Phase diagrams by strong coupling methods:QCD at finite temperature and density

Owe Philipsen

Bad Honnef, February 2012

Introduction: The QCD phase diagram

The deconfinement transition in Yang-Mills theory

The deconfinement transition in QCD with heavy quarks

in collaboration with M. Fromm, J. Langelage, S. Lottini

Strong Interactions beyond the Standard Model

The (lattice) calculable region of the phase diagram

T

µ

confined

QGP

Color superconductor

Tc!

Sign problem prohibits direct simulation, circumvented by approximate methods:reweigthing, Taylor expansion, imaginary chem. pot., need

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

Here: phase diagram itself, so far based on models, most difficult!

µ/T <! 1 (µ = µB/3)

order of p.t.at zero densitydepends on Nf, quark mass

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

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

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 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

Much harder: is there a QCD critical point?

12

Some methods trying (1) give indications of critical point, but systematics not yet controlled

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

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

-Heavy quarks Fromm, Langelage, Lottini, O.P. 11

-Light quarks with finite isospin density Kogut, Sinclair 07

-Electroweak phase transition with finite lepton density Gynther 03

Larger densities? Try effective theories!

Example e.w. phase transition: success with dimensional reduction!

Scale “separation”Integrate hard scale perturbatively, treat eff. 3d theory on lattice, valid for sufficiently weak coupling

Does not work for the QCD transition, breaks Z(3) symmetry of Yang-Mills theory

Bottom up construction of Z(N)-invariant theory by matching couplings: works for SU(2), not for SU(3) Vuorinen, Yaffe; de Forcrand, Kurkela; ....

Here: solution by strong coupling expansion!

Convergent series within finite convergence radius, valid in confined phase

Starting point: Wilson’s lattice action

Plaquette:

The strong coupling expansion

Here: effective lattice theory, general strategy

The effective theory for SU(2)

c

c

Generalisation to SU(3)

L

Numerical evaluation of effective theories

Monte Carlo simulation of scalar model, Metropolis update

Search for criticality:

Binder cumulant:

Susceptibility:

Finite size scaling:

Numerical results for SU(3)

0

0.002

0.004

0.006

0.008

0.01

0.175 0.18 0.185 0.19 0.195 0.2 0.205

χ |L|

λ1

Ns = 06Ns = 08Ns = 10Ns = 12Ns = 14

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

0.175 0.18 0.185 0.19 0.195 0.2 0.205

|L|

λ1

Ns = 06Ns = 08Ns = 10Ns = 12Ns = 14

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Im(L

)

Re(L)

λ1 = 0.178λ1 = 0.202

Order-disorder transition

!c

!c!c=

>

<!!

!

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8A

bund

ance

|L|

First order phase transition for SU(3) in the thermodynamic limit!

0.64

0.645

0.65

0.655

0.66

0.665

0.67

6 8 10 12 14

B |L|

min

imum

Ns

DataFit

2/3Asymptotic value

Histogram estimate

The influence of a second coupling

...gets very small for large !N!

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19

0 0.002 0.004 0.006 0.008 0.01

λ 1

λ2

from χ|L|from B|L|second-order fit on B|L|Nτ = 2Nτ = 3Nτ = 4Nτ = 6Nτ = 8

0.14 0.145 0.15

0.155 0.16

0.165 0.17

0.175 0.18

0.185 0.19

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035λ 1

λa

from χ|L|from B|L|third-order fit on χ|L|Nτ = 1Nτ = 2Nτ = 3Nτ = 4Nτ = 6Nτ = 10

NLO-couplings: next-to-nearest neighbour, adjoint rep. loops

Numerical results for SU(2), one coupling

0.92 0.94 0.96 0.98 1 1.02 1.04

Ab

un

da

nce

|L|

!1 = 0.1940

!1 = 0.1944

!1 = 0.1948

!1 = 0.1952

!1 = 0.1956

!1 = 0.1960

!1 = 0.1964

!1 = 0.1968

!1 = 0.1972

!1 = 0.1976

!1 = 0.1980

0.196

0.198

0.2

0.202

0.204

10 15 20 25

!1

Ns

datasecond order fit ["=0.63002]

TD limit 0.195374(42)

Second order (3d Ising) phase transition for SU(2) in the thermodynamic limit!

Mapping back to 4d finite T Yang-Mills

Inverting

!1(N! , ")! "c(!1,c, N! ) ...points at reasonable convergence

SU(3)

5 5.2 5.4 5.6 5.8

6 6.2 6.4 6.6 6.8

7 7.2

2 4 6 8 10 12 14 16

β c

Nτ

Order 10Order 9Order 8

Comparison with 4d Monte Carlo

-7

-6

-5

-4

-3

-2

-1

0

1

2 4 6 8 10 12 14 16

% d

evia

tio

n

N!

M=1 resultsM=infinity results

Relative accuracy for compared to the full theory!c

SU(3)SU(2)

-8

-6

-4

-2

0

2

4

6

8

2 4 6 8 10 12 14 16

% d

evia

tion

of β

c

Nτ

From (1)From (1,2)From (1,a)

Note: influence of some couplings checked explicitly!

Extrapolation to continuum limit!

-error bars: difference between last two orders in strong coupling exp.

-using non-perturbative beta-function (4d T=0 lattice)

-all data points from one single 3d MC simulation!

200

300

400

500

600

700

800

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

T c [M

eV]

1/N!2

One-coupling effective theory270 MeV

Linear fit: Tc = 250(14) MeV

Including fermions

QCD: first order deconf. transition region

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

deconfinement p.t.: breaking of global symmetry;explicitly broken by quark massestransition weakens

Z(3)

Phase diagram in eff. theory:

c!,c

h II

ordered (deconfined)

disordered (confined)

h

!0

!

crossover

I

Phase boundary in two-coupling space

0.1855

0.186

0.1865

0.187

0.1875

0.188

0 0.0003 0.0006 0.0009 0.0012

!

h

!pc(h)Linear fit

Critical point

Observable to identify order of p.t.:

!BQ = B4(!Q) =!(!Q)4"!(!Q)2"2

B4(x) = 1.604 + bL1/!(x! xc) + . . .

1.4

1.5

1.6

1.7

1.8

1.9

2

-0.02 -0.01 0 0.01 0.02

B 4, Q

x

Ns = 16Ns = 18Ns = 20Ns = 22Ns = 24

Scaling function

1.4

1.5

1.6

1.7

1.8

1.9

2

0.0004 0.0006 0.0008 0.001 0.0012

B 4, Q

h

Ns = 16Ns = 18Ns = 20Ns = 22Ns = 24

Critical point

Mapping back to QCD:

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

N!

" c(N!)

"chiral

"0

"2

"2, resum."4

"4, resum.

Convergence properties:

Finite density: sign problem solvable

Critical quark mass as function of chemical potential

The fully calculated deconfinement transition

0 0.5 1 1.5 2 2.5 3 6 7 8 9 10 11 12 13 14 15

0.9

0.95

1T

T0

µ

T M!

2T

1st

1st

Mtric

T! 6.3

1st

-1-0.5

0 0.5

1 1.5

2 2.5

3

Mu,d/T

Ms/T

-1-0.5

0 0.5

1 1.5

2 2.5

3

MtMM ric

T! 6.3

1st

!T( )2

Mtric

T! 5.6Mtric

T! 6.7

tricrit. Roberge-Weiss

2nd, Z2

Mtric

T+ K

!

3

2+

µ

T

2 2 5

deconfinement critical surface

phase diagram for Nf=2, Nt=6

OutlookHeavy quarks delay baryon condensation , want physical quark masses

Light quarks: need very large order hopping expansion, difficult

Instead: compute corrections to the strong coupling limitof the chiral staggered quark action, U(1) chiral symmetry

Links can be integrated, resulting monomer-dimer action simulated, worm algorithm

0 0.1 0.2 0.3 0.4 0.5a µ

0

0.5

1

1.5

aT =

!2 / N "

#$$% & 0

#$$% =

TCP

2nd order

1st orde

Liquid-gas transitionat infinite coupling, m=0de Forcrand, Fromm 09

Conclusions

Proposal for two-step treatment of QCD phase transition:

I. Derivation of effective action by strong-coupling expansionII. Simulation of effective theory

Z(N)-invariant effective theory for Yang-Mills, correct order of p.t.,crit. temperature ~10% accurate in the continuum limit

Deconfinement for heavy fermions and all chemical potentials

Hope for finite density QCD: Treatment of light fermions?