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
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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