QCD Special thanks to Flow QCD Collab Asakawa

勾配流を用いた格子QCD数値解析北沢正清（大阪大学） Special thanks to: Flow. QCD Collab. (柳原良亮, Asakawa, Hatsuda, Iritani, Itou, Suzuki) WHOT Collab. (Kanaya, Ejiri, Taniguchi, Umeda, Shirogane, Suzuki, … ) Hadron Spectroscopy Café, 東大, 2017年 7月10日

格子QCD数値解析量子色力学（QCD）を第一原理的に取り扱う現状唯一の手段 hadron spectra, chiral symmetry, phase transition, etc.

energy momentum Poincare symmetry stress Einstein Equation pressure Hydrodynamic Eq.

: nontrivial observable on the lattice Definition of the operator is nontrivial ①because of the explicit breaking of Lorentz symmetry ex: Its measurement is extremely noisy ②due to high dimensionality and etc.

If we have

Thermodynamics direct measurement of expectation values If we have

Thermodynamics direct measurement of expectation values Fluctuations and Correlations viscosity, specific heat, . . . If we have

Thermodynamics direct measurement of expectation values Fluctuations and Correlations viscosity, specific heat, . . . If we have Ø confinement string Ø EM distribution in hadrons Hadron Structure

Fluctuations and Correlations Thermodynamics direct measurement of expectation values viscosity, specific heat, . . . If we have Ø confinement string Ø EM distribution in hadrons Hadron Structure Ø vacuum configuration Ø mixed state on 1 st transition Vacuum Structure

Gaussian ① ②

Gaussian ① ② YM Gradient Flow Gauge invariant version of 4 -dim. diffusion equation

呼称について • • • gradient flow Yang-Mills gradient flow Wilson flow Symanzik, Iwasaki flow, … 勾配流（変換）？

勾配流を用いたスケール設定 Luscher, 2010 : 適当な無次元演算子の期待値格子間隔には依存しない : an observable p standard choice of O: p perturbative formula:

A Dimensionless Choice: t 2<E> one-loop perturbation violated lattice discretization effect Another choice: w 0 Budapest-Wuppertal 2012 weaker a dep.

スケール設定 by Flow. QCD Ø SU(3) YM theory Ø Wilson gauge action Ø w 0 scaling Ø 6. 3<b<7. 4 Ø 少ない配位で高統計 b size Nconf 6. 3 644 30 6. 4 644 6. 5 b size Nconf 6. 9 644 30 100 7. 0 964 60 644 49 7. 2 964 53 6. 6 644 100 7. 4 1284 40 6. 7 644 30 6. 8 644 100 Flow. QCD, PRD 94 (2016); see also 1503. 06516

微少フロー時間展開 original 4 -dim theory an operator at t>0 t 0 limit Luescher, Weisz, 2011 remormalized operators of original theory

EMTの構築 Suzuki, 2013 Del. Debbio, Patella, Rago, 2013 p gauge-invariant dimension 4 operators

EMTの構築 2 Suzuki coeffs. Suzuki, 2013

EMTの構築 2 Suzuki coeffs. Remormalized EMT Suzuki, 2013

Rough Idea coarse graining no translational invariance translational symmetry is recovered!

勾配流によるEMT測定 lattice regularized gauge theory gradient flow measurement on the lattice continuum theory (with dim. reg. ) analytic (perturbative) gradient flow continuum theory (with dim. reg. )

QCD Eo. S (Energy Density, Pressure) BNL-Bielefeld 2011 hadronic QGP • Rapid increase of e/T 4 around T=150 -200 Me. V • Crossover transition • Low T: hadron resonance gas model / High T: perturbative QCD

格子間隔を変化させてみる Changing lattice spacing 1/T and V change e-3 pが求まる

積分法 Boyd+ 1996 Ø measurements of e-3 p for many T Ø vacuum subtraction for each T Ø information on beta function

Numerical Simulation Ø SU(3) YM theory Ø Wilson gauge action Ø Parameters: • Nt = 12, 16, 20 -24 • aspect ratio 5. 3<Ns/Nt<8 • 1500~2000 configurations Ø Scale from gradient flow Flow. QCD 1503. 06516

Gradient Flow Method lattice regularized gauge theory gradient flow measurement on the lattice continuum theory (with dim. reg. ) analytic (perturbative) gradient flow continuum theory (with dim. reg. )

Caveats Gauge field has to be sufficiently smeared! lattice regularized gauge theory Perturbative relation gradient flow has to be applicable! measurement on the lattice continuum theory (with dim. reg. ) analytic (perturbative) gradient flow continuum theory (with dim. reg. )

in continuum non-perturbative region O(t) effect classical

in continuum non-perturbative region O(t) effect classical on the lattice p Double extrapolation t 0, a 0 required p Extrapolation has to be taken keeping t>>a 2

t Dependence clover+plaq clover : strong discretization effect : oversmeared : Linear t dependence

Double Extrapolation Continuum extrapolation strong discretization effect Note: Flow. QCD, 2014: continuum extrapolation only WHOT-QCD, 2016: small t limit only

Double Extrapolation Continuum extrapolation strong discretization effect Note: Small t extrapolation Flow. QCD, 2014: continuum extrapolation only WHOT-QCD, 2016: small t limit only

Double Extrapolation Black band: continuum extrapolated p range of t for fitting:

T Dependence Error includes Ø statistical error Ø choice of t range for t 0 limit Ø uncertainty in a. LMS total error <1. 5% for T>1. 1 Tc Flow. QCD, PRD, 2016 p Excellent agreement with integral method p High accuracy only with ~2000 confs.

Nf=2+1 QCD 熱力学の解析 Taniguchi+, WHOT-QCD, PRD, 2017 • Nf=2+1 QCD, Iwasaki gauge + NP-clover • m. PS/m. V ≈0. 63 with ≈physical s quark • T=0: CP-PACS+JLQCD (ß=2. 05, 283 x 56, a≈0. 07 fm) • T>0: 323 x. Nt, Nt = 4, 6, . . . , 14, 16): • T≈174 -697 Me. V • t 0 extrapolation only (No continuum limit)

物理点シミュレーション • • Nf=2+1 QCD, Iwasaki gauge + NP-clover T=0: PACS-CS (ß=1. 9, 323 x 64, a≈0. 09 fm) Fine-tuned to the phys. pt. by reweighting. T>0: 323 x. Nt, Nt = 4, 5, . . . , 14, T≈157 --549 Me. V Budapest-Wuppertal 2010 WHOT-QCD, Preliminary

なぜEMT相関関数？ p Kubo Formula: T 12 correlator shear viscosity Ø Hydrodynamics describes long range behavior of Tmn p Energy fluctuation specific heat

EMT Correlator : Noisy… With naïve EMT operators Nakamura, Sakai, PRL, 2005 Nt=8 improved action ~106 configurations Nt=16 standard action 5 x 104 configurations … no signal

Linear Response Relations Specific heat entropy density Giusti, Meyer, 2011 enthalpy density Minami, Hidaka, 2012 導出

格子解析 • • Flow. QCD, to appear soon SU(3) ゲージ理論 Wilson作用／クローバー演算子アスペクト比：Ns/Nt=4 統計数： 180, 000 β 483 x 12 643 x 16 963 x 24 T=1. 66 Tc 6. 719 6. 941 7. 265 T=2. 22 Tc 6. 943 7. 170 7. 500 数値解析：Bluegene/Q @KEK

Nf=2+1 QCD WHOT-QCD, Preliminary Slide from Y. Taniguchi, LATTICE 2017/XQCD 2017

ｑｑ系の応力分布 R=0. 684 fm beta=6. 6 R/a=18 APE: 180 (alpha=2. 3) MF temporal link t/a 2=1. 5 (no t 0 limit)