Stress tensor distribution around static quarks in hot medium - PowerPoint PPT Presentation

Stress tensor distribution around static quarks in hot medium Ryosuke Yanagihara (Osaka University) For FlowQCD collaboration : Takumi Iritani, Masakiyo Kitazawa, Masayuki Asakawa, Tetsuo Hatsuda FLQCD 2019 @ YITP (2019/04/18)

Confined vs. Deconfined gluon quark 𝑈 0 Critical temperature 𝑈 𝑑 Confined Deconfined 1 FLQCD 2019 @ YITP (2019/04/18)

Confined vs. Deconfined Pressure distribution inside Hadrons gluon Exp. quark Burkert et al ., Nature 557 (2018) 396. Th. 温度 𝑈 0 𝑈 𝑑 Confined 非閉じ込め相 Shanahan et al ., PRL 122 (2019) no7, 072003. Kumano et al ., PRD 97 (2018) 014020. 1 FLQCD 2019 @ YITP (2019/04/18)

Pressure distribution inside hadrons vs. Our study Pressure distribution Our study inside Hadrons Exp. ത 𝑅 𝑅 Burkert et al ., Nature 557 (2018) 396. Th. Shanahan et al ., PRL 122 (2019) no7, 072003. Kumano et al ., PRD 97 (2018) 014020. 2 FLQCD 2019 @ YITP (2019/04/18)

Flux tube QCD QED ത 𝑅 𝑅  Flux tube, squeezed one-dimensionally  Electric field spreads all over the space  Confinement potential  Coulomb potential 3 FLQCD 2019 @ YITP (2019/04/18)

Flux tube QCD QED ത 𝑅 𝑅  Flux tube, squeezed one-dimensionally  Electric field spreads all over the space  Confinement potential  Coulomb potential Local interaction ？ Maxwell stress 3 FLQCD 2019 @ YITP (2019/04/18)

𝑓 𝑠 𝑦 𝑧 mid A lot of previous studies 𝑨 𝑓 𝑨 𝑓 𝜄 𝑆 𝑧𝑨 Color electric field Action density Cea et al. , PRD 88 (2012) 054504. Cardoso et al. , PRD 86 (2013) 054501. 4 FLQCD 2019 @ YITP (2019/04/18)

𝑓 𝑠 𝑦 𝑧 mid A lot of previous studies 𝑨 𝑓 𝑨 𝑓 𝜄 𝑆 𝑧𝑨 Color electric field Action density Cea et al. , PRD 88 (2012) 054504. Cardoso et al. , PRD 86 (2013) 054501. More direct physical quantity : Stress tensor !! 4 FLQCD 2019 @ YITP (2019/04/18)

Energy momentum tensor (EMT) Momentum density Energy density 𝑈 02 𝑈 03 𝑈 00 𝑈 01 𝑈 𝑈 𝑈 𝑈 12 13 10 11 𝑈 𝜈𝜉 = 𝑈 20 𝑈 21 𝑈 22 𝑈 23 Pressure 𝑈 30 𝑈 31 𝑈 32 𝑈 33 Stress tensor  Stress is force per unit area 𝑔 𝑗 = 𝜏 𝑗𝑘 𝑜 𝑘 ; 𝜏 𝑗𝑘 = −𝑈 𝑗𝑘 Landau and Lifshitz 5 FLQCD 2019 @ YITP (2019/04/18)

Energy momentum tensor (EMT) Momentum density Energy density 𝑈 02 𝑈 03 𝑈 00 𝑈 01 𝑈 𝑈 𝑈 𝑈 12 13 10 11 𝑈 𝜈𝜉 = 𝑈 20 𝑈 21 𝑈 22 𝑈 23 Pressure 𝑈 30 𝑈 31 𝑈 32 𝑈 33 Stress tensor  Stress is force per unit area 𝑔 𝑗 = 𝜏 𝑗𝑘 𝑜 𝑘 ; 𝜏 𝑗𝑘 = −𝑈 𝑗𝑘 Landau and Lifshitz rubber 5 FLQCD 2019 @ YITP (2019/04/18)

Maxwell stress 𝑘 − 𝜀 𝑗𝑘 + 1 𝑘 − 𝜀 𝑗𝑘 2 𝐹 2 2 𝐶 2 𝑈 𝑗𝑘 = 𝜗 0 𝐹 𝑗 𝐹 𝐶 𝑗 𝐶 𝜈 0 𝐹 Ԧ 𝑔 Ԧ 𝑔  Perpendicular plane: 𝜇 𝑙 < 0  Parallel plane: 𝜇 𝑙 > 0  Stress tensor (𝑙) = 𝜇 𝑙 𝑜 𝑗 (𝑙) 𝑈 𝑗𝑘 𝑜 𝑘 Length of arrows = 𝜇 𝑙 (𝑗, 𝑘 = 1,2,3 ; 𝑙 = 1,2,3) 6 FLQCD 2019 @ YITP (2019/04/18)

Measurement on the lattice To do ① Prepare 𝑅 ത ② Measure EMT around 𝑅 ത 𝑅 on the lattice 𝑅 7 FLQCD 2019 @ YITP (2019/04/18)

Measurement on the lattice To do ① Prepare 𝑅 ത ② Measure EMT around 𝑅 ത 𝑅 on the lattice 𝑅 Confinement potential Wilson Loop ⟨𝑋 𝑆, 𝑈 ⟩ = 𝐷 0 exp −𝑊 𝑆 𝑈 + 𝐷 1 exp −𝑊 1 𝑆 𝑈 + ⋯ 1 𝑊 𝑆 = − lim 𝑈 log⟨𝑋 𝑆, 𝑈 ⟩  quenched SU(3) Yang-Mills 𝑈→∞  𝛾 = 6.600 ( 𝑏 = 0.038 fm ) Ground state potential 7 FLQCD 2019 @ YITP (2019/04/18)

Measurement on the lattice To do ① Prepare 𝑅 ത ② Measure EMT around 𝑅 ത 𝑅 on the lattice 𝑅 Iritani et al . (2018) Gradient flow Flow eq. L ሷ uscher (2010) 𝜖𝐶 𝜈 𝑢, 𝑦 𝜀𝑇[𝐶] 2 = −𝑕 0 𝜖𝑢 𝜀𝐶 𝜈 (𝑢, 𝑦) 𝐶 𝜈 : smeared field EMT defined via gradient flow Suzuki (2013) Entropy density vs. temperature 1 𝜀 𝜈𝜉 𝑈 𝜈𝜉 𝑢, 𝑦 = 𝛽 𝑉 𝑢 𝑉 𝜈𝜉 𝑢, 𝑦 + 4𝛽 𝐹 (𝑢) 𝐹 𝑢, 𝑦 − 𝐹 𝑢, 𝑦 + 𝑃(𝑢)  2-loop coefficient is now available ! Harlander et al . (2018) 7 FLQCD 2019 @ YITP (2019/04/18)

Set up  Quenched SU(3) Yang-Mills gauge theory  Wilson gauge action  Clover operator 0.92 fm 0.69 fm  Continuum limit 0.46 fm  APE smearing for spatial links  Multihit improvement in temporal links  Simulation using BlueGene/Q @ KEK 𝜸 Lattice spacing Lattice size # of statistics 48 4 0.057 fm 6.304 140 48 4 0.046 fm 6.465 440 48 4 6.513 0.043 fm 600 48 4 6.600 0.038 fm 1500 64 4 0.029 fm 6.819 1000 8 FLQCD 2019 @ YITP (2019/04/18)

A lattice study of stress distribution around 𝑅 ത 𝑅 in vacuum Stress distribution in terms of local interaction FlowQCD, PLB 789 (2019) 210. 𝑦 mid-plane 𝑧 𝑨 𝑃 𝑃 ത 𝑅 𝑅 𝑆 𝑧𝑨 -plane 9 FLQCD 2019 @ YITP (2019/04/18)

𝑦 𝑧 Stress distribution around 𝑅 ത 𝑅 mid 𝑨 FlowQCD, PLB 789 (2019) 210. 𝑆 𝑧𝑨  𝑏 = 0.029 fm  𝑢/𝑏 2 = 2.0  𝑆 = 0.69 fm  Length of arrows = 𝜇 𝑙  Gauge invariant  Local interaction  squeezed 10 FLQCD 2019 @ YITP (2019/04/18)

Stress distribution around 𝑅 ത 𝑅 : Cylindrical coordinates 𝑈 44 𝑃 𝑈 𝑓 𝑠 𝑨𝑨 𝑈 𝜈𝜉 = 𝑈 𝑃 𝑠𝑠 𝑓 𝑨 𝑈 𝜄𝜄 𝑓 𝜄 Diagonalized EMT ത (Cylindrical / Parity symmetry) 𝑅 𝑅 Degeneracy (Maxwell Theory) 𝑈 44 = 𝑈 𝑨𝑨 = 𝑈 𝑠𝑠 = |𝑈 𝜄𝜄 | 11 FLQCD 2019 @ YITP (2019/04/18)

Stress distribution around 𝑅 ത 𝑦 𝑧 𝑅 mid 𝑨 𝑓 𝑠 Properties in non-Abelian theory 𝑆 𝑧𝑨 𝑓 𝑨  𝑈 44 ≈ 𝑈 𝑨𝑨 , 𝑈 𝑠𝑠 ≈ 𝑈 𝜄𝜄 (Degeneracy) 𝑓 𝜄  𝑈 44 ≠ 𝑈 𝑠𝑠 (Separation)  σ 𝜈 𝑈 𝜈𝜈 ≠ 0 (Trace anomaly ≠ 0 ) FlowQCD, PLB 789 (2019) 210. ( Note : after double limit ) 12 FLQCD 2019 @ YITP (2019/04/18)

EMT and confinement potential confinement potential From EMT 𝑊 𝑆 = 𝑏 + 𝑐𝑆 + Τ 𝑑 𝑆 ここに数式を入力します。 𝐺pot ≔ − 𝑒𝑊(𝑆) 𝑅 𝑒 2 𝑦 𝜍 ≔ −𝐺stress ≔ න 𝑈 𝑨𝑨 𝑅 ത 𝑒𝑆 mid 13 FLQCD 2019 @ YITP (2019/04/18)

EMT and confinement potential confinement potential From EMT Good agreement !! 𝑊 𝑆 = 𝑏 + 𝑐𝑆 + Τ 𝑑 𝑆 ここに数式を入力します。 𝐺pot ≔ − 𝑒𝑊(𝑆) 𝑅 𝑒 2 𝑦 𝜍 ≔ −𝐺stress ≔ න 𝑈 𝑨𝑨 𝑅 ത 𝑒𝑆 mid 13 FLQCD 2019 @ YITP (2019/04/18)

Toward analysis at nonzero temperature Stress distribution around 𝑅 ത 𝑅 / 𝑅 at nonzero temperature ？ 0 Critical temperature 𝑈 𝑈 𝑑 14 FLQCD 2019 @ YITP (2019/04/18)

Measurement on the lattice To do ① Prepare 𝑅 ത ② Measure EMT around 𝑅/ ത 𝑅 on the lattice 𝑅 15 FLQCD 2019 @ YITP (2019/04/18)

Measurement on the lattice To do ① Prepare 𝑅 ത ② Measure EMT around 𝑅/ ത 𝑅 on the lattice 𝑅 Free energy Polyakov Loop ത 𝑅 𝑅 𝑦 4 𝑦 Ԧ 𝑓 −𝐺 𝑆 /𝑈 = 1 3 Tr Ω † Ԧ 𝑦 Ω ( Ԧ 𝑧) Color singlet free energy  quenched SU(3) Yang-Mills (We use Coulomb gauge fixing)  𝛾 = 6.600 ( 𝑏 = 0.038 fm ) 15 FLQCD 2019 @ YITP (2019/04/18)

Measurement on the lattice To do ① Prepare 𝑅 ത ② Measure EMT around 𝑅/ ത 𝑅 on the lattice 𝑅 Iritani et al . (2018) Gradient flow Flow eq. L ሷ uscher (2010) 𝜖𝐶 𝜈 𝑢, 𝑦 𝜀𝑇[𝐶] 2 = −𝑕 0 𝜖𝑢 𝜀𝐶 𝜈 (𝑢, 𝑦) 𝐶 𝜈 : smeared field EMT defined via gradient flow Suzuki (2013) Entropy density vs. temperature 1 𝜀 𝜈𝜉 𝑈 𝜈𝜉 𝑢, 𝑦 = 𝛽 𝑉 𝑢 𝑉 𝜈𝜉 𝑢, 𝑦 + 4𝛽 𝐹 (𝑢) 𝐹 𝑢, 𝑦 − 𝐹 𝑢, 𝑦 + 𝑃(𝑢)  2-loop coefficient is now available ! Harlander et al . (2018) 15 FLQCD 2019 @ YITP (2019/04/18)

Set up (quark — anti-quark, single quark)  Quenched SU(3) Yang-Mills gauge theory 𝑅 ത 𝑅  Wilson gauge action  Clover operator  Fixed 𝑏, 𝑢 0.69 fm 0.46 fm  Multihit improvement in temporal links  Simulation using OCTOPUS, Reedbush Lattice Temporal # of 𝜸 Spatial size 𝑼/𝑼 𝒅 spacing size statistics 48 3 0.038 fm 6.600 12 1.44 640 16 FLQCD 2019 @ YITP (2019/04/18)

Maxwell stress (revisit) 𝑘 − 𝜀 𝑗𝑘 + 1 𝑘 − 𝜀 𝑗𝑘 2 𝐹 2 2 𝐶 2 𝑈 𝑗𝑘 = 𝜗 0 𝐹 𝑗 𝐹 𝐶 𝑗 𝐶 𝜈 0 𝐹 Ԧ 𝑔 Ԧ 𝑔  Perpendicular plane: 𝜇 𝑙 < 0  Parallel plane: 𝜇 𝑙 > 0  Stress tensor (𝑙) = 𝜇 𝑙 𝑜 𝑗 (𝑙) 𝑈 𝑗𝑘 𝑜 𝑘 Length of arrows = 𝜇 𝑙 (𝑗, 𝑘 = 1,2,3 ; 𝑙 = 1,2,3) 6 ’ FLQCD 2019 @ YITP (2019/04/18)

𝑦 𝑧 Stress distribution around 𝑅 ത 𝑅 mid 𝑨 𝑆 𝑧𝑨 Preliminary  singlet  𝑏 = 0.038 fm (fixed)  𝑢/𝑏 2 = 2.0 (fixed)  𝑆 = 0.69 fm  Length of arrows = 𝜇 𝑙 17 FLQCD 2019 @ YITP (2019/04/18)

𝑦 𝑧 Stress distribution around 𝑅 ത 𝑅 mid 𝑨 𝑆 𝑧𝑨 Preliminary Critical temperature 𝑈 𝑈 0 𝑑 18 FLQCD 2019 @ YITP (2019/04/18)