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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
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?