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
QED vs QCD
QED QCD
flux tube, squeezed one-dimentionally !
confinement potential
Electric field spreads all over the space
Coulomb potential
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
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
Measurement of the Stress on the Lattice
①prepare 𝑞 ത𝑞 on the lattice and ②measure EMT around 𝑞 ത𝑞
To Do
Imaginary
time
𝑇/2
𝑅
space ത 𝑞 𝑞
𝑊(𝑅, 𝑇)
= 𝐶
0exp −𝑉
0𝑅 𝑇 + 𝐶
1exp −𝑉
1𝑅 𝑇 + ⋯
𝑉
0𝑅 = − 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
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
0exp −𝑉
0𝑅 𝑇 + 𝐶
1exp −𝑉
1𝑅 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
0𝑅 = − lim
𝑇→∞
1
𝑇 log 𝑊(𝑅, 𝑇)
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
0exp −𝑉
0𝑅 𝑇 + 𝐶
1exp −𝑉
1𝑅 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
0𝑅 = − lim
𝑇→∞
1
𝑇 log 𝑊(𝑅, 𝑇) 𝑇
𝜇𝜈𝑡, 𝑥
𝑊
= ⟨𝑇
𝜇𝜈𝑡, 𝑥 𝑊 𝑅, 𝑇 ⟩
⟨𝑊 𝑅, 𝑇 ⟩ − 𝑇
𝜇𝜈𝑡, 𝑥
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
4140
6.465 0.046 fm 5 48
4440
6.600 0.038 fm 6 48
41500
6.819 0.029 fm 8 64
41000
0.912fm 0.684fm
0.456fm
Stress Distribution in Maxwell Theory
𝑇
𝑖𝑗= 𝜖
0𝐸
𝑖𝐸
𝑗− 𝛿
𝑖𝑗2 𝐸
2+ 1
𝜇
0𝐵
𝑖𝐵
𝑗− 𝛿
𝑖𝑗2 𝐵
2 Perpendicular plane : attractive
Parallel plane : repulsive
𝐸
stress tensor
𝑇
𝑖𝑗𝑛
𝑗(𝜆𝑘)= 𝜆
𝑘𝑛
𝑖(𝜆𝑘)(𝑖, 𝑗 = 𝑥, 𝑦, 𝑧; 𝑘 = 𝑥, 𝑦, 𝑧) Colored : attractive
Gray : repulsive
Stress Distribution in SU(3) YM Theory
mid 𝑦 𝑥𝑧𝑧𝑥 𝑅
𝛽 = 6.819 (no continuum limit)
𝑅 = 0.456[fm], 𝑎 = 0.029[fm]
Only 𝑡 → 0(linear fit)
Preliminary
SU(3) YM Theory vs Maxwell Theory
SU(3) YM Theory Maxwell Theory
𝑦
𝑧 mid 𝑥
𝑧𝑥 𝑅
Preliminary
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
𝑇
𝑖𝑗= 𝜖
0𝐸
𝑖𝐸
𝑗− 𝛿
𝑖𝑗2 𝐸
2+ 1
𝜇
0𝐵
𝑖𝐵
𝑗− 𝛿
𝑖𝑗2 𝐵
2EMT in Maxwell Theory (revisit)
𝑧 𝑟
𝜃
− 𝑇
00 𝑊= − 𝑇
𝑧𝑧 𝑊= 𝑇
𝑟𝑟 𝑊= 𝑇
𝜃𝜃 𝑊degenerate
(In this case, 𝐸 = 0,0, 𝐸 , 𝐵 = 0)
𝑦
𝑧 mid 𝑥
𝑧𝑥 𝑅
Double Extrapolation @ mid point
𝑎 2 𝑡
Strong
discretization effect
①
②
Double extrapolation
𝑂lat = 𝑂cont + 𝐶
0𝑡 + 𝐶
1𝑎
2𝑡 + ⋯
①
② continuum
Preliminary
𝑦
𝑧 mid 𝑥
𝑧𝑥 𝑅
𝑎 2 𝑡
Strong
discretization effet
①
②
Double extrapolation
𝑂lat = 𝑂cont + 𝐶
0𝑡 + 𝐶
1𝑎
2𝑡 + ⋯
①
② continuum
Preliminary
FlowQCD (2016)
Double Extrapolation @ mid point
Separation !!
Profile of 𝑻 𝒊𝒊 𝑾 (𝒊 = 𝟎, 𝒛, 𝒓, 𝜽) (mid plane)
𝑧 𝑟
𝜃
𝑦
𝑧 mid 𝑥
𝑧𝑥 𝑅