Energy-Momentum Tensor

(1)

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

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Energy-Momentum Tensor

One of the most fundamental quantities in physics

(3)

stress

energy momentum

Energy-Momentum Tensor

One of the most fundamental quantities in physics

(4)

: 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:

(5)

stress

energy momentum

Thermodynamics

direct measurement of expectation values

Fluctuations and Correlations

viscosity, specific heat, ...

Hadron Structure

• flux tube / hadrons

• stress distribution

(6)

1. Constructing EMT on the lattice 2. Thermodynamics

3. Correlation Function

4. Stress distribution in qq system 5. Summary

ー

(7)

(Yang-Mills) Gradient Flow

t: “flow time”

dim:[length²] leading

• diffusion equation in 4-dim space

• diffusion distance

• “continuous” cooling/smearing

Luscher 2010

Narayanan, Neuberger, 2006 Luscher, Weiss, 2011

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(Yang-Mills) Gradient Flow

t: “flow time”

dim:[length²] leading

• diffusion equation in 4-dim space

• diffusion distance

• “continuous” cooling/smearing

Luscher 2010

Narayanan, Neuberger, 2006 Luscher, Weiss, 2011

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(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 a0 limit

Luscher,Weisz,2011

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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

(11)

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

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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.

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Small Flow-time Expansion

Luescher, Weisz, 2011 Suzuki, 2013

remormalized operators of original theory

an operator at t>0

t0 limit

original 4-dim theory

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Constructing EMT

Suzuki, 2013

DelDebbio,Patella,Rago,2013

 gauge-invariant dimension 4 operators

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Constructing EMT 2

Suzuki coeffs.

Suzuki, 2013

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Constructing EMT 2

Suzuki coeffs.

Remormalized EMT

Suzuki, 2013

(17)

EMT with Fermions

Makino, Suzuki, 2014

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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

(19)

QCD EoS

(Energy Density, Pressure)

• Rapid increase of e/T⁴ around T=150-200 MeV

• Crossover transition

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

2011

hadronic

QGP

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QCD Thermodynamics

Statistical Mechanics

Changing lattice spacing 1/T and V change

e-3p

(21)

Integral Method

 measurements of e-3p for many T

 vacuum subtraction for each T

 information on beta function

Boyd+ 1996

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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)

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t Dependence

: strong discretization effect : over smeared

: Linear t dependence

clover+plaq clover

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Double Extrapolation

Continuum extrapolation

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

Note:

strong

discretization effect

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Double Extrapolation

Continuum extrapolation

Small t extrapolation

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

Note:

strong

discretization effect

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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

(27)

T Dependence

Error includes

 statistical error

 choice of t range for t0 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

(28)

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, 28³x56, a≈0.07fm)

• T>0: 32³xN_t, N_t = 4, 6, ... , 14, 16):

• T≈174-697MeV

• t0 extrapolation only (No continuum limit)

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

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Fermion Propagator

t=0 gauge field

fermion field

• propagator of flow equation

• Inverse propagator is needed

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t0 Extrapolation

 ”linear window” for Nt>6

 Checked: fit range, a²/t term

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N _f =2+1 Thermodynamics

 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

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Chiral Condensate / Suceptibility

 Chiral condensate decreases for T>Tc.

 Chiral susceptibility has a sharp peak around T=Tc.

Subtracted condensate

Chiral susceptibility

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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, ...

(34)

Why EMT Correlation Func.?

Kubo Formula: T

₁₂

correlator shear viscosity

Energy fluctuation specific heat

 Hydrodynamics describes long range behavior of T_mn

(35)

EMT Correlator: Extremely Noisy…

Nakamura, Sakai, PRL,2005

N_t=8

improved action

~10⁶ configurations … no signal

Nt=16

standard action

5x10⁴ configurations

With naïve EMT operators

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Conservation Law

t independent constant

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Linear Response Relations

Specific heat

entropy density

Giusti, Meyer, 2011

enthalpy density

Derivation

Minami, Hidaka, 2012

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Numerical Simulation

• SU(3) pure gauge

• Wilson gauge action / clover operator

• Ns/Nt=4

• Statistics: 18-20x10⁴

β T=1.66T_c T=2.22T_c 48³x12 6.719 6.943 64³x16 6.941 7.170 96³x24 7.265 7.500

on Bluegene/Q @KEK FlowQCD, arXiv:1708.01415

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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

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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

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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

(42)

Stress

Pressure

force on a surface per unit area

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Stress

Pressure

force on a surface per unit area

Stress

Generally, F and n are not parallel

Stress Tensor

v

F

(44)

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

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Maxwell Stress

attractive eigenvector

(≈Line of electric field)

E

Parallel to field: Attractive Vertical to field: Repulsive

Coulomb force

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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

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Stress Distribution in qq System

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

t0 limit

No continuum limit

attractive eigenvector

R. Yanagihara+ (FlowQCD) to appear soon!

First visualization of distortion in space due to color charges

Preliminary

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Stress Distribution in qq System

Yang-Mills SU(3) Maxwell

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Stress Distribution in qq System

Yang-Mills SU(3) Maxwell

New insights into physics of confinement

qq force?

Don’t miss

Yanagihara’s talk

this afternoon

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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

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EMT (Naïve Constructuion)

 Z₁, Z₂, Z₃ have to be determined non-perturbatively.

 Accurate determinations of Z₁, Z₃: Giusti, Pepe, 2014-; BW, 2016

 So far, only for pure gauge theory

 multi-level algorithm

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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

(53)

gradient flow

continuum theory

(with dim. reg.)

continuum theory

Caveats

Gauge field has to be sufficiently smeared!

analytic (perturbative) Perturbative relation

has to be applicable!

(54)

gradient flow

continuum theory

(with dim. reg.)

continuum theory

Caveats

Gauge field has to be sufficiently smeared!

analytic (perturbative) Perturbative relation

has to be applicable!

(55)

Energy-Momentum Tensor

Lattice Study of

Energy-Momentum Tensor

with Gradient Flow:

Thermodynamics, Correlations, and Stress

Masakiyo Kitazawa (Osaka U.)

In Collaboration with:

Energy-Momentum Tensor

One of the most fundamental quantities in physics

Energy-Momentum Tensor

One of the most fundamental quantities in physics

: nontrivial observable on the lattice

Definition of the operator is nontrivial

①

② Its measurement is extremely noisy

Thermodynamics

Fluctuations and Correlations

Hadron Structure

Contents

1. Constructing EMT on the lattice 2. Thermodynamics

3. Correlation Function

4. Stress distribution in qq system 5. Summary

(Yang-Mills) Gradient Flow

(Yang-Mills) Gradient Flow

(YM) Gradient Flow: Properties

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

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

Gradient Flow for Fermions

Small Flow-time Expansion

Constructing EMT

 gauge-invariant dimension 4 operators

Constructing EMT 2

Constructing EMT 2

Remormalized EMT

EMT with Fermions

Contents

1. Constructing EMT on the lattice 2. Thermodynamics

3. Correlation Function

4. Stress distribution in qq system 5. Summary

Thermodynamics

QCD EoS

(Energy Density, Pressure)

hadronic

QGP

QCD Thermodynamics

Statistical Mechanics

e-3p

Integral Method

Numerical Simulation

 Expectation values of T

 SU(3) YM theory

 Wilson gauge action

 Parameters:

 Scale from gradient flow

t Dependence

: Linear t dependence

Double Extrapolation

Double Extrapolation

Double Extrapolation

≈

T Dependence

Excellent agreement with integral method

High accuracy only with

~2000 confs.

N f =2+1 QCD Thermodynamics

Fermion Propagator

t0 Extrapolation

N f =2+1 Thermodynamics

Chiral Condensate / Suceptibility

Contents

1. Constructing EMT on the lattice 2. Thermodynamics

N _f =2+1 QCD Thermodynamics

N _f =2+1 Thermodynamics