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