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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
Page 2
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}
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How experiment probes the phase transition & QGP....
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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
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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
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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
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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.?
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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?
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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
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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
+ . . .
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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
+ . . .
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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)
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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)
* QCD critical point
crossover 1rst0
∞
Real world
X
Heavy quarks
mu,dms
µ
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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
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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
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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)
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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 + . . .
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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
+ . . .
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A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
Conventional wisdom
confined
QGP
Color superconductor
m < mc(0)
Tc
Page 20
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
Conventional wisdom
confined
QGP
Color superconductor
m = mc(0)
Tc♥
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A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
Conventional wisdom
confined
QGP
Color superconductor
m > mc(0)
Tc ♥
Page 22
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
Conventional wisdom
confined
QGP
Color superconductor
m > > mc(0)
Tc♥
Page 23
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
Conventional wisdom
confined
QGP
Color superconductor
m > > > mc(0)
Tc
♥
Page 24
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
confined
QGP
Color superconductor
m < < < mc(0)
Tc
Page 25
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
confined
QGP
Color superconductor
m < < mc(0)
Tc
♥
Page 26
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
confined
QGP
Color superconductor
m < mc(0)
Tc
♥
Page 27
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
confined
QGP
Color superconductor
m = mc(0)
Tc♥
Page 28
A non-standard scenario: no critical point? sign ofdmc(µ)
dµ2 |µ=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
T
µ
confined
QGP
Color superconductor
m > mc(0)
Tc
Page 29
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 !
Page 30
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
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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!
Page 32
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
QCD critical point DISAPPEARED
crossover 1rst0
∞
Real world
X
Heavy quarks
mu,dms
µ
Real µ: first order region shrinking! de Forcrand, Kim, Takaishi
Page 33
Contradiction with other lattice studies? ...not necessar ily!
• Fodor & Katz:{TE , µE} = {162(2), 120(13)} MeV ?
different systematics, cutoff effects
µ
QCD critical point DISAPPEARED
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
Page 34
Are there other possibilities still....?
• critical point at finite µ not analytically connected to that at µ = 0
• additional critical structure logical possibility
Page 35
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 !