Exploring the QCD phase diagram
Transcript of 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)
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?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
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
* 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
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♥
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 ♥
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♥
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
♥
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
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
♥
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
♥
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♥
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
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
QCD critical point DISAPPEARED
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
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
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 !