

Non-pert. problem ⇒Lattice 1975-2001: µ 6= 0 impossible ⇒sign problem

Where does this picture come from?

• Simulations on T -axis (light quarks only now)

• models for T = 0, µ 6= 0

Since recently: phase diagram, µq/T <∼ 1: Reweighting, Taylor expansion, imaginary µ

Take more general view ⇒parameter space {mu,d,ms, T, µ}

How experiment probes the phase transition & QGP....

Nf = 3 phase diagram 3d: {m,T, µ}

5.2 5.3 5.40

0.1

0.2

0.3

0.4

0.5

0.6

m

m

0

mu

T

m

1.O.

•confined/deconfined ⇒pseudo-crit. surface T0(µ,m) (from susceptibilities)

•1.O./crossover ⇒line of crit.points TE(µ) = T0(µ,mc(µ)) (from finite size scaling)

Projection onto (pseudo-) critical surface:

⇒µc(m) or mc(µ)

0

m

mu

m_c

The case Nf = 2 + 1, µ = 0:

?

?phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1storder

2nd orderO(4) ?

2nd orderZ(2)

crossover

1storder

d

tric

∞

∞Pure

•

⇒mc(µ = 0) (unimproved KS)

Bielefeld; Columbia; de Forcrand, O.P.

Nf = 3 universality: 3d Ising model Bielefeld

N.B: mc has strong cut-off effects!

(factor 1/4?) Bielefeld, MILC

Nf = 2,m = 0: is it O(4)/O(2) or first order?

•previous evidence inconclusive cf. old proceedings

New investigations:

D’Elia, Di Giacomo, Pica: FSS onL3×4, L = 16−32, standard KS, R-algorithm,m/T >∼ 0.055

⇒prefers first order

Kogut, Sinclair: FSS , χQCD, standard KS, fitting to small volume O(2) model ⇒O(2) scaling

The case Nf = 2 + 1, µ = 0:

?

?phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1storder

2nd orderO(4) ?

2nd orderZ(2)

crossover

1storder

d

tric

∞

∞Pure

•

⇒mc(µ = 0) (unimproved KS)

Bielefeld; Columbia; de Forcrand, O.P.

Nf = 3 universality: 3d Ising model Bielefeld

N.B: mc has strong cut-off effects!

(factor 1/4?) Bielefeld, MILC

Finite density, µ 6= 0:

* QCD critical point

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

QCD critical point DISAPPEARED

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

Lattice QCD at finite temperature and density

Difficult (impossible?): sign problem of lattice QCD

Z =

∫

DU [detM(µ)]fe−Sg[U ], Sf =∑

f

ψMψ

det(M) complex for SU(3), µ = µB/3 6= 0 ⇒ no Monte Carlo importance sampling

Evading the sign problem:

I. Two-parameter reweighting in (µ, β) Fodor, Katz

Z =

⟨

e−Sg(β) det(M(µ))

e−Sg(β0) det(M(µ = 0))

⟩

µ=0,β0

= e∆F/T ∼ e−const.V

idea: simulate at β0 = βc(0), better overlap

by sampling both phases; errors? ovlp.?

II. Taylor expansion

idea: for small µ/T , compute coeffs. of Taylor series ⇒local ops.

⇒gain V convergence?

Bielefeld/Swansea: Nf = 2,m/T0 = 0.4, improved KS, no critical signal at O(µ6)

Gavai/Gupta: Nf = 2,m/T0 = 0.1, standard KS, critical point at µcB/T = 1.1 ± 0.2

III.a Imaginary µ + analytic continuation

de Forcrand, O.P.

D’Elia, Lombardo

Azcoiti et al.

Chen, Luofermion determinant positive ⇒no sign problem

idea: for small µ/T , fit full simulation results of imag. µ by Taylor series

• vary two parameters (µ, T ) ⇒controlled continuation?

III.b Imaginary µ + Fourier transformationde Forcrand, Kratochvila

Alexandru et al

idea: canonical partition function at fixed baryon density

•no analytic continuation, but determinant needed ⇒thermodynamic limit?

QCD at complex µ: general properties

Z(V, µ, T ) = Tr(

e−(H−µQ)/T)

; µ = µr + iµi; µ = µ/T

exact symmetries: µ-reflection and µi-periodicity Roberge, Weiss

Z(µ) = Z(−µ), Z(µr, µi) = Z(µr, µi + 2π/Nc)

Imaginary µ phase diagram:

Z(3)-transitions: µci = 2π

3

(

n+ 12

)

1rst order for high T, crossover for low T

analytic continuation within:

|µ|/T ≤ π/3 ⇒µB <∼ 550MeV 0 1/3 2/3µΙ/(πT)

T

〈O〉 =

N∑

n

cnµ2ni ⇒ µi −→ −iµi

Analyticity of T0(µ) on finite V de Forcrand, O.P.

location of phase transition from

susceptibilites:

χ(β, aµ, V ) = V Nt

⟨

(O − 〈O〉)2⟩

,

• finite volume:

suscept. always finite and analytic5.2 5.3 5.40

0.1

0.2

0.3

0.4

0.5

0.6

m

Critical line βc(aµ) defined by peak χmax ≡ χ(aµc, βc)

• implicit function theorem: χ(β, aµ) analytic ⇒βc(aµ) analytic!

symmetries: ⇒ βc(aµ) =∑

n cn(aµ)2n

What to expect in physical units?

Natural expansion parameter is µπT : • thermal perturbation theory

⇒ T0(µ)T0(µ=0) = 1 −O(1)

(

µπT0(0,m)

)2

+O(1)(

µπT0(0,m)

)4

+ . . .

T0(µ) : Nf = 3 results, quark mass dependence , unimproved KS, 83 − 163 × 4

β0(aµ, am) =∑

k,l=0

ckl (aµ)2k (am− amc(0))l

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

5.2

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

β 0-c

01(a

m-a

mc(

0))

(aµi)2

O(µ2)O(µ4)

0 0.5 1 1.5 2 2.5 3µ

B/T

c

0.9

0.92

0.94

0.96

0.98

1

T/Tc

Nf=3

Nf=2+1 Fodor,Katz

⇒data well described by µ2 fit ! similar for pressure, screening masses

T0(µ,m)

T0(µ = 0,mc(0))= 1 + 2.111(17)

m−mc(0)

πT0− 0.667(6)

(

µ

πT0(0,m)

)2

+ . . .

Comparing different approaches: Nf = 2 (left), Nf = 4 (right):

Reweighting vs. imag. µ (FK, FP) Rew., imag. µ, canonical ensemble ...

4.84.824.844.864.88

4.94.924.944.964.98

55.025.045.06

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

β

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

de Forcrand, Kratochvila, 63

All agree on T0(m,µ)!!! (µ/T <∼ 1)

The critical endpoint and its quark mass dependence in Nf = 3

0

µ2

T

m>mc(0)

m=mc(0)

m<mc(0)

?

?phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1storder

2nd orderO(4) ?

2nd orderZ(2)

crossover

1storder

d

tric

∞

∞Pure

•

Expect: mc(µ)mc(µ=0) = 1 + c1

(

µπT

)2+ . . .

Inverted: curvature of critical surface µc(m)


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

Criticality: cumulant ratios, Binder cumulant

B4(m,µ) =〈(δψψ)4〉

〈(δψψ)2〉2, δψψ = ψψ − 〈ψψ〉

3d Ising universality :

V → ∞, step function

B4(m,µ) → 3 crossover

B4(mc, µc) → 1.604 critical

B4(m,µ) → 1 first order

finite V + FSS

B_4

mm_c

Ising infinite V

finite V

Redoing Nf = 3 with an exact algorithm

controlling algorithmic step size errors prohibitively expensive for smallm

⇒use exact algorithm here: Rational Hybrid MC Clark, Kennedy

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

0 0.2 0.4 0.6 0.8 1 1.2

B4

(δτ/am)2

RHMC-alg.

am=0.034am=0.030am=0.025

ising

critical massmc(0) with RHMC:

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.015 0.02 0.025 0.03 0.035 0.04

B4

am

R-alg.

⇒25% change in mc(0)

⇒∼10% change in mcπ

cf. Kogut, Sinclair

Finite size scaling:

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=8L=12L=16Ising

Fit: ν = 0.67(13)

ξ ∼ L−ν , νIsing = 0.63

⇒B4(m,L) = b0 + bL1/ν(m−mc0)

Computing mc(µ) for Nf = 3

1.45

1.5

1.55

1.6

1.65

1.7

1.75

0.02 0.024 0.028 0.032 0.036

B4

(aµi)2

aµi=0.00aµi=0.14aµi=0.17aµi=0.20aµi=0.24 0.022

0.024

0.026

0.028

0.03

0.032

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

mc

(aµi)2

O(µ2)

About 300k trajectories per data point!

Fitting to Taylor series B4(am, aµ) = 1.604 +B(

am− amc(0) + A(aµ)2)

+ . . .

⇒amc(µ)

amc(µ = 0)= 1 + c′1(aµ)2 + . . .

N.B. finite a effect:amc(µ)

amc(0)6=mc(µ)

mc(0); a =

1

T (µ)Nt= a(µ)

Continuum conversion:

c1 =1

mc(0)

dmc

d(µ/πT )2=

π2

N2t (amc)(0)

d(amc)

d(aµ)2+

1

T0

dT0

d(µ/πT )2.

....leads to negative curvature of critical mass:

⇒mc(µ)

mc(µ = 0)= 1 − 0.7(4)

( µ

πT

)2

+ . . .

A non-standard scenario: no critical point? sign ofdmc(µ)

dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

Conventional wisdom

confined

QGP

Color superconductor

m < mc(0)

Tc


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

Conventional wisdom

confined

QGP


m = mc(0)

Tc♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

Conventional wisdom

confined

QGP


m > mc(0)

Tc ♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

Conventional wisdom

confined

QGP


m > > mc(0)

Tc♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

Conventional wisdom

confined

QGP


m > > > mc(0)

Tc

♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

confined

QGP


m < < < mc(0)

Tc


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

confined

QGP


m < < mc(0)

Tc

♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

confined

QGP


m < mc(0)

Tc

♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

confined

QGP


m = mc(0)

Tc♥


dµ2 |µ=0


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

confined

QGP


m > mc(0)

Tc

In either case: mc(µ) slowly varying

smallness of dmc(µ)dµ2 |µ=0⇒very high quark mass sensitivity of µc

Can one expect the critical point to be at “small” µ?

If µcB ∼ 360 MeV (FK), then

1 <m

mc(µ = 0)<∼ 1.05

fine tuning of quark masses !

Nf = 2 + 1 : (ms,mu,d) phase-diagram at µ = 0

?

?phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1storder

2nd orderO(4) ?

2nd orderZ(2)

crossover

1storder

d

tric

∞

∞Pure

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

If there is a tricritical point mtrics ≈ 2.8T0<∼ 500 MeV

Setting the scale

(marked by arrows):

(amu,d, ams) mπ/mρ mK/mρ

(0.0265,0.0265) 0.304(2) 0.304(2)

(0.005,0.25) 0.148(2) 0.626(9)

physical 0.18 0.6456

Nf = 2 + 1 : (ms,mu,d) phase-diagram at µB = i2.4T

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.005 0.01 0.015 0.02 0.025 0.03

ams

amu,d

µB=0µB=i 2.4 T

Real µ: first order region shrinking!

Unusual? ...the same happens for heavy quark masses!

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

(µ/Τ)^2

0

2

4

6

8

10

12

M/T

from µ_ιfrom µM_infinity limit

Potts, 72^3

first order transition

cross-over


crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

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

Contradiction with other lattice studies? ...not necessar ily!

• Fodor & Katz:{TE , µE} = {162(2), 120(13)} MeV ?

different systematics, cutoff effects

µ


crossover

mu,d

ms

X

1rst

0

∞

Real worldHeavy quarks

Nf=3F-K

µ

F&K keep (amq) fixed, while a(T (µ)) increases with µ

⇒Lighter quarks at larger µ may cause the phase transition

• Gavai & Gupta: µE/TE <∼ 1 ? different theory,Nf = 2

Are there other possibilities still....?

• critical point at finite µ not analytically connected to that at µ = 0

• additional critical structure logical possibility

Conclusions

• Mapping of QCD phase diagram possible for µq/T <∼ 1

• Critical endpoint is extremely quark mass sensitive

⇒µc<∼ 400 MeV requires nature to fine tune mq ’s close to µ = 0 critical line

• existence of QCD critical point as yet inconclusive

• so far a ∼ 0.3 fm, staggered only

• Need finer lattices to understand even qualitative features !

Documents

Exploring the QCD phase diagram