### Lattice Study of

**Energy-Momentum Tensor **

### with Gradient Flow:

### Thermodynamics, Correlations, and Stress

### Masakiyo Kitazawa (Osaka U.)

### In Collaboration with:

FlowQCD Collab.(R. Yanagihara, T. Iritani, Asakawa, Hatsuda, Suzuki) WHOT Collab.(Taniguchi, Kanaya, Ejiri, Suzuki, Umeda, Shirogane, … )

Workshop of Recent Developments in QCD and QFT, Nov. 10, 2017, Taipei

### Energy-Momentum Tensor

### One of the most fundamental quantities in physics

stress

energy momentum

### Energy-Momentum Tensor

### One of the most fundamental quantities in physics

### : nontrivial observable on the lattice

### Definition of the operator is nontrivial

because of the explicit breaking of Lorentz symmetry

## ①

## ② Its measurement is extremely noisy

due to high dimensionality and etc.

ex:

stress

energy momentum

**Thermodynamics**

direct measurement of expectation values

**Fluctuations and** **Correlations**

viscosity, specific heat, ...

**Hadron Structure**

• flux tube / hadrons

• stress distribution

### Contents

### 1. Constructing EMT on the lattice 2. Thermodynamics

### 3. Correlation Function

### 4. Stress distribution in qq system 5. Summary

ー

### (Yang-Mills) Gradient Flow

t: “flow time”

dim:[length^{2}] leading

• diffusion equation in 4-dim space

• diffusion distance

• “continuous” cooling/smearing

Luscher 2010

Narayanan, Neuberger, 2006 Luscher, Weiss, 2011

### (Yang-Mills) Gradient Flow

t: “flow time”

dim:[length^{2}] leading

• diffusion equation in 4-dim space

• diffusion distance

• “continuous” cooling/smearing

Luscher 2010

Narayanan, Neuberger, 2006 Luscher, Weiss, 2011

### (YM) Gradient Flow: Properties

• All (composite) operators are finite at t>0

• Quite effective in reducing statistical error.

• Flowed field ≠ original field

• Gradient flow is not an approximation method.

• Applications

• scale setting

• topological charge/susceptibility

• Structure Func. / PDF

All operators are renormalized at t>0 Safe a0 limit

Luscher,Weisz,2011

### 孔子

### confucius

### （論語、子路 13）

## Miss the wood for the trees

### 小利を見ればすなわち大事成らず

見小利則大事不成

### 孔子

### confucius

### （論語、子路 13）

### UV fluctuations Physics

## Miss the wood for the trees

### 小利を見ればすなわち大事成らず

見小利則大事不成

### Gradient Flow for Fermions

Luscher, 2013

Makino, Suzuki, 2014 Taniguchi+ (WHOT) 2016

Not “gradient” flow but a “diffusion” equation.

Flow for gauge field is independent of fermion field.

Divergence in field renormalization of fermions.

All observables becomes finite once Z(t) is determined.

### Small Flow-time Expansion

Luescher, Weisz, 2011 Suzuki, 2013

remormalized operators of original theory

an operator at t>0

t0 limit

original 4-dim theory

### Constructing EMT

Suzuki, 2013DelDebbio,Patella,Rago,2013

** gauge-invariant dimension 4 operators**

### Constructing EMT 2

Suzuki coeffs.

Suzuki, 2013

### Constructing EMT 2

Suzuki coeffs.

**Remormalized EMT**

Suzuki, 2013

### EMT with Fermions

Makino, Suzuki, 2014

### Contents

### 1. Constructing EMT on the lattice 2. Thermodynamics

### 3. Correlation Function

### 4. Stress distribution in qq system 5. Summary

ー

**Thermodynamics**

direct measurement of expectation values

### QCD EoS

### (Energy Density, Pressure)

• Rapid increase of e/T^{4} around T=150-200 MeV

• Crossover transition

• Low T: hadron resonance gas model / High T: perturbative QCD BNL-Bielefeld

2011

### hadronic

### QGP

### QCD Thermodynamics

### Statistical Mechanics

Changing lattice spacing 1/T and V change

### e-3p

### Integral Method

measurements of e-3p for many T

vacuum subtraction for each T

information on beta function

Boyd+ 1996

### Numerical Simulation

* Expectation values of T*

_{mn}

### SU(3) YM theory

### Wilson gauge action

### Parameters:

### Scale from gradient flow

• N_{t} = 12, 16, 20-24

• aspect ratio 5.3<N_{s}/N_{t}<8

• 1500~2000 configurations

FlowQCD 1503.06516

MK+ (FlowQCD),

**PRD94, 114512 (2016)**

### t Dependence

: strong discretization effect : over smeared

### : Linear t dependence

clover+plaq clover

### Double Extrapolation

Continuum extrapolation

FlowQCD, 2014: continuum extrapolation only WHOT-QCD, 2016: small t limit only

Note:

strong

discretization effect

### Double Extrapolation

Continuum extrapolation

Small t extrapolation

FlowQCD, 2014: continuum extrapolation only WHOT-QCD, 2016: small t limit only

Note:

strong

discretization effect

### Double Extrapolation

Fitting ranges:

range-1:

range-2:

range-3:

Black line: continuum extrapolated

Range1 Range2

Range3

Systematic error from the choice of fitting range

### ≈

statistical error### T Dependence

Error includes

statistical error

choice of t range for t0 limit

uncertainty in aL_{MS}

total error <1.5% for T>1.1T_{c}

### Excellent agreement with integral method

### High accuracy only with

### ~2000 confs.

FlowQCD, PRD, 2016

### N _{f} =2+1 QCD Thermodynamics

• N_{f}=2+1 QCD, Iwasaki gauge + NP-clover

• m_{PS}/m_{V} ≈0.63 / almost physical s quark mass

• T=0: CP-PACS+JLQCD (ß=2.05, 28^{3}x56, a≈0.07fm)

• T>0: 32^{3}xN_{t}, N_{t} = 4, 6, ... , 14, 16):

• T≈174-697MeV

• t0 extrapolation only (No continuum limit)

Taniguchi+ (WHOT-QCD),
**PRD96, 014509 (2017)**

### Fermion Propagator

t=0 gauge field

fermion field

• propagator of flow equation

• Inverse propagator is needed

### t0 Extrapolation

”linear window” for Nt>6

Checked: fit range, a^{2}/t term

Taniguchi+ (WHOT-QCD),
**PRD96, 014509 (2017)**

### N _{f} =2+1 Thermodynamics

Taniguchi+ (WHOT-QCD),
**PRD96, 014509 (2017)**

Agreement with integral method except for N_{t}=4, 6

No stable extrapolation for N_{t}=4, 6

Suppression of statistical error

Physical mass: Kanaya+ (WHOT-QCD), 1710.10015

no linear window

### Chiral Condensate / Suceptibility

Chiral condensate decreases for T>Tc.

Chiral susceptibility has a sharp peak around T=Tc.

Subtracted condensate

Chiral susceptibility

Taniguchi+ (WHOT-QCD),
**PRD96, 014509 (2017)**

### Contents

### 1. Constructing EMT on the lattice 2. Thermodynamics

### 3. Correlation Function

### 4. Stress distribution in qq system 5. Summary

ー

**Fluctuations and** **Correlations**

viscosity, specific heat, ...

### Why EMT Correlation Func.?

### Kubo Formula: T

_{12}

### correlator shear viscosity

### Energy fluctuation specific heat

Hydrodynamics describes long range behavior of T_{mn}

### EMT Correlator: Extremely Noisy…

Nakamura, Sakai, PRL,2005

N_{t}=8

improved action

~10^{6} configurations … no signal

Nt=16

standard action

5x10^{4} configurations

### With naïve EMT operators

### Conservation Law

### t independent constant

### Linear Response Relations

### Specific heat

### entropy density

Giusti, Meyer, 2011

### enthalpy density

### Derivation

Minami, Hidaka, 2012

### Numerical Simulation

• SU(3) pure gauge

• Wilson gauge action / clover operator

• Ns/Nt=4

• Statistics: 18-20x10^{4}

**β** **T=1.66T**_{c}**T=2.22T**** _{c}**
48

^{3}x12 6.719 6.943 64

^{3}x16 6.941 7.170 96

^{3}x24 7.265 7.500

on Bluegene/Q @KEK FlowQCD, arXiv:1708.01415

### Euclidean Correlator @T=2.24Tc

t-independent plateau in all channels conservation law

small t region: artificial enhancement due to overlap of operators

linear response relations for 4411, 4141 channels

FlowQCD, arXiv:1708.01415

### Mid-Point Correlator @T=2.24Tc

• (44;11), (41;41) channels : confirmation of LRR

• (44;44) channel: new measurement of c_{V}

2+1 QCD:

Taniguchi+ (WHOT-QCD), 1711.02262

### Contents

### 1. Constructing EMT on the lattice 2. Thermodynamics

### 3. Correlation Function

### 4. Stress distribution in qq system 5. Summary

ー

**Hadron Structure**

• flux tube / hadrons

• stress distribution

### Stress

### Pressure

force on a surface per unit area

### Stress

### Pressure

force on a surface per unit area

### Stress

Generally, F and n are not parallel

Stress Tensor

*v*

*F*

### Maxwell Stress

### E

Parallel to field: Attractive Vertical to field: Repulsive

### 𝑇 = 1 2

### 𝐸

^{2}

### 0 0

### 0 −𝐸

^{2}

### 0

### 0 0 −𝐸

^{2}

### Maxwell Stress

### attractive eigenvector

(≈Line of electric field)

### E

Parallel to field: Attractive Vertical to field: Repulsive

Coulomb force

**Previous Studies**

• action density

• color electric field

• (color electric field)^{2}

### q-qbar System in YM Theory

### ＝Flux Tube Formation

### Color electric field is

### confined into a flux tube

### Linear potential

**This Study: stress**

• gauge invariant!

• definite physical meaning!

• establish action thr. medium

### Stress Distribution in qq System

beta=6.819 (a=0.029fm) R/a=16

t0 limit

No continuum limit

### attractive eigenvector

R. Yanagihara+ (FlowQCD) to appear soon!

### First visualization of distortion in space due to color charges

Preliminary

### Stress Distribution in qq System

R. Yanagihara+ (FlowQCD) to appear soon!

### Yang-Mills SU(3) Maxwell

### Stress Distribution in qq System

R. Yanagihara+ (FlowQCD) to appear soon!

### Yang-Mills SU(3) Maxwell

New insights into physics of confinement

### qq force?

**Don’t miss**

**Yanagihara’s talk **

**this afternoon**

### Summary

• EMT operator on the lattice is now available!

• Correctly renormalized operator

• Statistical error is suppressed thanks to gradient flow.

• The operator is applied to various analyses:

**Many** **future studies**

• transport coefficient

• EM/stress distribution in hadrons

• Flux tube: T dependence

### EMT (Naïve Constructuion)

Z_{1}, Z_{2}, Z_{3} have to be determined non-perturbatively.

Accurate determinations of Z_{1}, Z_{3}: Giusti, Pepe, 2014-; BW, 2016

So far, only for pure gauge theory

multi-level algorithm

### Gradient Flow Method

**lattice regularized**
**gauge theory**

gradient flow

continuum theory

(with dim. reg.)

continuum theory

(with dim. reg.) gradient flow

analytic (perturbative)

**measurement **
**on the lattice**

**lattice regularized**
**gauge theory**

gradient flow

continuum theory

(with dim. reg.)

continuum theory

(with dim. reg.) gradient flow

### Caveats

Gauge field has to be sufficiently smeared!**measurement **
**on the lattice**

analytic (perturbative) Perturbative relation

has to be applicable!

**lattice regularized**
**gauge theory**

gradient flow

continuum theory

(with dim. reg.)

continuum theory

(with dim. reg.) gradient flow

### Caveats

Gauge field has to be sufficiently smeared!analytic (perturbative) Perturbative relation

has to be applicable!

**measurement **
**on the lattice**