KEK理論センター研究会 「熱場の量子論とその応用」 2020/Aug./26

MK, Mogliacci, Kolbe, Horowitz, Phys.Rev.D 99 (2019) 094507[arXiv:1904.00241[hep-lat]]attractive force between two conductive platesattractive force between two conductive platesattractive force between two conductive plates

Brown, Maclay
1969Brown, Maclay
1969Brown, Maclay
1969Brown, Maclay
1969

T

MK, Mogliacci, Kolbe,
Horowitz, PRD(2019)

Free scalar field L2=L3=∞
 Periodic BC
Mogliacci+, 1807.07871

MK, Mogliacci, Kolbe,Horowitz, PRD(2019)

Free scalar field L2=L3=∞ Periodic BC
Mogliacci+, 1807.07871

Lattice result
 Periodic BC
 Only t0 limit
 Error: stat.+sys.

Medium near Tc is remarkably insensitive to finite size!

Various Methods
 Integral, differential, moving frame, non-equilibrium, …
 rely on thermodynamic relations valid in V∞Not applicable toanisotropic systems

Various Methods
 Integral, differential, moving frame, non-equilibrium, …
 rely on thermodynamic relations valid in V∞Not applicable toanisotropic systems

We employ Gradient Flow MethodComponents of EMT are directly accessible!

Luscher 2010
Narayanan, Neuberger, 2006
Luscher, Weiss, 2011

t: “flow time”
dim:[length2]

leading diffusion equation in 4-dim space diffusion distance “continuous” cooling/smearing No UV divergence at t>0

Luescher, Weisz, 2011
Suzuki, 2013

original 4-dim theory

an operator at t>0t0 limit

remormalized operatorsof original theory

Suzuki, 2013vacuum subtr.

Remormalized EMT

Perturbative coefficient:Suzuki (2013); Makino, Suzuki (2014); Harlander+ (2018); Iritani, MK, Suzuki, Takaura (2019)

LO

1-loop 2-loop 3-loop

Suzuki, PTEP 2013, 083B03
Harlander+, 1808.09837
Iritani, MK, Suzuki, Takaura,PTEP 2019

×Iritani, MK, Suzuki,zeroTakaura, 2019
Suzuki (2013) Harlander+(2018)

 Choice of the scale of g2
Previous:
Improved:
Harlander+ (2018)

N2LO (2-loop)
Range2Range1
Range3

Iritani, MK, Suzuki, Takaura, PTEP 2019

 Stable t0 extrapolation with higher order coeff.
 Systematic error: fit range, m0 or md, uncertaintyof L (±3%)
 Extrapolation func: linear, higher order term in c1 (~g6)

Iritani, MK, Suzuki, Takaura, 2019□: 2-loop△: Flow2016
Systematic error: m0 or md, L, t0 function, fit range

More stable extrapolation with higher order c1 & c2
(pure gauge) SU(3) YM theory
 Wilson gauge action
 Nt = 16, 12
 Nz/Nt=6
 2000~4000 confs.
 Even Nx

 No Continuum extrap.
 Same Spatial volume• 12x722x12 ~ 16x962x16• 18x722x12 ~ 24x962x16

Simulations on

OCTOPUS/ReedbushFilled: Nt=16 / Open: Nt=12

Small-t extrapolation• Solid: Nt=16, Range-1• Dotted: Nt=16, Range-2,3• Dashed: Nt=12, Range-1

Stable small-t extrapolation
No Nt dependence within statistics for LxT=1, 1.5

Filled: Nt=16 / Open: Nt=12

Small-t extrapolation• Solid: Nt=16, Range-1• Dotted: Nt=16, Range-2,3• Dashed: Nt=12, Range-1

Stable small-t extrapolation
No Nt dependence within statistics for LxT=1, 1.5

MK, Mogliacci, Kolbe,
Horowitz, PRD(2019)

Free scalar field L2=L3=∞
 Periodic BCMogliacci+, 1807.07871

Lattice result
 Periodic BC
 Only t0 limit
 Error: stat.+sys.

Medium near Tc is remarkably insensitive to finite size!

MK, Mogliacci, Kolbe,Horowitz, PRD(2019)

Free scalar field L2=L3=∞
 Periodic BC
Mogliacci+, 1807.07871

Lattice result
 Periodic BC
 Only t0 limit
 Error: stat.+sys.

Medium near Tc is remarkably insensitive to finite size!

High-T limit: massless free gluons
How does the anisotropy approach this limit?

Difficulties
 Vacuum subtraction requires large-volume simulations.
 Lattice spacing not available  c1(t), c2(t) are not determined.

High-T limit: massless free gluons
How does the anisotropy approach this limit?

Difficulties
 Vacuum subtraction requires large-volume simulations.
 Lattice spacing not available  c1(t), c2(t) are not determined.

We study
No vacuum subtr.
nor Suzuki coeffs.
necessary!

TT/Tc≅8.1 (b=8.0) / T/Tc≅25 (b=9.0)

Ratio approaches the asymptotic value.
But, large deviation exists even at T/Tc~25.

First numerical simulation ofanisotropic pressure in SU(3) YM
with periodic BC.
Medium at 1.4<T/Tc<2.1 is remarkably
insensitive to the existence of boundary.

FutureAnti-periodic / Dirichlet BCs
BC for two directions, magnetic field, below Tc, …

And, many other problems related to EMT!!

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O(t) terms in SFTE lattice discretization

FlowQCD2016

This Study②Small t extrapol.

①Continuum

strongdiscretization
effectstrongdiscretization
effectEnergy Density

Transverse Pressure Pz

lattice regularizedgauge theory
gradient flow

Smearedworldby gradient flow

continuum
theorygradient flow
analytic relation

Measure on
the latticeTake Extrapolation (t,a)(0,0)O(t) terms in SFTE lattice discretization

In conformal (SmTmm=0)

Lattice EMT Operator Caracciolo+, 1990

 Fit to thermodynamics: Z3, Z1

 Shifted-boundary method: Z6, Z3 Giusti, Meyer, 2011; 2013;

Giusti, Pepe, 2014~; Borsanyi+, 2018

Multi-level algorithm effective in reducing statisticalerror of correlator

Meyer, 2007;
Borsanyi, 2018;
Astrakhantsev+, 2018