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Lattice QCD @ nonzero temperature and finite density
Heng-Tong Ding (丁亨通)
Central China Normal University
34th International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK
Outline
• QCD phase structure
• QCD phase structure
T>0 & µ =0
T>0 & µ > 0• Equation of State
• Properties of QCD medium
Milestone: transition temperature from hadronic phase to QGP phase
See also the continuum extrapolated results of HISQ, stout & overlap in: Wuppertal-Budapest: Nature 443(2006)675, JHEP 1009 (2010) 073 , HotQCD: PRD 85 (2012)054503
HotQCD: PRL 113 (2014) 082001
Calculations with Domain wall, HISQ, stout fermions consistently give Tpc ~155 MeV
Not a true phase transition but a crossover
Domain wall fermions
Borsanyi et al., [WB collaboration], arXiv: 1510.03376, Phys.Lett. B713 (2012) 342
HotQCD, PRD 90 (2014) 094503, Wuppertal-Budapest, Phys. Lett. B730 (2014) 99
Milestone: QCD Equation of State
QCD phase structure in the quark mass plane
mu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
order
mc
HTD, F. Karsch, S. Mukherjee, arXiv:1504.0527
mq=0 or ∞ with Nf=3: a first order phase transition
Critical lines of second order transition Nf=2: O(4) universality class Nf=3: Z(2) universality class
The location of 2nd order Z(2) lines ?
The value of tri-critical point (ms ) ?tri
columbia plot, PRL 65(1990)2491
The influence of criticalities to the physical point ?
Pisarski & Wilczek, PRD29 (1984) 338
Gavin, Gocksch & Pisarski, PRD 49 (1994) 3079
K. Rajagopal & F. Wilczek, NPB 399 (1993) 395
F. Wilczek, Int. J. Mod. Phys. A 7(1992) 3911,6951
RG arguments:
Lattice QCD calculations:
scenarios of QCD phase transition at ml=0
mu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
order
mc
μ
1st order
criticalpoint
cross over
Tmphy
s < mtris
T
μ
2nd order
1st order
tri-critical point
mphys > mtri
s
mphys = mtri
s
T
μ
tri-critical point
1st order
QCD phase transition at the physical point
hadronic matter
quark mattercross over
critical point
1st order
m=mphy
de Forcrand & Philipsen, ’07
N = 2f
N = 3f
m
m
µ
s
phys. lineCROSSOVER
0u,d
FIRS
T O
RDER
o
o
0 o
o
Karsch et al., ’03, Nakamura et al., 15’
mu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
order
mc
mu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
order
mc
Nt=4, naive stag.[1,2,3]
Nt=6, naive stag. [2]
Nt=4, p4fat3[1]
Nt=6, stout[4]
Nt=6, HISQ[5]
mc⇡[MeV]
[4]G. Endrodi et al., PoS LAT2007 (2007) 228 [2] P. de Forcrand et al, PoS LATTICE2007 (2007) 178 [1]F. Karsch et al., Nucl.Phys.Proc.Suppl. 129 (2004) 614
[5] HTD et al., Lattice 15’, arXiv: 1511.00553 [6]Y. Nakamura, Lattice 15’,PRD92 (2015) no.11, 114511 [3]D. Smith & C. Schmidt, Lattice 2011
1st order chiral phase transition region
0
50
100
150
200
250
300
350
Nt=6, 8& 10, Wilson clover[6]
[Philippe De Forcrand, Monday]
Rooting issue? Nf=4 staggered QCD
Unimproved staggered fermions, Nt=4, Nf=4
preliminary
Nt↑ mc ↓
1st order chiral phase transition region
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
Nt=4, Naive staggered fermionsNt=6, Wilson-Clover
Philippe de Forcrand and Owe Philipsen, JHEP 0701 (2007) 077
Location of the physical point: Inconsistency between results from Wilson-clover and Naive
staggered fermions on coarse lattices
[Yoshifumi Nakamura, Monday]
750 MeV
Proper order parameter of chiral phase transition in Nf=3 QCD [Shinji Takeda, Tuesday]
Karsch, Laermann & Schmidt Phys.Lett. B520 (2001) 41
Cross point moves to the Z(2) critical line with:
Estimate of critical end point at mu=0: analytical continuation from imaginary mu
O. Philipsen and C. Pinke, PRD93 11(2016)114507 Bonati et al.,PRD90 7(2014) 074030
⇣ µ
T
⌘2
unimproved Wilson fermions, Nt=6
unimproved Wilson fermions: mc⇡(µ = 0, N⌧ = 4) ⇡ 560 MeV
estimate: mc⇡(µ = 0, N⌧ = 6) ⇠ 400 MeV
[Alessandro Sciarra, Monday]
Chiral phase transition region in Nf=3 QCD
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
ordermu,d
ms
∞
∞
cross over
2nd orderZ(2)
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mtris
1st
order
mc
??
Whether the 1st order chiral phase transition is relevant for the physical point at all?
ml/ms=1/201/271/401/601/80
Scaling window becomes smaller from Nt=4 p4fat3 to Nt=6 HISQ results
Nt=4, p4fat3 Nt=6, HISQ
0 0.2 0.4 0.6 0.8
1 1.2 1.4 1.6 1.8
2 2.2
-2 -1 0 1 2 3
χ2/dof is: 14.4133
z=t/h1/ βδ
M/h1/δ
mπ=160MeVmπ=140MeVmπ=110MeV
mπ=90MeVmπ=80MeV
O(2)
Ejiri et al., PRD 09’
Universal behavior of chiral phase transition in Nf=2+1 QCD[Sheng-Tai Li, Wednesday]
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
?
Universal behavior of chiral phase transition in Nf=2+1 QCD
Best evidence of the O(2) scaling
[Sheng-Tai Li, Wednesday]
singular part of chiral condensate singular part of total susceptibility
Nt=6, HISQ
O(2)O(2)
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
?
mc ≈ 0 from Z(2) scaling analysis
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-2 -1 0 1 2 3
χ2/dof is: 4.59549
z=t/h1/ βδ
(χM-χreg)h0/h(1/δ-1)
mπ=160MeVmπ=140MeVmπ=110MeV
mπ=90MeVmπ=80MeV
Z(2)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-2 -1 0 1 2 3
χ2/dof is: 4.59549
z=t/h1/ βδ
(M-freg)/h1/δ mπ=160MeVmπ=140MeVmπ=110MeV
mπ=90MeVmπ=80MeV
Z(2)Z(2) Z(2)
singular part of chiral condensate singular part of total susceptibility
Nt=6, HISQ
msphy > ms
tri
symmetry breaking field: ml - mc
Universal behavior of chiral phase transition in Nf=2+1 QCD[Sheng-Tai Li, Wednesday]
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
mu,d
ms
∞
∞
cross over
2nd orderO(4)
Nf=2
Nf=1
2nd orderZ(2)
1st
order
PUREGAUGE
Nf=3physical point
?
O(4) scaling behavior in Nf=2 QCD[Takashi Umeda, Wednesday]
Chiral condensate defined from Ward identity
t/h1/�
h ̄ iWI/h1/�
163 x 4, Clover improved Wilson Consistent with O(4) scaling
See similar conclusion from the many flavor approach: Ejiri, Iwami & Yamada, 1511.06126, PRL 110(2013)no. 17, 172001
Indication of a 2nd order phase transition in the
massless two flavor QCD
fluctuation of Goldstone modes at T<Tc ?
role of UA(1) symmetry in Nf=2 QCD
UA(1) symmetry:
• broken, 2nd order (O(4)) phase transition• restored, 1st or 2nd order (U(2)L⊗U(2)R/U(2)V)
Pisarski and Wilczek, PRD 29(1984)338Butti, Pelissetto and Vicar, JHEP 08 (2003)029
UA(1) symmetry on the lattice:
• always broken in the Wilson/ Staggered discretization scheme
Fate of chiral symmetries at T=/=0: Nf=2+1 QCD
π
δ
τ2: q γ
5q
: q τ2 q
: q
: q γ5
σ
ηL RSU(2) x SU(2)
SU(2) x SU(2)L R
U(1)AU(1)A
q
q
χχ
χχcon
con
5,con
5,con + χ
− χ 5,disc
disc
At the physical point, U(1)A does not restore at TχSB~170 MeV, remains broken up to 195 MeV ~ 1.16TχSB
Domain Wall fermions, 323x8, Ls=24,16
HotQCD&RBC/LLNL, PRL 113 (2014) 082001,PRD 89 (2014) 054514
0
50
100
150
200
250
130 140 150 160 170 180 190 200T [MeV]
(χMSπ - χ
MSσ )/T2
(χMSπ - χ
MSδ )/T
2
mπ=140 MeV
Underlying mechanism of UA(1) breaking
black lines: 163 lattices; red histograms: 323 lattices
N0: total # of near zero modes
N+: # of near zero modes with positive chirality
323x8, T=177 MeV
# of configurations with N0 and N+
Dirac Eigenvalue spectrum: Chirality distribution
HotQCD&RBC/LLNL, PRL 113 (2014) 082001,PRD 89 (2014) 054514
Density of near zero modes prefers to be independent of V rather than to shrink with 1/sqrt(323/163)
Chirality distribution shows a binomial distribution more than a bimodal one
Underlying mechanism of UA(1) breaking
black lines: 163 lattices; red histograms: 323 lattices
N0: total # of near zero modes
N+: # of near zero modes with positive chirality
323x8, T=177 MeV
# of configurations with N0 and N+
Dirac Eigenvalue spectrum: Chirality distribution
HotQCD&RBC/LLNL, PRL 113 (2014) 082001,PRD 89 (2014) 054514
Density of near zero modes prefers to be independent of V rather than to shrink with 1/sqrt(323/163)
Chirality distribution shows a binomial distribution more than a bimodal one
A dilute instanton gas model can describe the non-zero UA(1) breaking above Tc !
Fate of chiral symmetries from HISQ calculations
0
0.4
0.8
1.2
1.6
100 120 140 160 180 200 220 240 260
0.8 1 1.2 1.4 1.6
T [MeV]
T/Tc
M(T) [GeV], ud-
Nτ 0
− 0
+ 1
− 1
+
810
12
courtesy of Yu Maezewa,
YITP, work in progress
Indication of the breaking of UA(1) symmetry up to ~1.2 Tc in the continuum limit
at mπ = 160 MeV
mπ = 160 MeV, Nf=2+1 QCD
mπ = 160 MeV, Nf=2+1 QCD
-10
0
10
20
30
40
50
60
0.8 1 1.2 1.4 1.6 1.8
T/Tc
(χπ-χδ)/T2 Nτ=12
Nτ=8Nτ=6
Petreczky, Schadler & Sharma, arXiv:1606.03145
UA(1) symmetry from HISQ calculations
arXiv:1606.03145 Bonati etl., JHEP 1603 (2016) 155 Petreczky et al., arXiv:1606.03145
Indication of the breaking of UA(1) symmetry up to ~1.2 Tc in the continuum limit
at mπ = 160 MeVSee similar conclusions from measurements of overlap operators on DWF,
S. Sharma, lattice 2015, arXiv: 1510.03930
Topological susceptibility up to very high T
b= d
logχ
/dlo
gTT[MeV]
contDIGA
Nt=4Nt=6
Nt=8Nt=10
4
5
6
7
8
9
10
300 600 1200 240010-12
10-10
10-8
10-6
10-4
10-2
100
102
100 200 500 1000 2000
χ[fm
-4]
T[MeV]
10-4
10-3
10-2
10-1
100 150 200 250
The fall-off exponent agrees with Dilute Instanton Gas Approximation (DIGA) /Stefan Boltzmann limit for
temperatures above T ∼ 1GeV
[Kalman Szabo , Monday]
Borsanyi et al., [WB collaboration],1606.07494
4stout, continuum extrapolated
Marching to the chiral limit…
-1200
-1000
-800
-600
-400
-200
0
200
0 10 20 30 40 50
∆M
PS[M
eV]
mud [MeV]
T = TClinear chiral extrapolation
T = 0 physical pointT = 0 chiral extrapolated
courtesy of Bastian Brandt, Univ. of Frankfurt, updated results of 1310.8326 (lattice 2013), work in progress
clover-improved Wilson fermions on Nt=16 lattices, Nf=2
Violation of Ginsparg-Wilson relation
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
∆G
W⁄/∆
m (MeV)
323x8 β=4.07, T=203MeV323x8 β=4.10, T=210MeV
323x12 β=4.23, T=191MeV323x12 β=4.24, T=195MeV
163x8 β=4.07, T=203MeV163x8 β=4.10, T=210MeV
Courtesy of Guido Cossu, University of Edinburgh, JLQCD, G. Cossu et al., PRD93 (2016) no.3, 034507,1511.05691, A. Tomiya, 1412.7306
Mobius Domain Wall fermions
Marching to the chiral limit…
Courtesy of Guido Cossu, University of Edinburgh, JLQCD, G. Cossu et al., PRD93 (2016) no.3, 034507,1511.05691, A. Tomiya, 1412.7306
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25
∆(G
eV2 )
m (MeV)
323x8 β=4.07, T=203MeV323x8 β=4.10, T=210MeV
323x12 β=4.23, T=191MeV323x12 β=4.24, T=195MeV
163x8 β=4.07, T=203MeV163x8 β=4.10, T=210MeVchiral zero-modes included
Nf=2 QCD, Reweighting to Overlap
Columbia plot in the heavy quark mass region[Christopher Czaban, Monday]
Nt=8, unimproved Wilson fermions
B4=1.604, 2nd order of Z(2)
C.f. WHOT collaboration results on 243x4 lattices using standard Wilson fermions, PRD84 (2011) 054502, Erratum: PRD85 (2012) 079902
Possible connections between deconfinement & chiral aspects of the cross over
0
1
2
3
4
5
6
7
8
9
10
0.8 1 1.2 1.4 1.6 1.8 2
SQ
T/Tc
Nf=2+1, mπ=161 MeVNf=3, mπ=440 MeVNf=2, mπ=800 MeV
Nf=0
A. Bazavov et al., PRD93 (2016) no.11, 114502
G. Cossu and S. Hashimoto, JHEP 1606 (2016) 056
Causes of transitions? Analogy to Anderson Localization
Similar topics [Matteo Giordano, Friday]
[Guido Cossu@Friday]
Near zero modes are correlated with Polyakov loop Conjecture: localization of low modes causes the restoration of chiral symmetry
QCD thermodynamics at very high temperature
0
0.5
1
1.5
2
2.5
3
300 600 1200 24003
m − ψψ
1−0
T[MeV]
Nt=10Nt=8Nt=6
std Nt=4
Nt=10Nt=8Nt=6
rew+zm Nt=4
Borsanyi et al., [WB collaboration],1606.07494
For the method see also Frison et al., 1606.07175
[Kalman Szabo , Monday]
Fixed Q Integration approach
Topological susceptibility up to very high T
10-12
10-10
10-8
10-6
10-4
10-2
100
102
100 200 500 1000 2000
χ[fm
-4]
T[MeV]
10-4
10-3
10-2
10-1
100 150 200 250
[Kalman Szabo , Monday]
Borsanyi et al., [WB collaboration],1606.07494
See also Petreczky, Schadler & Sharma, arXiv:1606.03145, Bonati et al., JHEP 1603 (2016) 155
An axion mass of 50(4)μeV
Relevance for Dark Matter [Enrico Rinaldi, Saturday]
0.95
1
1.05
1.1
1.15gρ/gs
10
30
50
70
90
110
100 101 102 103 104 105
g(T)
T[MeV]
gρ(T)
gs(T)
0
1
2
3
4
5
6
7
200 400 600 800 1000 1200 1400 1600 1800 2000
p/T
4
T [MeV]
O(g6) Nf=3+1 qc=-3000O(g6) Nf=3+1+1 qc=-3000
2+1+1 flavor EoS from lattice
0
1
2
3
4
5
100 200 300 400 500 600 700 800 900 1000
(ρ-3
p)/
T4
T [MeV]
HTLpt2+1 flavor continuum
2+1+1 flavor continuum
0
1
2
3
4
5
100 200 300 400 500 600 700 800 900 1000
(ρ-3
p)/
T4
T [MeV]
HTLpt
Equation of State up to very high T[Szabolcs Borsanyi, Monday]
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
200 400 600 800 1000 1200 1400 1600 1800 2000
(ρ-3
p)/
T4
T [MeV]
O(g6) Nf=3+1 qc=-3000O(g6) Nf=3+1+1 qc=-3000
2+1+1 flavor EoS from lattice
SU(3) thermodynamics from Gradient flow
Courtesy of Masakiyo Kitazawa, Osaka Univ., work in progress
Numerical analysis is performed on Nt=12 - 24 lattices.
Relatively low cost compared to the standard method
EoS of full QCD from Gradient flow[Kazuyuki KANAYA ,Wednesday]
Results agree with T-integration method at T<=300 MeV
See [Yusuke Taniguchi, Friday] on topological susceptibility from Gradient FlowSee [Saumen Datta, Friday],arXiv:1512.04892 on the deconfinement transition from GF
Larger Nt-s at high temperature are needed
Properties of QGP through spectral functions
Methods to solve the ill-posed problem:
Spectral functions: in-medium heavy hadron properties ([Seyong Kim, Today’s plenary]), transport properties, dissociation T of hadron, electromagnetic properties of QGP
GH(⌧, ~p, T ) =
Z 1
0
d!
2⇡⇢H(!, ~p, T )
cosh(!(⌧ � 1/2T ))
sinh(!/2T )
• Maximum Entropy Method (MEM): Based on Bayesian theorem using Shannon-Jaynes Entropy
• Improved Maximum Entropy Method: Similar with MEM but with a different Entropy term
• Stochastic Analytical Interference (SAI) & Stochastic optimization method (SOM) [Hai-Tao Shu, Thursday]
Y. Burnier & A. Rothkopf, PRL. 111(2013)182003
M. Asakawa, T. Hatsuda & Y. Nakahara, Prog.Part.Nucl.Phys. 46 (2001) 459
• Backus-Gilbert Method [Daniel Robaina, Tuesday]
Charm quark diffusion coefficient1923x48, quenched QCD
[Hiroshi Ohno, Thursday]
T=1.5 Tc
2πTD≲2
Debye mass for a complex heavy quark potential
Y. Burnier & A. Rothkopf. Phys.Lett. B753 (2016) 232, arXiv.1607.04049
T=105MeV T=210MeV T=252MeV
T=280MeV T=295MeV T=315MeV
T=334MeV T=360MeV T=419MeV
��� ��� ��� ��� ��� ��� ���� [��]
�
�
�
�
�
�
��[�]SU(3) β=6.1 ξr=4 Ns=32
AC
TC
mD/T
��� ��� ��� ��� ���� [���]���
���
���
���
�����/�
SU(3) β=6.1 ξr=4 Ns=32
[A. Rothkopf, Thursday]
� � � � � � � � [��]���
���
���
���
���
�����[�]
SU(3) β=6.1 ξr=4 Ns=32
mD includes both screening & scattering effects
Indirect evidence of experimentally not yet observed strange states hinted from QCD thermodynamics
PDG-HRG: Hadron Resonance Gas model calculations with spectrum from PDG
QM-HRG: Similar as PDG-HRG but with spectrum from Quark Model
cont. est.PDG-HRG
QM-HRG
0.15
0.20
0.25
0.30 - r11BS/r2
S
No=6: open symbolsNo=8: filled symbols
B1S/M1
S
B2S/M2
S
B2S/M1
S
0.15
0.25
0.35
0.45
140 150 160 170 180 190
T [MeV]
A. Bazavov et al.[BNL-Bielefeld-CCNU], Phys. Rev. Lett. 113 (2014)072001
strange-baryon correlations
partial pressures
HISQ, mπ=160 MeV
Missing strange states?
25/32
! Some observables are in agreement with the PDG 2014 but not with the Quark Model:
WB collaboration, in preparation
Quark ModelPDG 2014
0.12 0.14 0.16 0.18 0.20
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
T [GeV ]
χ 4S
χ 2S
Quark ModelPDG 2014
0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
-0.04
-0.02
0.00
0.02
0.04
0.06
T [GeV ]
χ 11us
! χ4S/χ2
S is proportional to <S2> in the system ! It seems to indicate that the quark model predicts too many multi-
strange states
Strange mesons in PDG & Quark Model[Szabolcs Borsanyi, Monday]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
120 130 140 150 160 170 180 190
Sector: |B|=0, |S|=1
p/T4
T [MeV]
lattice continuum limitHRG using PDG2014
HRG using the Quark Model
Relative abundance in strange baryons to strange mesons are well described by QM-HRG
Some single-strange mesons are missing in QM
T>0 & µ > 0
Fluctuations of conserved charges
Taylor expansion coefficients at μ=0
�BQSijk ⌘ �BQS
ijk (T ) =1
V T 3
@P (T, µ̂)/T 4
@µ̂iB@µ̂
jQ@µ̂
kS
���µ̂=0
p
T 4=
1
V T 3lnZ(T, V, µ̂u, µ̂d, µ̂s) =
1X
i,j,k=0
�BQSijk
i!j!k!
⇣µB
T
⌘i ⇣µQ
T
⌘j ⇣µS
T
⌘k
✏� 3p
T 4= T
@P/T 4
@T=
1X
i,j,k=0
T d�BQSijk /dT
i!j!k!
⇣µB
T
⌘i ⇣µQ
T
⌘j ⇣µS
T
⌘k
Thermodynamic relations
Pressure of hadron resonance gas (HRG)
Taylor expansion of the QCD pressure:
p
T 4=
X
m2meson,baryon
lnZ(T, V, µ) ⇠ exp(�mH/T ) exp((BµB + Sµs +QµQ)/T )
Allton et al., Phys.Rev. D66 (2002) 074507Gavai & Gupta et al., Phys.Rev. D68 (2003) 034506
Pressure of QCD at nonzero muB
�(P/T 4) =P (T, µB)� P (T, 0)
T 4=
1X
n=1
�B2n(T )
(2n)!
⇣µB
T
⌘2n
=1
2�B2 (T )µ̂
2B
⇣1 +
1
12
�B4 (T )
�B2 (T )
µ̂2B +
1
360
�B6 (T )
�B2 (T )
µ̂4B + · · ·
⌘
LO expansion coefficient variance of net-baryon number distribution
NLO expansion coefficient kurtosis * variance
• HRG describes well on the LO expansion coefficient up to ~160 MeV while it deviates from NLO expansion coefficient ~ 40% in the crossover region
• For small muB/T the LO contribution dominates
NNLO expansion coefficient
T [MeV]
free quark gas
Tc=(154 +/-9) MeV
BNL-Bielefeld-CCNUpreliminary
ms/ml=20 (open)27 (filled)
continuum extrap.PDG-HRG
Nτ=68
1216
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
120 140 160 180 200 220 240 260 280
χ2B
[Edwin Laermann, Monday]
χ 4B /χ2B
T [MeV]
HRG
free quark gas
ms/ml=20 (open)27 (filled)
continuum est.Nτ=6
8
0
0.2
0.4
0.6
0.8
1
1.2
120 140 160 180 200 220 240 260 280
BNL-Bielefeld-CCNUpreliminary
-2
-1
0
1
2
3
4
5
130 140 150 160 170 180 190 200
T [MeV]
χ6B/χ2
B
HRG
ms/ml=20 (open)27 (filled)
continuum est.Nτ=6
8
BNL-Bielefeld-CCNUpreliminary
Explore the QCD phase diagram in Heavy Ion collisions
=�B4
�B2
1 +
✓�B6
�B4
� �B4
�B2
◆⇣µB
T
⌘2+ · · ·
�(�2)B =
�B4,µ
�B2,µ
STAR
In the O(4) universality class:�B6 < 0 , T ⇠ Tc
T [MeV]
free quark gas
Tc=(154 +/-9) MeV
BNL-Bielefeld-CCNUpreliminary
ms/ml=20 (open)27 (filled)
continuum extrap.PDG-HRG
Nτ=68
1216
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
120 140 160 180 200 220 240 260 280
χ2B
χ 4B /χ2B
T [MeV]
HRG
free quark gas
ms/ml=20 (open)27 (filled)
continuum est.Nτ=6
8
0
0.2
0.4
0.6
0.8
1
1.2
120 140 160 180 200 220 240 260 280
BNL-Bielefeld-CCNUpreliminary
-2
-1
0
1
2
3
4
5
130 140 150 160 170 180 190 200
T [MeV]
χ6B/χ2
B
HRG
ms/ml=20 (open)27 (filled)
continuum est.Nτ=6
8
BNL-Bielefeld-CCNUpreliminary
Pressure of QCD at nonzero μB
�(P/T 4) =P (T, µB)� P (T, 0)
T 4=
1X
n=1
�B2n(T )
(2n)!
⇣µB
T
⌘2n
=1
2�B2 (T )µ̂
2B
⇣1 +
1
12
�B4 (T )
�B2 (T )
µ̂2B +
1
360
�B6 (T )
�B2 (T )
µ̂4B + · · ·
⌘μQ=μs=0
[Edwin Laermann, Monday]
0
0.5
1
1.5
2
2.5
3
3.5
4
120 140 160 180 200 220 240 260 280
[ε(T
,µB)
-ε(T
,0)]/
T4
T [MeV]
BNL-Bielefeld-CCNUpreliminary
µB/T=2
µB/T=2.5
µB/T=1HRG
µQ=µS=0
O(µB6)
O(µB4)
O(µB2)
0
0.2
0.4
0.6
0.8
1
120 140 160 180 200 220 240 260 280
[P(T
,µB)
-P(T
,0)]/
T4
T [MeV]
BNL-Bielefeld-CCNUpreliminary
µB/T=2
µB/T=2.5
µB/T=1
HRG
µQ=µS=0
O(µB6)
O(µB4)
O(µB2)
Equation of State well under control at μB/T ≤2
EoS in the strangeness neutral case: conditions in Heavy Ion Collisions
0
0.5
1
1.5
2
2.5
3
120 140 160 180 200 220 240 260 280
[ε(T
,µB)
-ε(T
,0)]/
T4
T [MeV]
BNL-Bielefeld-CCNUpreliminary
µB/T=2
µB/T=2.5
µB/T=1
HRG
NS=0, NQ/NB=0.4
O(µB6)
O(µB4)
O(µB2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
120 140 160 180 200 220 240 260 280
[P(T
,µB)
-P(T
,0)]/
T4
T [MeV]
BNL-Bielefeld-CCNUpreliminary
µB/T=2
µB/T=2.5
µB/T=1
NS=0, NQ/NB=0.4
O(µB6)
O(µB4)
O(µB2)
[Edwin Laermann, Monday]
Equation of State well under control at μB/T ≤2, i.e. sqrt(SNN) > 12 GeV in Heavy Ion Collisions
At LHC and RHIC: <nS>=0, <NQ>/<NB>=0.4
[Jana Günther, Wednesday]
Taylor expansion of pressure in real μB:
T
µB
d(p/T 4)
d(µB/T )
���<nS>=0,r=0.4,T=const.
= 2c̃2(T ) + 4c̃4(T )⇣µB
T
⌘2+ 6c̃6(T )
⇣µB
T
⌘4+O(µ8
B)
= 2c̃2(T ) + 4c̃4(T )⇣µB
T
⌘2+ 6c̃6(T )
⇣µB
T
⌘4+O(µ8
B)
Taylor expansion of pressure calculated in imaginary μB:
J. Günther et al.[WB collaboration], 1607.02493
Taylor expansion coefficients from analytic continuation
Analytic continuation from analytic continuation[Jana Günther, Wednesday]
Intercept ->c2 slope -> c4
curvature ->c6 c8 ?
T
µB
d(p/T 4)
d(µB/T )
���<nS>=0,r=0.4,T=const.
= 2c̃2(T ) + 4c̃4(T )⇣µB
T
⌘2+ 6c̃6(T )
⇣µB
T
⌘4+O(µ8
B)
= 2c̃2(T ) + 4c̃4(T )⇣µB
T
⌘2+ 6c̃6(T )
⇣µB
T
⌘4+O(µ8
B)
[Jana Günther Wednesday]
Taylor expansion coefficients from analytic continuation
continuum extrapolated results from 4stout results on Nt=10,12,16
Compatible with the preliminary results of Taylor expansion coefficients from direct calculations (see Laermann’s talk, Monday)
Roberge-Weiss phase transition temperature
Bonati et al., Phys.Rev. D93 (2016) no.7, 074504
Nt=6 Nt=4
Nf=2+1 QCD, stout fermions with physical pion mass
Evidence found for on Nt=4 & 6 latticesmtriL < mphy
l < mtriH
[Michele Mesiti, Monday]
Roberge-Weiss phase transition temperature
Bonati et al., Phys.Rev. D93 (2016) no.7, 074504
Continuum extrapolated: TRW = 208(5) MeV
[Michele Mesiti, Monday]
Nf=2+1 QCD, stout fermions with physical pion mass
Nt=4,6,8,10 Nt=8
The location of RW endpoint is obtained
Roberge-Weiss phase transition temperature
Czaban et al., Phys.Rev. D93 (2016) no.5, 054507
tri-critical pion mass values shift considerably when lattice cutoff is reduced
Nf=2 QCD, standard Wilson fermions, Nt=6,8
[Christopher Czaban, Monday]
Phase diagram of QCD with heavy quarks (HDQCD) from Complex Langevin
[Felipe Attanasio, Tuesday]
• To cure the sign problem: Gauge links SU(3) -> SL(3,𝕔)
• Gauge cooling: Gauge transformations between Langevin updates to minimize the distance from SU(3)
Instabilities in Complex Langevin simulations for Heavy Dense QCD
See also [Felipe Attanasio, Tuesday][Benjamin Jager, Tuesday]
Gauge cooling is essential, however, it fails at some circumstances
Dynamic stabilization[Benjamin Jager, Tuesday]
M: SU(3) gauge invariant, ~ a7
Dynamic stabilization[Benjamin Jager, Tuesday]
M: SU(3) gauge invariant, ~ a7
Dynamic stabilization improves convergenceMore tests need for full QCD
Comparisons of Complex Langevin with reweighting for full QCD
[D. Sexty, Tuesday]
Nt=4 Nt=4
Fodor et al., PRD92 (2015) no.9, 094516
Similar to HDQCD, in the low temperate region CLE simulation instable
Comparisons of Complex Langevin with reweighting for full QCD [D. Sexty, Tuesday]
Nt=6 Nt=8
Fodor et al., PRD92 (2015) no.9, 094516
Issues in singularities of the drift force of Langevin dynamics: see talks on Tuesday by e.g. Gert Aarts, Keitaro Nagata
Similar to HDQCD, in the low temperate region CLE simulation instable
QCD at nonzero isospin density Bastian Brandt & Gergely Endrodi (Thursday)
Son & Stephanov, PRL86 (2001)
Kogut, Sinclair, PRD66 (2002); PRD70 (2004) First lattice simulations Nt=4 with mπ larger than physical one:
1st order deconfinement and 2nd curve join? Existence of a tri-critical point
QCD at nonzero isospin density Bastian Brandt & Gergely Endrodi (Thursday)
Direct method:
Banks-Casher-type method:Kanazawa, Wettig, Yamamoto ’11
Leading reweighting:
QCD at nonzero isospin density Bastian Brandt & Gergely Endrodi (Thursday)
0
0.5
1
1.5
2
2.5
3
3.5
0 20 40 60 80 100 120 140
⟨nI⟩/
T3
µI [MeV]
mπ/2
T = 124 MeV6× 243
preliminaryTaylor exp. O(µI)
O(µ3I)
simulation
0
0.5
1
1.5
2
2.5
3
3.5
0 20 40 60 80 100 120 140
⟨nI⟩/
T3
µI [MeV]
mπ/2
T = 162 MeV6× 243
preliminaryTaylor exp. O(µI)
O(µ3I)
simulation
Thanks to• Members of Bielefeld-BNL-CCNU collaboration
• People who sent me materials/plots: Pedro Bicudo, Jacques Bloch, Szabolcs Borsanyi, Bastian Brandt, Falk Bruckmann,Guido Cossu, Gergely Endrodi, Philippe de Forcrand, Yoichi Iwasaki, Kazuyuki Kanaya, Sandor Katz, Masakiyo Kitazawa, Yu Maezawa, Atsushi Nakamura, Yoshifumi Nakamura, Alexander Rothkopf, Jonivar Skullerud, Hélvio Vairinhos, Takashi Umeda
Apologies to those whose achievements were not mentioned in my talk
many talks on the sign problem, e.g. Lefschetz thimbles, canonical method, complex Langevin, subsets etc and QC2D, strong coupling as well as some topics in the
sessions of this afternoon are not covered
