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(1)

Stress and energy distribution in quark-anti-quark systems using gradient flow

Ryosuke Yanagihara (Osaka University)

for FlowQCD Collaboration :

Masayuki Asakawa, Takumi Iritani,

Masakiyo Kitazawa, Tetsuo Hatsuda

(2)

QED vs QCD

QED QCD

 flux tube, squeezed one-dimentionally !

 confinement potential

 Electric field spreads all over the space

 Coulomb potential

(3)

QED vs QCD QED

 Electric field spreads all over the space.

 Coulomb potential

QCD

Action through medium

 flux tube, squeezed one-dimentionally !

 confinement potential Maxwell stress

 Perpendicular plane : attractive

 Parallel plane : repulsive

(4)

Energy Momentum Tensor (EMT)

Physics around flux tube in terms of energy and stress

goal

𝑇 𝜇𝜈 =

𝑇 00 𝑇 01 𝑇 02 𝑇 03 𝑇 10 𝑇 11 𝑇 12 𝑇 13 𝑇 20

𝑇 30

𝑇 21 𝑇 31

𝑇 22 𝑇 23 𝑇 32 𝑇 33

Energy density

Stress tensor

Momentum density

 Gauge invariant !

 Determine absolute values of all components

Action through

medium

(5)

Measurement of the Stress on the Lattice

①prepare 𝑞 ത𝑞 on the lattice and ②measure EMT around 𝑞 ത𝑞

To Do

(6)

Imaginary

time

𝑇/2

𝑅

space ത 𝑞 𝑞

𝑊(𝑅, 𝑇)

= 𝐶

0

exp −𝑉

0

𝑅 𝑇 + 𝐶

1

exp −𝑉

1

𝑅 𝑇 + ⋯

𝑉

0

𝑅 = − lim

𝑇→∞

1

𝑇 log 𝑊(𝑅, 𝑇)

Ground state potential

Wilson Loop

Confinement potential

① prepare 𝑞 ത 𝑞 on the lattice and ②measure EMT around 𝑞 ത 𝑞

To Do

Measurement of the Stress on the Lattice

(7)

𝑡

8𝑡

𝐵𝜇(𝑡 ≠ 0, 𝑥)

𝜕𝐵𝜇 𝑡, 𝑥

𝜕𝑡 = −𝑔02 𝛿𝑆[𝐵]

𝛿𝐵𝜇(𝑡, 𝑥) Flow eq. L ሷuscher (2010)

𝐵

𝜇

: smeared field

EMT defined via gradient flow

𝑇𝜇𝜈 𝑡, 𝑥 = 1

𝛼𝑈 𝑡 𝑈𝜇𝜈 𝑡, 𝑥 + 𝛿𝜇𝜈

4𝛼𝐸(𝑡) 𝐸 𝑡, 𝑥 − 𝐸 𝑡, 𝑥 + 𝑂(𝑡) Suzuki (2013)

Gradient flow

Imaginary

time

𝑇/2

𝑅

space ത 𝑞 𝑞

𝑊(𝑅, 𝑇)

= 𝐶

0

exp −𝑉

0

𝑅 𝑇 + 𝐶

1

exp −𝑉

1

𝑅 𝑇 + ⋯

Ground state potential

Wilson Loop

① prepare 𝑞 ത 𝑞 on the lattice and ②measure EMT around 𝑞 ത 𝑞

To Do

Measurement of the Stress on the Lattice

𝑉

0

𝑅 = − lim

𝑇→∞

1

𝑇 log 𝑊(𝑅, 𝑇)

(8)

𝑡

8𝑡

𝐵𝜇(𝑡 ≠ 0, 𝑥)

𝜕𝐵𝜇 𝑡, 𝑥

𝜕𝑡 = −𝑔02 𝛿𝑆[𝐵]

𝛿𝐵𝜇(𝑡, 𝑥) Flow eq. L ሷuscher (2010)

𝐵

𝜇

: smeared field

EMT defined via gradient flow

𝑇𝜇𝜈 𝑡, 𝑥 = 1

𝛼𝑈 𝑡 𝑈𝜇𝜈 𝑡, 𝑥 + 𝛿𝜇𝜈

4𝛼𝐸(𝑡) 𝐸 𝑡, 𝑥 − 𝐸 𝑡, 𝑥 + 𝑂(𝑡) Suzuki (2013)

Gradient flow

Imaginary

time

𝑇/2

𝑅

space ത 𝑞 𝑞

𝑊(𝑅, 𝑇)

= 𝐶

0

exp −𝑉

0

𝑅 𝑇 + 𝐶

1

exp −𝑉

1

𝑅 𝑇 + ⋯

Ground state potential

Wilson Loop

① prepare 𝑞 ത 𝑞 on the lattice and ②measure EMT around 𝑞 ത 𝑞

To Do

Measurement of the Stress on the Lattice

𝑉

0

𝑅 = − lim

𝑇→∞

1

𝑇 log 𝑊(𝑅, 𝑇) 𝑇

𝜇𝜈

𝑡, 𝑥

𝑊

= ⟨𝑇

𝜇𝜈

𝑡, 𝑥 𝑊 𝑅, 𝑇 ⟩

⟨𝑊 𝑅, 𝑇 ⟩ − 𝑇

𝜇𝜈

𝑡, 𝑥

(9)

Setup

 Quenched SU(3)

 Wilson gauge action

 Clover operator

 APE smearing for spatial links

 Multihit improvement in temporal link

 Simulation using BlueGene/Q @ KEK

𝜷 lattice spacing ratio lattice size # of statistics

6.304 0.057 fm 4 48

4

140

6.465 0.046 fm 5 48

4

440

6.600 0.038 fm 6 48

4

1500

6.819 0.029 fm 8 64

4

1000

0.912fm 0.684fm

0.456fm

(10)

Stress Distribution in Maxwell Theory

𝑇

𝑖𝑗

= 𝜖

0

𝐸

𝑖

𝐸

𝑗

− 𝛿

𝑖𝑗

2 𝐸

2

+ 1

𝜇

0

𝐵

𝑖

𝐵

𝑗

− 𝛿

𝑖𝑗

2 𝐵

2

 Perpendicular plane : attractive

 Parallel plane : repulsive

𝐸

 stress tensor

𝑇

𝑖𝑗

𝑛

𝑗(𝜆𝑘)

= 𝜆

𝑘

𝑛

𝑖(𝜆𝑘)

(𝑖, 𝑗 = 𝑥, 𝑦, 𝑧; 𝑘 = 𝑥, 𝑦, 𝑧) Colored : attractive

Gray : repulsive

(11)

Stress Distribution in SU(3) YM Theory

mid 𝑦 𝑥𝑧

𝑧𝑥 𝑅

 𝛽 = 6.819 (no continuum limit)

 𝑅 = 0.456[fm], 𝑎 = 0.029[fm]

 Only 𝑡 → 0(linear fit)

Preliminary

(12)

SU(3) YM Theory vs Maxwell Theory

SU(3) YM Theory Maxwell Theory

𝑦

𝑧 mid 𝑥

𝑧𝑥 𝑅

Preliminary

(13)

We focus on mid-plane.

next

𝑦

𝑧 mid 𝑥

𝑧𝑥 𝑅

Preliminary

SU(3) YM Theory vs Maxwell Theory

SU(3) YM Theory Maxwell Theory

(14)

𝑇

𝑖𝑗

= 𝜖

0

𝐸

𝑖

𝐸

𝑗

− 𝛿

𝑖𝑗

2 𝐸

2

+ 1

𝜇

0

𝐵

𝑖

𝐵

𝑗

− 𝛿

𝑖𝑗

2 𝐵

2

EMT in Maxwell Theory (revisit)

𝑧 𝑟

𝜃

− 𝑇

00 𝑊

= − 𝑇

𝑧𝑧 𝑊

= 𝑇

𝑟𝑟 𝑊

= 𝑇

𝜃𝜃 𝑊

degenerate

(In this case, 𝐸 = 0,0, 𝐸 , 𝐵 = 0)

(15)

𝑦

𝑧 mid 𝑥

𝑧𝑥 𝑅

Double Extrapolation @ mid point

𝑎 2 𝑡

Strong

discretization effect

Double extrapolation

𝑂lat = 𝑂cont + 𝐶

0

𝑡 + 𝐶

1

𝑎

2

𝑡 + ⋯

② continuum

Preliminary

(16)

𝑦

𝑧 mid 𝑥

𝑧𝑥 𝑅

𝑎 2 𝑡

Strong

discretization effet

Double extrapolation

𝑂lat = 𝑂cont + 𝐶

0

𝑡 + 𝐶

1

𝑎

2

𝑡 + ⋯

② continuum

Preliminary

FlowQCD (2016)

Double Extrapolation @ mid point

(17)

Separation !!

Profile of 𝑻 𝒊𝒊 𝑾 (𝒊 = 𝟎, 𝒛, 𝒓, 𝜽) (mid plane)

𝑧 𝑟

𝜃

𝑦

𝑧 mid 𝑥

𝑧𝑥 𝑅

Preliminary

(18)

Profile of 𝑻 𝒊𝒊 𝑾 (𝒊 = 𝟎, 𝒛, 𝒓, 𝜽) (mid plane)

Can confinement models describe this separation ?

Abelian dual monopole model Nambu-Goto string

Holographic QCD

Preliminary

(19)

From confinement potential

Energy density= ׬

𝑚𝑖𝑑

𝑇

00

𝑑

2

𝑥 Force = ׬

𝑚𝑖𝑑

𝑇

11

𝑑

2

𝑥

Potential vs EMT (mid plane)

From EMT

𝑉 𝑅 = 𝑎 + 𝜎𝑅 + Τ 𝑐 𝑅 ここに数式を入力します。

𝐹 = − 𝑑𝑉(𝑅) 𝑑𝑅

Preliminary

Preliminary

(20)

Summary and Outlook

summary First measurements of stress distribution on the lattice !!

Separation

outlook

 We need to explain the stress distribution

using abelian dual monopole model model, NG string…

 Application : two flux tube, finite temperature, excited states…

Preliminary Preliminary

(21)

backup

(22)

Enhancement of ground state

𝛽 = 6.819

(23)

𝒂 → 𝟎 limit

(24)

Fat flux tube

Cardoso et al. (2013)

Preliminary

(25)

Stress asymmetry

Preliminary

參考文獻

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