Properties of Thermal Matter: Conductivity, Parity Restoration and … · 2015-07-18 · Properties of Thermal Matter: Conductivity, Parity Restoration and the Charmonium Potential

Properties of Thermal Matter:Conductivity, Parity Restorationand the Charmonium Potential

Chris AlltonSwansea University, U.K.

LHPV 2015 Workshop, Cairns, July 2015

FASTSUM Collaboration

Gert Aarts1, CRA1, Alessandro Amato1,2, Davide de Boni1, Wynne Evans1,3,Pietro Giudice4, Simon Hands1, Benjamin Jäger1, Aoife Kelly5, Seyong Kim6,Maria-Paola Lombardo7, Dhagash Mehta8, Bugra Oktay9, Chrisanthi Praki1,

Sinead Ryan10, Jon-Ivar Skullerud5, Tim Harris10,11

1 Swansea University 7Frascati, INFN2 University of Helsinki 8 North Carolina State University3 University of Bern 9 University of Utah4 Münster University 10 Trinity College Dublin5 Maynooth University 11 University of Mainz6Sejong University

Setting the scene

hadronsmassesmx elsatomic physics

[http://www.bnl.gov/rhic/news]

quarks & gluonspressureviscosityplasma physics

Setting the scene

hadronsmassesmx elsatomic physics

Correlation Functions↔ Spectral Functions

quarks & gluonspressureviscosityplasma physics

Particle Data Book

∼ 1,500 pages

zero pages on Quark-Gluon Plasma...

Overview

I Parity Restoration in theBaryon Sector

SYMMETRIES

arXiv/1505.06616 0 0.5 1 1.5 2T/T

c

0

0.2

0.4

0.6

0.8

1

R

I Charmonium PotentialINTERACTIONS

arXiv/1502.03603

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

I Conductivity, Susceptibility andDiffusion Coefficient

PHENOMENOLOGY

arXiv/1412.6411100 150 200 250 300 350

T [MeV]

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

C−1

emσ/T

light+strange

0.50 0.75 1.00 1.25 1.50 1.75 2.00

T/Tc

Other Work

I Bottomonium and CharmoniumSpectral Functions

MELTING

arXiv:1402.6210

0

2

4

6

8

10

12

0

1

2

3

4

5

9 10 11 12 13 14 15

ρ(ω

)/m

2 b

9 10 11 12 13 14 15ω (GeV)

9 10 11 12 13 14 15

ΥT/Tc = 0.76

0.84

0.840.95

0.951.09

1.091.27

1.271.52

1.521.90

FASTSUM set up

I anisotropic lattices aτ < asI allowing better resolution, particularly at finite temperatures

since T =1

Nτaτ

τ

x

τ

x

I "2nd" generation lattice ensemblesI moving towards continuum, infinite volume, realistic light

quark masses

Physics/lattice parameters

2nd Generation2+1 flavourslarger volume: (3fm)3 – (4fm)3

finer lattices: as = 0.123 fmquark mass: Mπ/Mρ ∼ 0.45temporal cut-off: aτ ∼ 5.6 GeV

Ns Nτ T (MeV) T/Tc

24, 32 16 352 1.9024 20 281 1.52

24, 32 24 235 1.2724, 32 28 201 1.0924, 32 32 176 0.95

24 36 156 0.8424 40 141 0.7632 48 117 0.6316 128 44 0.24

Gauge Action:

Symanzik-improved, tree-level tadpole

Fermion Action:

clover, stout-links, tree-level tadpole

(Hadron Spectrum Collaboration)

Parity Restoration in the Baryon Spectrum

arXiv/1505.06616

Baryons at Finite Temperature

I little work on Baryons @ T 6= 0I DeTar and Kogut (1987) screening massesI QCD-TARO (2005) µ 6= 0I Datta et al (2013) quenched

We use a standard baryon operator:

ON(x, τ) = εabcua(x, τ)[uT

b (x, τ)Cγ5dc(x, τ)]

and parity project it:ON±(x, τ) = P±ON±(x, τ)

Forward (+ve) and backward (−ve) parity states in correlator:

G(τ) =

∫d3x 〈ON+ (x, τ)ON+ (0, 0)〉

=

∫ ∞0

dω2π

[e−ωτ

1 + e−ω/T ρ+(ω)− e−ω(1/T−τ)

1 + e−ω/T ρ−(ω)

]

Baryon Correlators(Using Gaussian smeared baryon operators)

−→ parity doubling for T ∼> TC observed at correlator level

T/Tc=1.90

T/Tc=1.52

T/Tc=1.27

T/Tc=1.09

0 0.2 0.4 0.6 0.8 1τT

10-3

10-2

10-1

100

G(τ

)/G

(0)

T/Tc=0.95

T/Tc=0.84

T/Tc=0.76



T/Tc=1.90

T/Tc=1.52

T/Tc=1.27

T/Tc=1.09

0 0.2 0.4 0.6 0.8 1τT

10-3

10-2

10-1

100

G(τ

)/G

(0)

T/Tc=0.95

T/Tc=0.84

T/Tc=0.76



T/Tc=1.90

T/Tc=1.52

T/Tc=1.27

T/Tc=1.09

0 0.2 0.4 0.6 0.8 1τT

10-3

10-2

10-1

100

G(τ

)/G

(0)

T/Tc=0.95

T/Tc=0.84

T/Tc=0.76

Correlators - Parity Comparison

0 8 16 2410-5

10-4

10-3

10-2

10-1

100

G(τ

)/G

(0)

T/Tc=0.95

T/Tc=0.84

T/Tc=0.76

T/Tc=0.24

0 8 16 24 32

τ/aτ

T/Tc=1.90

T/Tc=1.52

T/Tc=1.27

T/Tc=1.09

positive parity negative parity

Experiment:

+ve parity: MN = 939 MeV −ve parity: MN∗ = 1535 MeV

Correlators - Parity Comparison

0 8 16 2410-5

10-4

10-3

10-2

10-1

100

G(τ

)/G

(0)

T/Tc=1.90

T/Tc=0.24

0 8 16 24 32

τ/aτ

positive parity negative parity

Experiment:

+ve parity: MN = 939 MeV −ve parity: MN∗ = 1535 MeV

Naive Exponential Fits

T/Tc m+ [GeV] m− [GeV] m+ −m− [MeV]

0.24 1.20(3) 1.9(3) ∼700 cf expt: ∼6000.76 1.18(9) 1.6(2)0.84 1.08(9) 1.6(1)0.95 1.12(14) 1.3(2)

Parity Comparison

Define R(t) =G(τ)−G(Nτ − τ)

G(τ) + G(Nτ − τ)Datta et al, arXiv:1212.2927

Note: R(1/2T ) ≡ 0with: R(τ) ≡ 0 for parity symmetry

0 0.1 0.2 0.3 0.4 0.5τT

0

0.25

0.5

0.75

1

1.25

R(τ

)

T/Tc=0.24

T/Tc=0.76

T/Tc=0.84

T/Tc=0.95

T/Tc=1.09

T/Tc=1.27

T/Tc=1.52

T/Tc=1.90

Parity Restoration

Define R =

∑Nτ/2−1τ=1 R(τ)/σ2(τ)∑Nτ/2−1τ=1 1/σ2(τ)

0 0.5 1 1.5 2T/T

c

0

0.2

0.4

0.6

0.8

1

R

Effects of SmearingSystematics checks of smearing:

I vary nI vary τ -range

Implies parity doubling is:I ground state feature (recall Wilson term breaks chiral symmetry)I not an artefact of smearing

0 0.5 1 1.5 2T/T

c

0

0.2

0.4

0.6

0.8

1

R

point source, τ/aτ>5smearing, n=10smearing, n=10, τ/aτ>5smearing, n=60smearing, n=140

Maximum Entropy Method (MEM)

Cont: G(τ) =

∫K (τ, ω)ρ(ω)dω Lat: G(τi ) =

∑j

K (τi , ωj ) ρ(ωj )

Input data: τi , i = {1, . . . ,O(10)} Output data : ωj , j = {1, . . . ,O(103)}

−→ ill-posed

Bayes Th’m: P[ρ|DH] =P[D|ρH]P[ρ|H]

P[D|H]∝ exp(−χ2 + αS)

H = prior knowledge D = data

Shannon-Jaynes entropy: S =

∫ ∞0

dω2π

[ρ(ω)−m(ω)− ρ(ω) ln

ρ(ω)

m(ω)

]Competition between minimising χ2 and maximising S

Asakawa, Hatsuda, Nakahara, Prog.Part.Nucl.Phys. 46 (2001) 459

Maximum Entropy Method (MEM)

Cont: G(τ) =

∫K (τ, ω)ρ(ω)dω Lat: G(τi ) =

∑j

K (τi , ωj ) ρ(ωj )

Input data: τi , i = {1, . . . ,O(10)} Output data : ωj , j = {1, . . . ,O(103)}

−→ ill-posed

Bayes Th’m: P[ρ|DH] =P[D|ρH]P[ρ|H]

P[D|H]∝ exp(−χ2 + αS)

H = prior knowledge D = data

Shannon-Jaynes entropy: S =

∫ ∞0

dω2π

[ρ(ω)−m(ω)− ρ(ω) ln

ρ(ω)

m(ω)

]Competition between minimising χ2 and maximising S

Asakawa, Hatsuda, Nakahara, Prog.Part.Nucl.Phys. 46 (2001) 459

Example Spectral Functions

G2(t) ∼∫ρ(ω) e−ωt dω

Melted/Plasma with Transport

ω

ρ(ω

)

No transport peak

Tra

nsp

ort

pea

k

Melted/Plasma

Bound (decaying) States

Stable




ω

ρ(ω

)

No transport peak

Tra

nsp

ort

pea

k

Melted/Plasma


Stable




ω

ρ(ω

)

Tra

nsp

ort

pea

k

Melted/Plasma


Stable

MEM for finite T baryons

Recall: G(τ) =

∫d3x 〈ON+ (x, τ)ON+ (0, 0)〉

=

∫ ∞0

dω2π

[e−ωτ

1 + e−ω/T ρ+(ω)− e−ω(1/T−τ)

1 + e−ω/T ρ−(ω)

]

So can define: K (τ, ω) =e−ωτ

1 + e−ω/T ω > 0

=e+ω(1/T−τ)

1 + e+ω/T ω < 0

and use MEM with G(τ) ≡∫ +∞

−∞K (τ, ω)ρ(ω)dω

giving: ρ+(ω) ≡ ρ(ω) ω > 0ρ−(−ω) ≡ −ρ(ω) ω < 0

(Need to assume ρ(ω) is positive definite for MEM to work)

Baryonic Spectral FunctionsPreliminary

-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 0.24TC

(Nτ = 128)

exponential fit


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 0.76TC

(Nτ = 40)


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 0.84TC

(Nτ = 36)


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 0.95TC

(Nτ = 32)


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 1.09TC

(Nτ = 28)


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 1.27TC

(Nτ = 24)


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 1.52TC

(Nτ = 20)


-3 -2 -1 0 1 2 3ω aτ

0

ρ(ω

)

T = 1.90TC

(Nτ = 16)


-3 -2 -1 0 1 2 3

ω

0

ρ(ω

)

T = 1.90TC

T = 1.52TC

T = 1.27TC

T = 1.09TC

T = 0.95TC

T = 0.84TC

T = 0.76TC

T = 0.24TC

Charmonium Potential charmonium potential

arXiv/1502.03603

Lattice goes Nuclear

N-N potential

-50

0

50

100

0.0 0.5 1.0 1.5 2.0

V(r

) [M

eV

]

r [fm]

mπ=529 MeV

VC(r)VT(r)

VC

eff(r)

HAL QCD Collaboration, Aoki, Doi, Hatsuda, Ikeda, Inoue, Ishii, Murano,Nemura, SasakiIida, Ikeda PoS LATTICE2011(2011)195

Schrödinger Equation ApproachHAL QCD Collaboration, S. Aoki et al. [arXiv:1206.5088]

Schrödinger equation used to “reverse engineer” the potential, V (r), giventhe Nambu- Bethe-Salpeter wavefunction, ψ(r):

input input

↓ ↓ ↓(p2

2M+ V (r)

)ψ(r) = E ψ(r)

↓

output

ψ(r) is determined from correlators of non-local operators,

J(x ;~r) = q(x) Γ U(x , x +~r)q(x +~r)

C(~r , t) =∑~x

〈 J(0;~r = ~0) J(x ;~r) 〉

−→ ψ(r) e−Mt where 〈 0 | J(x ;~r) | gnd 〉 ≈ ψ(r)

Schrödinger Equation ApproachHAL QCD Collaboration, S. Aoki et al. [arXiv:1206.5088]

Schrödinger equation used to “reverse engineer” the potential, V (r), giventhe Nambu- Bethe-Salpeter wavefunction, ψ(r):

input input

↓ ↓ ↓(p2

2M+ V (r)

)ψ(r) = E ψ(r)

↓

output

ψ(r) is determined from correlators of non-local operators,

J(x ;~r) = q(x) Γ U(x , x +~r)q(x +~r)

C(~r , t) =∑~x

〈 J(0;~r = ~0) J(x ;~r) 〉

−→ ψ(r) e−Mt where 〈 0 | J(x ;~r) | gnd 〉 ≈ ψ(r)

HAL QCD Time Dependent Method

(x, 0)

(x+r, τ)

(x, τ)

c

c

J†Γ(0;0) JΓ(x, τ ; r)

SOURCE SINK

JΓ(x ; r) = q(x) Γ U(x , x +r) q(x +r)

Local Extended Correlation Functions

CΓ(r, τ) =∑

x〈JΓ(x, τ ; r) J†Γ (0; 0)〉

CΓ(r, τ) =∑

j

ψ∗j (0)ψj (r)

2Ej

(e−Ejτ + e−Ej (Nτ−τ)

)≈∑

j

Ψj (r)e−Ejτignoring

backward mover

Schrödinger Eqn Ej Ψj (r) =

(−∇

2r

2µ+ VΓ(r)

)Ψj (r)

∂CΓ(r, τ)

∂τ= −

∑j

Ej Ψj (r)e−Ejτ =∑

j

(∇2

r

2µ− VΓ(r)

)Ψj (r)e−Ejτ

∂CΓ(r, τ)

∂τ=

(∇2

r

2µ− VΓ(r)

)CΓ(r, τ)

HAL QCD Time Dependent Method

(x, 0)

(x+r, τ)

(x, τ)

c

c

J†Γ(0;0) JΓ(x, τ ; r)

SOURCE SINK

JΓ(x ; r) = q(x) Γ U(x , x +r) q(x +r)

Local Extended Correlation Functions

CΓ(r, τ) =∑

x〈JΓ(x, τ ; r) J†Γ (0; 0)〉

CΓ(r, τ) =∑

j

ψ∗j (0)ψj (r)

2Ej

(e−Ejτ + e−Ej (Nτ−τ)

)≈∑

j

Ψj (r)e−Ejτignoring

backward mover

aaaaawSchrödinger Eqn Ej Ψj (r) =

(−∇

2r

2µ+ VΓ(r)

)Ψj (r)

∂CΓ(r, τ)

∂τ= −

∑j

Ej Ψj (r)e−Ejτ =∑

j

(∇2

r

2µ− VΓ(r)

)Ψj (r)e−Ejτ

∂CΓ(r, τ)

∂τ=

(∇2

r

2µ− VΓ(r)

)CΓ(r, τ)

Correlation Functions

0 5 10 15 20

τ/aτ

-25

-20

-15

-10

-5

0

log[

CPS

(τ)]

r/as=0

r/as=1

r/as=2

r/as=3

r/as=4

r/as=5

r/as=6

r/as=7

r/as=8

r/as=9

r/as=10

r/as=11

r/as=12

PS channel 0.76Tc (Nτ = 40)

Central Potentials - cold0 10 20 30 40 50 60

τ/aτ

2/5

3

4

5V

C(τ

) [G

eV]

r/as = 1

r/as = 2

r/as = 3

r/as = 4

r/as = 5

r/as = 6

r/as = 7

r/as = 8

r/as = 9

0 5 10 152

3

4

0 5 10 15 20

0.24TC

0.84TC

0.95TC

0.76TC

VΓ(r) = VC(r)+VS(r) s1 ·s2 −→ VC(r) =14

VPS +34

VV VS(r) = VV−VPS

Central Potentials - hot

2/5

3

4

5V

C(τ

) [G

eV]

0 5 10 15τ/aτ

2

3

4

0 5 10 15 20

1.09TC

1.27TC

1.90TC

1.52TC

VΓ(r) = VC(r)+VS(r) s1 ·s2 −→ VC(r) =14

VPS +34

VV VS(r) = VV−VPS

Fitting Ranges

T/TC Nτ Best Range Lower Range

0.24 128 30− 63 15− 190.76 40 15− 19 12− 170.84 36 12− 17 11− 150.95 32 11− 15 11− 131.09 28 11− 13 9− 111.27 24 9− 11 N/A

Central Potential Results

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

0.95TC

τ=11-15

0.95TC

τ=11-13

1.09TC

τ=11-13

1.09TC

τ=9-11

1.27TC

τ=9-11

0.24TC

τ=30-63

0.24TC

τ=15-19

0.76TC

τ=15-19

0.76TC

τ=12-17

0.84TC

τ=12-17

0.84TC

τ=11-15

0.6 GeV

r = as

r = 3as

r = 5as

r = 7as

r = 9as

Cornell Potential Comparison

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4V

C(r

) [G

eV]

0.24TC

0.76TC

0.84TC

0.95TC

1.09TC

1.27TC

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4V

C(r

) [G

eV]

Continuum Cornell0.24T

C Cornell Fit

1.27TC Cornell Fit

1.27TC Screened Fit

V (r) = −αc

r+ σr + C,

Karsch, hep-ph/0512217, “Continuum Cornell”: α = π/12,√σ = 445 GeV

String Tension

V (r) = −αc

r+ σr + C,

0 0.2 0.4 0.6 0.8 1 1.2 1.4T/T

C

100

150

200

250

300

350

400

450

500

sqrt

(σ)

[M

eV]

Central τ-rangeLower τ-range

Debye ScreeningKarsch, Mehr, Satz, Z.Phys. C37 (1988) 617

V (r ,T ) = −αs

re−mD(T )r +

σ

mD(T )

(1− e−mD(T )r

)+ C

mD(T ) = the Debye screening mass.

σ = 434 MeV (i.e. fixed to “zero” temperature value)

0 0.2 0.4 0.6 0.8 1 1.2 1.4T/T

C

0

50

100

150

200

250

300

350

400

mD

[M

eV]

Central τ-rangeLower τ-range

Spin-Dependent Potentials

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

-0.05

0

0.05

0.1

0.15

VS(r

) [G

eV]

0.24TC

0.76TC

0.84TC

0.95TC

1.09TC

1.27TC

VΓ(r) = VC(r)+VS(r) s1 ·s2 −→ VC(r) =14

VPS +34

VV VS(r) = VV−VPS

Comparison with 1st generation

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

0.24TC N

f=2+1

0.76TC N

f=2+1

0.84TC N

f=2+1

0.95TC N

f=2+1

1.09TC N

f=2+1

1.27TC N

f=2+1

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

0.42TC N

f=2

1.05TC N

f=2

1.20TC N

f=2

Comparison with Static Quark Potential

F1(r ,T )

T= − log [Tr (Lren(0)Lren(r))] Lren = renormalised Polyakov loop

Kaczmarek, arXiv:0710.0498

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

0.24TC

0.76TC

0.84TC

0.95TC

1.09TC

1.27TC

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

0.89TC Free Energy

0.95TC Free Energy

1.10TC Free Energy

Charmonia Properties from the Potential: Radii

Using the parameterised screened potential from Lattice

V (r ,T ) = −αs

re−mD(T )r +

σ

mD(T )

(1− e−mD(T )r

)+ C

and solve Schrödinger Equation

Preliminary

0 0.2 0.4 0.6 0.8 1 1.2 1.4T / T

C

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

Mas

s [

GeV

]

1S1P

0 0.2 0.4 0.6 0.8 1 1.2 1.4T / T

C

0.4

0.5

0.6

0.7

0.8

0.9

1

RM

S R

adiu

s [

fm] 1S

1P

Charmonia Properties from Potential: Binding Energy

Binding Energy = M(T )− V (r →∞,T ) = M(T )− mD

σ− C

Preliminary

0 0.2 0.4 0.6 0.8 1 1.2 1.4T / T

C

0

1

2

3

4

5

6

7

8

Bin

ding

Ene

rgy

[G

eV]

1S1P

I 1P melts ∼< 1.2TC

I 1S remains bound up to at least 1.2TC

Charmonia Properties from Potential: Binding Energy

Binding Energy = M(T )− V (r →∞,T ) = M(T )− mD

σ− C

Preliminary

0.9 1 1.1 1.2 1.3T / T

C

-0.2

0

0.2

0.4

0.6

0.8

1

Bin

ding

Ene

rgy

[G

eV]

1S1P

I 1P melts ∼< 1.2TC

I 1S remains bound up to at least 1.2TC

Conductivity & Light Quark Diffusivity

arXiv:1412.6411, arXiv:1307.6763

Electrical conductivity on the lattice

EM current: jemµ =

2e3

juµ −

e3

jdµ −

e3

jsµ,

EM Correlator: G emµν (τ) =

∫d3x 〈jem

µ (τ, x)jemν (0, 0)†〉

Spectral decomposition:

G emµν (τ) =

∫ ∞0

dω2π

K (τ, ω) ρemµν(ω) with K (τ, ω) =

cosh[ω(τ − 1/2T )]

sinh[ω/2T ]

Conductivity:σ

T=

16T

limω→0

ρem(ω)

ω

Relationship to Diffusivity: DχQ = σ

Conserved (lattice) vector current used for jemµ

V Cµ(x) =

[ψ(x + µ)(1 + γµ) U†µ(x)ψ(x)− ψ(x)(1− γµ) Uµ(x)ψ(x + µ)

]




ω

ρ(ω

)

No transport peak

Tra

nsp

ort

pea

k

Melted/Plasma


Stable

Conserved Vector Correlators

0.0 0.1 0.2 0.3 0.4 0.5

τT

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

G(τ)

mq = mu,d

T/Tc

1.90

1.52

1.27

1.09

0.0 0.1 0.2 0.3 0.4 0.5

τT

mq = ms

T/Tc

0.84

0.76

0.63

0.24

Vector Spectral Function

Using default model: m(ω) = m0(b + ω)ω

0 1 2 3 4 5 6

ω [GeV]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

ρ(ω

)/ω2

mq = mu,d

1.90Tc

1.09Tc

0.63Tc

0 1 2 3 4 5 6

ω [GeV]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4mq = ms

1.90Tc

1.09Tc

0.63Tc

Vector Spectral Functions

0.0 0.2 0.4 0.6 0.8 1.0 1.2

ω [GeV]

0

1

2

3

4

5

6

7

8

ρ(ω

)/ωT

mq = mu,d

1.90Tc

1.09Tc

0.63Tc

0.0 0.2 0.4 0.6 0.8 1.0 1.2

ω [GeV]

0

2

4

6

8

10

12mq = ms

1.90Tc

1.09Tc

0.63Tc

Recall σ ∼ limω→0

ρ(ω)

ω

Conductivity Result

100 150 200 250 300 350

T [MeV]

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

C−1

emσ/T

lightstrange

0.50 0.75 1.00 1.25 1.50 1.75 2.00

T/Tc

100 150 200 250 300 350

T [MeV]

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

C−1

emσ/T

light+strange

0.50 0.75 1.00 1.25 1.50 1.75 2.00

T/Tc

Useful to factor out charge:Cem = e2∑

f q2f

Rectangles = default model systematic(i.e. b). Recall :

m(ω) = m0(b + ω)ω

Whiskers = statistical error

Diffusion CoefficientRelationship to Diffusivity: D = σ/χQ

Summary 1

Baryonic Parity Restoration

I Signicant thermal effects in −ve paritynucleon

I No observed thermal modification of +veparity mass below TC

I Degeneracy in ground state of baryonicparity partners above Tc

I Finite temperature baryonic spectralfunctions determined

0 0.5 1 1.5 2T/T

c

0

0.2

0.4

0.6

0.8

1

R

-3 -2 -1 0 1 2 3

ω

0

ρ(ω

)

T = 1.90TC

T = 1.52TC

T = 1.27TC

T = 1.09TC

T = 0.95TC

T = 0.84TC

T = 0.76TC

T = 0.24TC

Summary 2

Charmonium Potential

I Relativistic quarks rather than static quarks

I Finite temperature rather than T = 0

I Clear temperature dependent effect

I Matches Debye-screened formula withmD ≈ 0 for T < TC

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

0.24TC

0.76TC

0.84TC

0.95TC

1.09TC

1.27TC

0 0.2 0.4 0.6 0.8 1 1.2

r [fm]

2.5

3

3.5

4

VC(r

) [G

eV]

Continuum Cornell0.24T

C Cornell Fit

1.27TC Cornell Fit

1.27TC Screened Fit

Conductivity & Light Quark Diffusivity

I 2+1 flavour conductivity calculated asfunction of temperature

I Finite temperature diffusion coefficientdetermined

100 150 200 250 300 350

T [MeV]

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

C−1

emσ/T

light+strange

0.50 0.75 1.00 1.25 1.50 1.75 2.00

T/Tc




Ns Nτ T (MeV) T/Tc

24, 32 16 352 1.9024 20 281 1.52

24, 32 24 235 1.2724, 32 28 201 1.0924, 32 32 176 0.95

24 36 156 0.8424 40 141 0.7632 48 117 0.6316 128 44 0.24

3rd Generation2+1 flavours

larger volume: (3fm)3 – (4fm)3


Ns Nτ T (MeV) T/Tc

24, 32 32 352 1.9024 40 281 1.52

24, 32 48 235 1.2724, 32 56 201 1.0924, 32 64 176 0.95

24 72 156 0.8424 80 141 0.7632 96 117 0.6316 256 44 0.24

Particle Data Book

∼ 1,500 pages

zero pages on Quark-Gluon Plasma...

SLIDES TO HELP ME ANSWERDUMB QUESTIONS

SLIDES TO HELP ME ANSWERTRICKY QUESTIONS


1st Generation2 flavours

smaller volume: (2fm)3

coarser lattices: as = 0.167 fmquark mass: Mπ/Mρ ∼ 0.55temporal cut-off: aτ ∼ 7.4 GeV

Ns Nτ T (MeV) T/Tc

12 16 460 2.0912 18 409 1.8612 20 368 1.6812 24 306 1.4012 28 263 1.2012 32 230 1.0512 80 90 0.42



Ns Nτ T (MeV) T/Tc

24, 32 16 352 1.9024 20 281 1.52

24, 32 24 235 1.2724, 32 28 201 1.0924, 32 32 176 0.95

24 36 156 0.8424 40 141 0.7632 48 117 0.6316 128 44 0.24

Effects of SmearingAbove results used Gaussian smearing with sources/sinks, η smeared with:

η′ = C (1 + κH)n η using κ = 8.7 and n = 140 Capitani et al [arXiv:1205.0180]

Systematics checks of smearingI vary nI vary τ -range

0 0.5 1 1.5 2T/T

c

0

0.2

0.4

0.6

0.8

1

R

smearing, n=10smearing, n=60smearing, n=140

Effects of SmearingAbove results used Gaussian smearing with sources/sinks, η smeared with:

η′ = C (1 + κH)n η using κ = 8.7 and n = 140 Capitani et al [arXiv:1205.0180]

Systematics checks of smearing:I vary nI vary τ -range

0 0.5 1 1.5 2T/T

c

0

0.2

0.4

0.6

0.8

1

R

point source, τ/aτ>5smearing, n=10, τ/aτ>5smearing, n=140

Baryonic Spectral Functions - Systematic ChecksChecking systematics by using MEM on fixed τ windows:

τ = 1, 2, . . . 7,Nτ − 7,Nτ − 6, . . . ,Nτ − 1Preliminary

-3 -2 -1 0 1 2 3

ω

0

ρ(ω

)

T = 1.90TC

T = 1.52TC

T = 1.27TC

T = 1.09TC

T = 0.95TC

T = 0.84TC

T = 0.76TC

T = 0.24TC

Volume Effects

0 1 2 3 4 5 6 7 8 9 10 11 12

τ/aτ

1.0

1.5

2.0

CPS

24(τ

) / C

PS

32(τ

)r/a

s= 0

r/as= 1

r/as= 2

r/as= 3

r/as= 4

r/as= 5

r/as= 6

r/as= 7

r/as= 8

r/as= 9

r/as= 10

r/as= 11

r/as= 12

0.95

1.00

1.05

1.10

Ns = 24 cf Ns = 32 for 1.27Tc (Nτ =24)

Naive Temporal Term in Potential

0 5 10 15 20

τ/aτ

-5

-4

-3

-2

-1

0

1(d

C/d

τ)/C

[G

eV]

r/as= 1

r/as= 2

r/as= 3

r/as= 4

r/as= 5

r/as= 6

r/as= 7

r/as= 8

r/as= 9

PS 0.76TC using the naive form

∂

∂τf (τ) −→

[f (τ + aτ )− f (τ − aτ )

2aτ

]

Improved Temporal Term in Potential

0 5 10 15 20

τ/aτ

-5

-4

-3

-2

-1

0

1E

0 [G

eV]

0 5 10 15 20

r/as= 1

r/as= 2

r/as= 3

r/as= 4

r/as= 5

r/as= 6

r/as= 7

r/as= 8

r/as= 9

~

PS 0.76TC using the improved form Durr 1203.2560

E0(τ) =12

log

(CΓ(τ − 1) +

√CΓ(τ − 1)2 − CΓ(Nτ/2)2

CΓ(τ + 1) +√

CΓ(τ + 1)2 − CΓ(Nτ/2)2

)

Renormalising the Polyakov Loop

Polyakov Loop, L, related to free energy, F , via:

L(T ) = e−F (T )/T

But F defined up to addivitive constant ∆F = f (β, κ).Imposing renormalisation condition:

LR(TR) ≡ some number

gives us

LR(T ) = e−FR(T )/T = e−(F0(T )+∆F )/T = L0(T )e−∆F/T = L0(T )ZLNτ

and ZL defined from renormalisation condition.

Wuppertal-Budapest, PLB713(2012)342 [1204.4089]

TC from Polyakov Loop

100 150 200 250 300 350

T [MeV]

0.0

0.2

0.4

0.6

0.8

1.0

1.2Scheme

ABC

0.75 1.00 1.25 1.50 1.75

T/Tc

Scheme A: LR(Nt = 16) = 1.0Scheme B: LR(Nt = 20) = 1.0

Scheme C: LR(Nt = 20) = 0.5

Cubic spline, solid = 323, open = 243

−→ Ncritτ = 30.4(7) or Tc = 171(4) MeV

Susceptibilities’ Definitions

ni =TV∂ ln Z∂µi

χij =TV∂2 ln Z∂µi∂µj

Q =TV∂ ln Z∂µQ

=3∑

i=1

qini χQ =∂Q∂µQ

=3∑

i=1

(qi )2χii +

3∑i 6=j

qiqjχij

B =TV∂ ln Z∂µB

=3∑

i=1

ni χB =∂B∂µB

=3∑

i=1

χii +3∑

i 6=j

χij

µI = µd − µu χI =TV∂2 ln Z∂µ2

I

SusceptibilitiesχSB is Stefan-Boltzman (free) result

100 150 200 250 300 350 400−0.2

0.0

0.2

0.4

0.6

0.8

1.0

χI

χQ

χB

100 150 200 250 300 350 400

T [MeV]

χll

χss

0.75 1.00 1.25 1.50 1.75 2.000.75 1.00 1.25 1.50 1.75 2.00

T/Tc T/Tc

T [MeV]

χ/χ

SB

Conserved Vector Correlators vs Free

0.0 0.1 0.2 0.3 0.4 0.5

τT

0.2

0.4

0.6

0.8

1.0

1.2

G(τ)/G

free

LAT(τ)

mq = mu,d

T/Tc

1.90

1.52

1.27

1.09

0.0 0.1 0.2 0.3 0.4 0.5

τT

mq = ms

T/Tc

0.95

0.84

0.76

0.63

MEM Systematics I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b (default Model)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

C−1

emσ/T

32x1624x2032x24

32x2832x3224x36

24x4032x4824x128

16202428323640Nτ

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

C−1

emσ/T

all1 in 21 in 3

Variation with default modelparameter b

Recall m(ω) = m0(b + ω)ω

Anisotropy check including:all or 1 in 2 or 1 in 3of the τ datapoints

MEM Systematics II

4 6 8 10 12 14 16 18 2010−4

10−3

10−2

nτ = 6

G(τ)

182022242628300.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40C−1

em σ/T

1.09Tc

1.27Tc

1.52Tc

4 6 8 10 12 14 16 18 2010−4

10−3

10−2

nτ = 8

222426283032340.00

0.05

0.10

0.15

0.20

0.25

0.300.95Tc

1.09Tc

1.27Tc

4 6 8 10 12 14 16 18 20

τ

10−4

10−3

10−2

nτ = 10

26283032343638Nτ

0.00

0.05

0.10

0.15

0.20

0.25

0.300.84Tc

0.95Tc

1.09Tc

Stability tests:discarding the last time slices:

Are we seeing a number-of-datapoints (Nτ ) systematicor a true thermal effect?

MEM systematics

I default modelI time rangeI energy discretisation: ω = {ωmin, ωmin + ∆ω . . . ωmax}I number of configsI numerical precision

(All true also for BR)

Recall I(ρ) ≤ Nt for MEM

Can vary this in free case by varying Nt

Feature ResolutionMEM can reproduce features smaller than the characteristicsize of its basis functions:

0 0.1 0.2

atω

-0.2

-0.1

0

0.1

0.2

"ρ(ω

)"

16th basis function17th basis functionρ(ω) MEM (scaled to fit)

MEM: more than you ever wanted to know

0 0.5 1 1.5

ω

0

10

20

30

40

50

60

70

ρ(ω

)gen2_NRQCD_40 sonia_40_ spp_i_000 K=.00000,.00000 # 2

t = 2-38 Err=J Sym=N #cfgs= 502 #cfg/clus= 1

0 10 20 30 40

t

-4

-3

-2

-1

0

1

2

log(

C(t

))

Data Default Model

0 5 10 15 20s

-100

-50

0

50

100

u(s)

. 0 0.5 1 1.5

ω

-0.3

-0.2

-0.1

0

0.1

0.2

Vij(ω

)

s = 1 s = 2 s = 3 s = 4

s = 5

s = 6 s = 7 s = 8 s = 9

s = 10 s = 11 s = 12 s = 13 s = 14

s = 15

s = 16 s = 17 s = 18 s = 19

s = 20 s = 21

dashed lines are fits ρα(ω)

dashed lines are fits Cα(t)















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































The Task

Given data D

Find fit F by maximising P(F |D)

Bayes Theorem

Need to maximise P(F |D)

Bayes Theorem:

P(F |D)P(D) = P(D|F )P(F )

i.e. P(F |D) =P(D|F )P(F )

P(D)

But P(D|F ) ∼ e−χ2 −→ minimising χ2 6= maximising P(F |D)

−→ Maximum Likelihood Method wrong??

No! Since for simple F (t) = Ze−Mt , P(F ) = P(Z ,M) ∼ const

Bayes Theorem

Need to maximise P(F |D)

Bayes Theorem:

P(F |D)P(D) = P(D|F )P(F )

i.e. P(F |D) =P(D|F )P(F )

P(D)

But P(D|F ) ∼ e−χ2 −→ minimising χ2 6= maximising P(F |D)

−→ Maximum Likelihood Method wrong??

No! Since for simple F (t) = Ze−Mt , P(F ) = P(Z ,M) ∼ const

Priors

Actually P(F = elephant) ≡ 0

−→ “priors” which encode any additional information

(a.k.a. predisposition, prejudices, impartialities, biases, prediliction, subjectivity, . . .)

E.g. in L.G.T. P(M < 0) ≡ 0

Maximum Likelihood Method applies this prior implicitly

Can encode prior information with “entropy”= S (dis-information)

Define I(F ) = “Information content” of F

“Bland” F has I(F ) ∼ 0 and S >> 0

“Spiky” F has I(F ) >> 0 and S ≡ 0

Priors

Actually P(F = elephant) ≡ 0

−→ “priors” which encode any additional information

(a.k.a. predisposition, prejudices, impartialities, biases, prediliction, subjectivity, . . .)

E.g. in L.G.T. P(M < 0) ≡ 0

Maximum Likelihood Method applies this prior implicitly

Can encode prior information with “entropy”= S (dis-information)

Define I(F ) = “Information content” of F

“Bland” F has I(F ) ∼ 0 and S >> 0

“Spiky” F has I(F ) >> 0 and S ≡ 0

Entropy

No Data Data

No Prior I(F ) ≡ 0 F from min χ2

Prior F ≡ prior F from max P(F |D)

P(F ) = e−S

Documents

Properties of Thermal Matter: Conductivity, Parity Restoration and … · 2015-07-18 · Properties of Thermal Matter: Conductivity, Parity Restoration and the Charmonium Potential