QCD energy momentum tensor at finite temperature using gradient - - PowerPoint PPT Presentation

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QCD energy momentum tensor at finite temperature using gradient - - PowerPoint PPT Presentation

Apr 01, 2023 373 likes •646 views

QCD energy momentum tensor at finite temperature using gradient flow Yusuke Taniguchi for WHOT QCD collaboration S.Ejiri, R.Iwami, K.Kanaya, M.Kitazawa, M.Shirogane, H.Suzuki, Y.T, T.Umeda, N.Wakabayashi Introduction Nuclear Matter

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SLIDE 1

QCD energy momentum tensor at finite temperature using gradient flow

Yusuke Taniguchi for WHOT QCD collaboration

S.Ejiri, R.Iwami, K.Kanaya, M.Kitazawa, M.Shirogane, H.Suzuki, Y.T, T.Umeda, N.Wakabayashi

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SLIDE 2

Introduction

Energy momentum tensor Poincare symmetry     T00 T01 T02 T03 T10 T11 T12 T13 T20 T21 T22 T23 T30 T31 T32 T33    

Tµν

energy pressure momentum stress

  • If we have Tµν

direct measurement of thermodynamic quantity

  • Fluctuations and correlations of Tµν

specific heat, viscosity, …

? ?

?

From Nuclear Matter to Quark Matter

Nuclear Matter Quark-Gluon-Plasma

Color Super- conductivity

Nucleus Neutronstar

Early Evolution

  • f the Universe

Temperature Density

hot topics in QGP

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SLIDE 3

How to calculate Tμν on lattice?

Measure expectation values on lattice Renormalization δµνF a

ρσ(x)F a ρσ(x)

terms in QCD Lagrangian δµν ¯ ψ(x)← → / D ψ(x) δµν ¯ ψ(x)ψ(x) terms in QCD Lagrangian when trace is taken F a

µρ(x)F a νρ(x)

¯ ψ(x) ⇣ γµ ← → D ν + γν ← → D µ ⌘ ψ(x) Well established for E and P Karsch coefficients problems non universal (No Poincare symmetry)

  • depends on: lattice action, operator

additive correction for δµν ¯ ψ(x)ψ(x)

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SLIDE 4

Easier method for renormalization?

Gradient Flow

∂tAµ(t, x) = −δSYM δAµ Aµ(t = 0, x) = Aµ(x) ∂tAµ(t, x) = DνGνµ Aµ(t, x) = Z d4yKt(x − y)Aµ(y) + interactions Kt(x) = e−x2/4t (4πt)D/2

Narayanan-Neuberger(2006) Lüscher(2009–)

Flow the gauge field t: flow time, dim=[length ]

2

A kind of diffusion equation Solution heat kernel smear field within √ 8t

How to calculate Tμν on lattice?

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SLIDE 5

Gradient Flow as a renormalization scheme

Narayanan-Neuberger(2006), Lüscher(2010), Lüscher-Weisz(2011)

Gauge operators with flowed field Aµ(t, x) does not have UV divergence does not have contact term singularity

  • perators are renormalized

A great view point: Re＜1ー ＞

flow + a→0

scale: √ 8t

lattice operator lattice action lattice operator differences in absorbed NP renormalized operator

F a

µνF a µν(x, t)

fi n i t e r e n .

universal

MS scheme

How to calculate Tμν on lattice?

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SLIDE 6

From flowed operator to proper operator

  • perator which satisfies

WT identity Matching coefficients are calculable perturbatively at small t region

H.Suzuki, PTEP 2013, 083B03 (2013)

O1µν(t, x) = F a

µρF a νρ(t, x)

O2µν(t, x) = δµνF a

ρσF a ρσ(t, x)

NP renormalized in flow scheme Need a matching factor like Wilson coefficient scale matching µ = 1 √ 8t

{Tµν}WT (x) = lim

t→0

⇢ c1(t)  ˜ O1µν(t, x) − 1 4 ˜ O2µν(t, x)

  • +c2(t)

h ˜ O2µν(t, x) − D ˜ O2µν(t, x) E

T =0

i

How to calculate Tμν on lattice?

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SLIDE 7

Matching coefficients are calculable perturbatively at small t region

H.Suzuki, PTEP 2013, 083B03 (2013)

O1µν(t, x) = F a

µρF a νρ(t, x)

O2µν(t, x) = δµνF a

ρσF a ρσ(t, x)

c1(t) = 1 ¯ g(1/ √ 8t)2 − 1 (4π)2 ✓ 9γ − 18 ln 2 + 19 4 ◆ c2(t) = 1 (4π)2 33 16 Matching coefficients at one loop

{Tµν}WT (x) = lim

t→0

⇢ c1(t)  ˜ O1µν(t, x) − 1 4 ˜ O2µν(t, x)

  • +c2(t)

h ˜ O2µν(t, x) − D ˜ O2µν(t, x) E

T =0

i

How to calculate Tμν on lattice?

From flowed operator to proper operator

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SLIDE 8

Three steps to calculate Tμν

  • 1. Flow the link variable
  • 2. Calculate expectation value of flowed operators

∂tUµ(t, x)U †

µ(t, x) = −g2 0∂x,µSlat(U)

O1µν(t, x) = F a

µρF a νρ(t, x)

O2µν(t, x) = δµνF a

ρσF a ρσ(t, x)

appropriately defined on lattice

  • 3. Multiply the coefficients and take t→0 limit

{Tµν}WT (x) = lim

t→0

⇢ c1(t)  ˜ O1µν(t, x) − 1 4 ˜ O2µν(t, x)

  • +c2(t)

h ˜ O2µν(t, x) − D ˜ O2µν(t, x) E

T =0

i

How to calculate Tμν on lattice?

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SLIDE 9

Previous works and lessons

FlowQCD Collaboration (2014-) BW12

T=1.66Tc

(E+P)/T4

O

  • t2

O ✓a2 t ◆ quench {Tµν} (x, t, a) = {Tµν}WT (x) (dim4 operator) +a2 t +t(dim6 operator) O (t) tamed at large t +t2(dim8 operator) why we need t→0 tamed at small t the window region +O(a2T 2, a2m2, a2Λ2

QCD)

need to take a→0 limit

flowed operator on lattice

How to calculate Tμν on lattice?

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SLIDE 10

What’s new?

Quarks included!

Flow of quark field

Lüscher, JHEP 1304, 123 (2013)

∂tχ(t, x) = DµDµχ(t, x) χ(t = 0, x) = ψ(x) ∂t ¯ χ(t, x) = ¯ χ(t, x)← − Dµ ← − Dµ ¯ χ(t = 0, x) = ¯ ψ(x) Renormalization is needed for quark field χR(t, x) = Zχχ0(t, x) No more renormalization is needed for composite op. (¯ χ(t, x)χ(t, x))R = Z2

χ (¯

χ(t, x)χ(t, x))0 flow the gauge field simultaneously

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SLIDE 11

Three steps to calculate Tμν

  • 1. Flow the gauge and quark field
  • 2. Calculate expectation value of flowed operators

How to calculate Tμν on lattice?

  • 3. Multiply the coefficients and take t→0 limit

+c4(t) X

r=u,d,s

⇣ ˜ Or

4µν(t, x) −

D ˜ Or

4µν(t, x)

E

T =0

⌘ + X

r=u,d,s

cr

5(t)

⇣ ˜ Or

5µν(t, x) −

D ˜ Or

5µν(t, x)

E

T =0

⌘

{Tµν}q

WT (x) = lim t→0

⇢ c3(t) X

r=u,d,s

⇣ ˜ Or

3µν(t, x) − 2 ˜

Or

4µν(t, x) −

D ˜ Or

3µν(t, x) − 2 ˜

Or

4µν(t, x)

E

T =0

⌘

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SLIDE 12

How to calculate Tμν on lattice?

  • 3. Multiply the coefficients and take t→0 limit

˜ Or

3µν(t, x) ≡ ϕr(t)¯

χr(t, x) ⇣ γµ ← → D ν + γν ← → D µ ⌘ χr(t, x)

˜ Or

4µν(t, x) ≡ ϕr(t)δµν ¯

χr(t, x)← → / D χr(t, x) ˜ Or

5µν(t, x) ≡ ϕr(t)δµν ¯

χr(t, x)χr(t, x)

+c4(t) X

r=u,d,s

⇣ ˜ Or

4µν(t, x) −

D ˜ Or

4µν(t, x)

E

T =0

⌘ + X

r=u,d,s

cr

5(t)

⇣ ˜ Or

5µν(t, x) −

D ˜ Or

5µν(t, x)

E

T =0

⌘

ϕr(t) ≡ −6 (4π)2t2 D ¯ χr(t, x)← → / D χr(t, x) E

T =0

2 D ˜ Or

3µν(t, x)

E

T =0 =

D ˜ Or

4µν(t, x)

E

T =0 =

−6 (4π)2t2 δµν

{Tµν}q

WT (x) = lim t→0

⇢ c3(t) X

r=u,d,s

⇣ ˜ Or

3µν(t, x) − 2 ˜

Or

4µν(t, x) −

D ˜ Or

3µν(t, x) − 2 ˜

Or

4µν(t, x)

E

T =0

⌘

wave function renormalization VEV sub.

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SLIDE 13

How to calculate Tμν on lattice?

  • 3. Multiply the coefficients and take t→0 limit

c3(t) = 1 4 ( 1 + ¯ g(1/ √ 8t)2 (4π)2 ✓ 2 + 4 3 ln(432) ◆)

c4(t) = 1 (4π)2 ¯ g(1/ √ 8t)2

cr

5(t) = − ¯

mr(1/ √ 8t) ( 1 + ¯ g(1/ √ 8t)2 (4π)2 ✓ 4γ − 8 ln 2 + 14 3 + 4 3 ln(432) ◆)

Makino-Suzuki, PTEP 2014, 063B02 (2014)

+c4(t) X

r=u,d,s

⇣ ˜ Or

4µν(t, x) −

D ˜ Or

4µν(t, x)

E

T =0

⌘ + X

r=u,d,s

cr

5(t)

⇣ ˜ Or

5µν(t, x) −

D ˜ Or

5µν(t, x)

E

T =0

⌘

{Tµν}q

WT (x) = lim t→0

⇢ c3(t) X

r=u,d,s

⇣ ˜ Or

3µν(t, x) − 2 ˜

Or

4µν(t, x) −

D ˜ Or

3µν(t, x) − 2 ˜

Or

4µν(t, x)

E

T =0

⌘

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SLIDE 14

Numerical setups

Iwasaki gauge action β=2.05 : a～0.07 [fm] Fixed scale method T=1/(aNt), Nt=16, 14, 12, 10, 8, 6, 4 Nf=2+1 NP improved Wilson fermion On an equal quark mass line

100 200 300 400 500 16 14 12 10 8 6 β (β=2.05) T[MeV]

mπ/mρ~0.6

Tc

mπ mρ ∼ 0.6 32 xNt for T≠0

3

28 x56 for T=0

3

aT~1/Nt artifact may be severe aT~1/Nt artifact is small enough

mηss mφ ∼ 0.74

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SLIDE 15

T=279MeV

t→0 limit by linear extrapolation

T=465MeV

p/T4

T=349MeV T=233MeV T=199MeV T=174MeV

t/a2

  • 4
  • 2

2 4 6 8 0.5 1 1.5 2 Tii/3T4 t/a2

  • 4
  • 2

2 4 6 8 0.5 1 1.5 2 Tii/3T4 t/a2

  • 4
  • 2

2 4 6 8 0.5 1 1.5 2 Tii/3T4 t/a2

  • 4
  • 2

2 4 6 8 0.5 1 1.5 2 Tii/3T4 t/a2

  • 4
  • 2

2 4 6 8 0.5 1 1.5 2 Tii/3T4 t/a2

  • 4
  • 2

2 4 6 8 0.5 1 1.5 2 Tii/3T4 t/a2

t/a2 t/a2

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SLIDE 16

p/T4

as a function of T

WHOT-QCD, Phys. Rev. D 85, 094508 (2012) integration method

gradient flow

2 4 6 8 10 100 200 300 400 500 600 p/T4 T (MeV) gradient flow T-integration

aT=1/Nt artifact is severe

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SLIDE 17

e/T4

  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 T00/T4 t/a2

  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 T00/T4 t/a2

t→0 limit by linear extrapolation

  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 T00/T4 t/a2

  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 T00/T4 t/a2

  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 T00/T4 t/a2

  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 T00/T4 t/a2

T=279MeV T=465MeV T=349MeV T=233MeV T=199MeV T=174MeV

t/a2 t/a2 t/a2

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SLIDE 18

e/T4

as a function of T

WHOT-QCD, Phys. Rev. D 85, 094508 (2012) integration method

gradient flow

5 10 15 20 25 100 200 300 400 500 600 e/T4 T (MeV) gradient flow T-integration

aT=1/Nt artifact is severe

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SLIDE 19

(e+p)/T4 and (e-3p)/T4

as a function of T

(e+p)/T4 (e-3p)/T4

WHOT-QCD, Phys. Rev. D 85, 094508 (2012) integration method

gradient flow

5 10 15 20 25 30 100 200 300 400 500 600 (e+p)/T4 T (MeV) gradient flow T-integration

  • 4
  • 2

2 4 6 8 10 12 100 200 300 400 500 600 (e-3p)/T4 T (MeV) gradient flow T-integration

aT=1/Nt artifact is severe

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SLIDE 20

Only three steps!

  • 1. Flow the gauge and quark field
  • 2. Calculate VEV of flowed operators

Measurement of chiral condensate

  • 3. Multiply the coefficients and take t→0 limit
  • ψψ
  • MS (2GeV) = lim

t→0 cS(t)mMS(1/

√ 8t) mMS(2GeV) ϕ(t)χ(t, x)χ(t, x) flowed operator

cS(t) = ( 1 + ¯ gMS(1/ √ 8t)2 (4π)2 ✓ 4γ − 8 ln 2 + 8 + 4 3 ln(432) ◆)

ϕ(t) = −6 (4π)2t2 D ¯ χ(t, x)← → / D χ(t, x) E

T =0

matching coefficient wave function renormalization

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SLIDE 21

Chiral condensate

  • 0.01
  • 0.005

0.005 0.01 0.5 1 1.5 2 chiral condensate t/a2 u quark s quark

  • 0.01
  • 0.005

0.005 0.01 0.5 1 1.5 2 chiral condensate t/a2 u quark s quark

  • 0.01
  • 0.005

0.005 0.01 0.5 1 1.5 2 chiral condensate t/a2 u quark s quark

  • 0.01
  • 0.005

0.005 0.01 0.5 1 1.5 2 chiral condensate t/a2 u quark s quark

  • 0.01
  • 0.005

0.005 0.01 0.5 1 1.5 2 chiral condensate t/a2 u quark s quark

  • 0.01
  • 0.005

0.005 0.01 0.5 1 1.5 2 chiral condensate t/a2 u quark s quark

T=279MeV T=349MeV T=233MeV T=199MeV T=174MeV

t/a2 t/a2 t/a2 t→0 limit by linear extrapolation

T=0

u quark s quark

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SLIDE 22

disconnected chiral susceptibility

2e-08 4e-08 6e-08 8e-08 1e-07 0.5 1 1.5 2 chiral susceptibility t/a2 u quark s quark

t→0 limit by linear extrapolation

2e-08 4e-08 6e-08 8e-08 1e-07 0.5 1 1.5 2 chiral susceptibility t/a2 u quark s quark 2e-08 4e-08 6e-08 8e-08 1e-07 0.5 1 1.5 2 chiral susceptibility t/a2 u quark s quark 2e-08 4e-08 6e-08 8e-08 1e-07 0.5 1 1.5 2 chiral susceptibility t/a2 u quark s quark 2e-08 4e-08 6e-08 8e-08 1e-07 0.5 1 1.5 2 chiral susceptibility t/a2 u quark s quark 2e-08 4e-08 6e-08 8e-08 1e-07 0.5 1 1.5 2 chiral susceptibility t/a2 u quark s quark

T=279MeV T=349MeV T=233MeV T=199MeV T=174MeV

t/a2 t/a2 t/a2

T=0

u quark s quark

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SLIDE 23

Chiral condensate

as a function of T

chiral condensate disconnected chiral susceptibility

0.1 0.2 0.3 0.4 0.5 0.6 0.7 100 200 300 400 500 600 chiral condensate T (MeV) u quark s quark 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 100 200 300 400 500 600 chiral susceptibility T (MeV) u quark s quark

⌦ ψψ ↵

MS (2GeV)

1/4 (GeV)6 (MeV)

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SLIDE 24

Summary

Flow method works well for EM tensor! as powerful as the derivative method. More suitable for Wilson fermion. We have exciting results:

(e+p)/T4 (e-3p)/T4

chiral condensate chiral susceptibility

Lattice artifact is severe for Nt=4, 6, 8 We want work with fluctuation and correlation function using the flow!

5 10 15 20 25 30 100 200 300 400 500 600 (e+p)/T T (MeV) gradient flow T-integration

  • 4
  • 2

2 4 6 8 10 12 100 200 300 400 500 600 (e-3p)/T4 T (MeV) gradient flow T-integration 0.1 0.2 0.3 0.4 0.5 0.6 0.7 100 200 300 400 500 600 chiral condensate T (MeV) u quark s quark 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 100 200 300 400 500 600 chiral susceptibility T (MeV) u quark s quark

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SLIDE 25

topological susceptibility

χT = 1 V4

  • hQ2i hQi2

⌦ Q2↵ = m2 N 2

f

⌦ P 0P 0↵

disc

Q = 1 64⇡2 Z d4x✏µνρσF a

µν(x)F a ρσ(x)

agrees well!

aT=1/Nt artifact is severe

(GeV4)

1×10-5 1×10-4 1×10-3 0.5 1 1.5 2 2.5 3 3.5 χT T/Tc gauge (T/Tc)-7.2(9) ud quark (T/Tc)-7.4(1.6)

Kanaya @ poster session

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SLIDE 26

Energy and Pressure

T=279MeV contributions from gauge and quarks

  • 2

2 4 6 8 10 12 0.5 1 1.5 2 total gauge ud quark s quark

  • 30
  • 25
  • 20
  • 15
  • 10
  • 5

0.5 1 1.5 2 total gauge ud quark s quark

P T 4 − E T 4 t t total gauge quark major contribution is from quarks