Gradient flow and the EMT on the lattice Hiroshi Suzuki Kyushu - - PowerPoint PPT Presentation

Gradient flow and the EMT on the lattice

鈴木 博 Hiroshi Suzuki

九州大学 Kyushu University

2019/04/18 @ FLQCD2019

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 1 / 42

References

Theory H.S., PTEP 2013 (2013) 083B03 [arXiv:1304.0533 [hep-lat]]

• H. Makino, H.S., PTEP 2014 (2014) 063B02 [arXiv:1403.4772 [hep-lat]]
• O. Morikawa, H.S., PTEP 2018 (2018) 073B02 [arXiv:1803.04132 [hep-th]]

Numerical experiment

• M. Asakawa, T. Hatsuda, E Itou, M. Kitazawa, H.S., PRD90 (2014) 011501

[arXiv:1312.7492 [hep-lat]]

• Y. Taniguchi, S. Ejiri, R. Iwami, K. Kanaya, M. Kitazawa, H.S., T. Umeda, N.

Wakabayashi, PRD96 (2017) 014509 [arXiv:1609.01417 [hep-lat]]

• M. Kitazawa, T. Iritani, M. Asakawa, T. Hatsuda, H.S., PRD94 (2016) 114512

[arXiv:1610.07810 [hep-lat]]

• Y. Taniguchi, K. Kanaya, H.S., T. Umeda, PRD95 (2017) 054502

[arXiv:1611.02411 [hep-lat]]

• K. Kanaya, S. Ejiri, R. Iwami, M. Kitazawa, H.S., Y. Taniguchi, T. Umeda,

EPJ Web Conf. 175 (2018) 07023 [arXiv:1710.10015 [hep-lat]]

• Y. Taniguchi, S. Ejiri, K. Kanaya, M. Kitazawa, A. Suzuki, H.S., T. Umeda,

EPJ Web Conf. 175 (2018) 07013 [arXiv:1711.02262 [hep-lat]]

• T. Iritani, M. Kitazawa, H.S., H. Takaura, PTEP 2019 (2019) 023B02

[arXiv:1812.06444 [hep-lat]]

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 2 / 42

Gradient flow (Narayanan–Neuberger, Lüscher)

One-parameter t ≥ 0 (the flow time) deformation of the gauge field Aµ(x), Aµ(x) → Bµ(t, x), Bµ(t = 0, x) = Aµ(x), according to (the flow equation) ∂tBµ(t, x) = −g2 δSYM[B] δBµ(t, x), = DνGνµ(t, x) = ∆Bµ(t, x) + · · · , Here, SYM is the Yang–Mills action and the RHS is the gradient in functional space. So the name of the Yang–Mills gradient flow. Since Dµ = ∂µ+[Bµ, ·], Gµν(t, x) = ∂µBν(t, x)−∂νBµ(t, x)+[Bµ(t, x), Bν(t, x)], this is a diffusion-type equation with the diffusion length, x ∼ √ 8t. The flow time t has the mass dimension −2.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 3 / 42

Yang–Mills gradient flow

Yang–Mills gradient flow (continuum) ∂tBµ(t, x) = −g2 δSYM[B] δBµ(t, x), Bµ(t = 0, x) = Aµ(x). Wilson flow (lattice) ∂tV(t, x, µ)V(t, x, µ)−1 = −g2

0∂x,µSWilson[V],

V(t = 0, x, µ) = U(x, µ). Applications in lattice gauge theory (the citation of the Lüscher’s original paper is ≳ 500)

Topological charge Scale setting Non-perturbative gauge coupling constant Chiral condensate Various renormalized operators, including the energy–momentum tensor Supersymmetric theory Chiral gauge theory etc.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 4 / 42

Finiteness of the gradient flow (Lüscher, Weisz (2011))

Correlation function of the flowed gauge field, ⟨Bµ1(t1, x1) · · · Bµn(tn, xn)⟩ = 1 Z ˆ DAµ Bµ1(t1, x1) · · · Bµn(tn, xn) e−SYM[A], when expressed in terms of renormalized coupling, g2 = g2

0µ−2εZ −1,

is UV finite without the wave function renormalization. This is quite contrast to the conventional gauge field, for which ⟨Aµ1(x1) · · · Aµn(xn)⟩ , requires the wave function renormalization (AR)a

µ = Z −1/2Z −1/2 3

Aa

µ.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 5 / 42

Finiteness of the gradient flow

This finiteness persists even for the equal-point product, ⟨Bµ1(t1, x1)Bµ2(t1, x1) · · · Bµn(tn, xn)⟩ , t1 > 0, . . . , tn > 0. Any composite operator of the flowed gauge field is automatically UV finite. All order proof of the finiteness uses a local D + 1-dimensional field theory:

t

Because of the gaussian damping factor ∼ e−tp2 in the propagator ⇒ No bulk (t > 0) counterterm. BRS symmetry ⇒ No boundary (t = 0) counterterm besides Yang–Mills

• nes.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 6 / 42

Small flow-time expansion (Lüscher, Weisz (2011))

Generally, the relation between a composite operator in t > 0 and that in 4D can be quite complicated. The relation becomes tractable, however, in the small flow time limit t → 0. Small flow-time expansion ˜ Oiµν(t, x) x √ 8t ˜ Oiµν(t, x) = ⟨ ˜ Oiµν(t, x) ⟩ 1 + ∑

j

ζij(t) [ORjµν(x) − VEV] + O(t). This is quite analogous to the OPE, but the continuous flow time t is more suitable for lattice gauge theory.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 7 / 42

Small flow-time expansion

Small flow-time expansion: ˜ Oiµν(t, x) = ⟨ ˜ Oiµν(t, x) ⟩ 1 + ∑

j

ζij(t) [ORjµν(x) − VEV] + O(t). Inverting this, ORiµν(x) − VEV = lim

t→0

   ∑

j

( ζ−1)

ij (t)

[ ˜ Ojµν(t, x) − ⟨ ˜ Ojµν(t, x) ⟩ 1 ]    , we have a representation of the (renormalized) operator in terms of flowed field. Furthermore, the t → 0 behavior of the coefficients ζij(t) can be determined by perturbation theory, thanks to the asymptotic freedom (cf. OPE). We use these facts to find a universal representation of the EMT.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 8 / 42

Lattice gauge theory (LGT) and the energy–momentum tensor (EMT)

LGT is very nice. . . a This however breaks spacetime symmetries (translation, Poincaré, SUSY, . . . ) for a ̸= 0. For a ̸= 0, one cannot define the Noether current associated with the translational invariance, EMT Tµν(x). Even for the continuum limit a → 0, this is difficult, because EMT is a composite operator which generally contains UV divergences: a × 1 a

a→0

→ 1.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 9 / 42

EMT in LGT?

We want to construct EMT on the lattice, which becomes the correct EMT, automatically in the continuum limit a → 0. The correct EMT is characterized by the translation Ward–Takahashi relation ⟨ Oext ˆ

D

dDx ∂µTµν(x) Oint ⟩ = − ⟨Oext ∂νOint⟩ . x D Oint Oext This contains the correct normalization and the conservation law. Applications to physics related to spacetime symmetries: QCD thermodynamics, transport coefficients in gauge theory, momentum/spin structure of baryons, conformal field theory, dilaton physics, . . .

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 10 / 42

Conventional approach (Caracciolo et al. (1989–))

Under the hypercubic symmetry, the operator reproducing the correct EMT of QCD for a → 0 is given by Tµν(x) =

7

∑

i=1

ZiOiµν(x)|lattice − VEV, where O1µν(x) ≡ ∑

ρ

F a

µρ(x)F a νρ(x),

O2µν(x) ≡ δµν ∑

ρ,σ

F a

ρσ(x)F a ρσ(x),

O3µν(x) ≡ ¯ ψ(x) ( γµ ← → D ν + γν ← → D µ ) ψ(x), O4µν(x) ≡ δµν ¯ ψ(x)← → / D ψ(x), O5µν(x) ≡ δµνm0 ¯ ψ(x)ψ(x), and, Lorentz non-covariant ones: O6µν(x) ≡ δµν ∑

ρ

F a

µρ(x)F a µρ(x),

O7µν(x) ≡ δµν ¯ ψ(x)γµ ← → D µψ(x) Seven non-universal coefficients Zi must be determined by lattice perturbation theory or non-perturbatively.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 11 / 42

Our approach (arXiv:1304.0533)

We bridge lattice regularization and dimensional regularization, which preserves the translational invariance, by the gradient flow. Schematically, regularization independent flowed composite operator dimensional lattice correct EMT low energy correlation functions

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 12 / 42

EMT in dimensional regularization

Vector-like gauge theory: S = − 1 2g2 ˆ dDx tr [Fµν(x)Fµν(x)] + ˆ dDx ¯ ψ(x)( / D + m0)ψ(x). By the Noether method, Tµν(x) = 1 g2 { O1µν(x) − 1 4O2µν(x) } + 1 4O3µν(x) − 1 2O4µν(x) − O5µν(x) − VEV, where O1µν(x) ≡ ∑

ρ

F a

µρ(x)F a νρ(x),

O2µν(x) ≡ δµν ∑

ρ,σ

F a

ρσ(x)F a ρσ(x),

O3µν(x) ≡ ¯ ψ(x) ( γµ ← → D ν + γν ← → D µ ) ψ(x), O4µν(x) ≡ δµν ¯ ψ(x)← → / D ψ(x), O5µν(x) ≡ δµνm0 ¯ ψ(x)ψ(x). Under the dimensional regularization, this simple combination is the correct EMT.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 13 / 42

EMT from the gradient flow

We consider following composite operators of flowed fields: ˜ O1µν(t, x) ≡ Ga

µρ(t, x)Ga νρ(t, x),

˜ O2µν(t, x) ≡ δµνGa

ρσ(t, x)Ga ρσ(t, x),

˜ O3µν(t, x) ≡ ˚ ¯ χ(t, x) ( γµ ← → D ν + γν ← → D µ ) ˚ χ(t, x), ˜ O4µν(t, x) ≡ δµν˚ ¯ χ(t, x)← → / D ˚ χ(t, x), ˜ O5µν(t, x) ≡ δµνm˚ ¯ χ(t, x)˚ χ(t, x), and then the small flow-time expansion reads, ˜ Oiµν(t, x) = ⟨ ˜ Oiµν(t, x) ⟩ 1 + ∑

j

ζij(t) [Ojµν(x) − ⟨Ojµν(x)⟩ 1] + O(t). We compute ζij(t) with dimensional regularization. We then substitute Oiµν(x)−⟨Oiµν(x)⟩ 1 = lim

t→0

   ∑

j

( ζ−1)

ij (t)

[ ˜ Ojµν(t, x) − ⟨ ˜ Ojµν(t, x) ⟩ 1 ]    , in the expression of the EMT in dimensional regularization.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 14 / 42

Fermion flow

We also introduce the fermion flow (Lüscher (2013)) ∂tχ(t, x) = [∆ − α0∂µBµ(t, x)] χ(t, x), χ(t = 0, x) = ψ(x), ∂t ¯ χ(t, x) = ¯ χ(t, x) [← − ∆ + α0∂µBµ(t, x) ] , ¯ χ(t = 0, x) = ¯ ψ(x), where ∆ = DµDµ, Dµ = ∂µ + Bµ, ← − ∆ = ← − D µ ← − D µ, ← − D µ ≡ ← − ∂ µ − Bµ. It turns out that the flowed fermion field requires the wave function renormalization: χR(t, x) = Z 1/2

χ

χ(t, x), ¯ χR(t, x) = Z 1/2

χ

¯ χ(t, x), Zχ = 1 + g2 (4π)2 C2(R)31 ϵ + O(g4). Still, any composite operators of χR(t, x) are UV finite.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 15 / 42

Ringed fermion fields

To avoid the complication associated with the wave function renormalization, we introduce the variable, ˚ χ(t, x) = C χ(t, x) √ t2 ⟨ ¯ χ(t, x)← → / D χ(t, x) ⟩ = χR(t, x) + O(g2), where C ≡ √ −2 dim(R) (4π)2 , and similarly for ¯ χ(t, x). Since Zχ is canceled out in ˚ χ(t, x), any composite operators of ˚ χ(t, x) and ˚ ¯ χ(t, x) are UV finite.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 16 / 42

Universal formula for EMT

In this way, (Makino, H.S., arXiv:1403.4772) Tµν(x) = lim

t→0

{ c1(t) [ ˜ O1,µν(t, x) − 1 4 ˜ O2,µν(t, x) ] + c2(t) ˜ O2,µν(t, x) + c3(t) [ ˜ O3,µν(t, x) − 2 ˜ O4,µν(t, x) ] + c4(t) ˜ O4,µν(t, x) + c5(t) ˜ O5,µν(t, x) − VEV } , where, to the one-loop order (TF = Tnf ) c1(t) = 1 g(µ)2 + [ −β0L(µ, t) − 7 3 CA + 3 2 TF ] 1 (4π)2 , c2(t) = 1 4 ( 11 6 CA + 11 6 TF ) 1 (4π)2 , c3(t) = 1 4 + [ 1 4 ( 3 2 + ln 432 ) CF ] g(µ)2 (4π)2 , c4(t) = 3 4 CF g(µ)2 (4π)2 , c5(t) = −1 − [ 3L(µ, t) + 7 2 + ln 432 ] CF g(µ)2 (4π)2 , where β0 = 11

3 CA − 4 3TF and L(µ, t) = ln(2µ2t) + γE. We set µ ∝ 1/

√ t → ∞.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 17 / 42

Universal formula for EMT

This is manifestly finite, as it should be for EMT! This is universal: ci(t) are independnt of the regularization. In the limit of infinite cutoff, the formula holds irrespective of the regularization. We have to first take the continuum limit a → 0 and then the small flow time limit t → 0. Practically, we cannot simply take a → 0 and may take t as small as possible in the fiducial window, a ≪ √ 8t ≪ 1 Λ. The usefulness with presently-accessible lattice parameters is not

• bvious a priori. . .

In the last year, ci(t) were obtained to the two-loop order! (Harlander, Kluth, Lange, arXiv:1808.09837)

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 18 / 42

First trial: Thermodynamics in the quenched QCD (FlowQCD Collaboration, arXiv:1312.7492)

The finite temperature expectation value of the EMT, Tµν(x). The entropy density as the traceless part: ε + p = −4 3 ⟨ T00(x) − 1 4Tµν(x) ⟩ , and the “trace anomaly” as the trace part: ε − 3p = − ⟨Tµµ(x)⟩ . Considered T = 0.99Tc, 1.24Tc, and 1.65Tc by 323 × (6, 8, 10) lattices. 300 configurations for each temperature. 324 lattice for the vacuum. For the quenched QCD, the two-loop order coefficient for the trace part is available.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 19 / 42

FlowQCD Collaboration, arXiv:1312.7492

Thermal expectation values as a function of the flow time √ 8t for T = 1.65Tc:

0.5 1 1.5 2 2.5 3

(ε-3P)/T

4

0.1 0.2 0.3 0.4 0.5 1 2 3 4 5

(ε+P)/T

4

beta=6.20 Nτ=6 beta=6.40 Nτ=8 beta=6.56 Nτ=10

/8t T ^

• ver

smeared 2a > sqrt(8t) for Nτ =10 for Nτ =8 for Nτ =6

Stable behavior in the fiducial window, 2a < √ 8t < 1/(2T).

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 20 / 42

FlowQCD Collaboration, arXiv:1312.7492

In the continuum limit, from the values at √ 8tT = 0.40,

0.5 1 1.5 2 2.5 3

(ε-3P)/T

4

1 1.5 2

T / Tc

1 2 3 4 5 6

(ε+P)/T

4

• ur result

Borsanyi et al. Okamoto et al. Boyd et al.

Although the error bars were rather large, this encouraged us very much!

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 21 / 42

FlowQCD Collaboration, arXiv:1610.07810

More systematic study: a = 0.013–0.061 fm, Ns = 64–128, Nτ = 12–24, ∼ 1000–2000 configurations:

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

tT 2

4.8 4.9 5.0 5.1 5.2 5.3 5.4

s/T 3 T/Tc = 1. 68

continuum Range-1 Range-2 Range-3 643 × 12 963 × 16 1283 × 20 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

tT 2

1.05 1.10 1.15 1.20 1.25 1.30 1.35

∆/T 4 T/Tc = 1. 68

continuum Range-1 Range-2 Range-3 643 × 12 963 × 16 1283 × 20

The gray band: the continuum limit at each flow time.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 22 / 42

FlowQCD Collaboration, arXiv:1610.07810

The double limit, a → 0 first and then t → 0 yields

0.5 1.0 1.5 2.0 2.5

T/Tc

1 2 3 4 5 6 7

s/T 3

FlowQCD Ref.[1] Ref.[4]

0.5 1.0 1.5 2.0 2.5

T/Tc

0.0 0.5 1.0 1.5 2.0 2.5 3.0

∆/T 4

FlowQCD Ref.[1] Ref.[4]

Figure: [1] Boyd, et al., hep-lat/9602007. [4] Borsanyi, et al., arXiv:1204.6184.

It appears that no room for doubt.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 23 / 42

More recently, Iritani, Kitazawa, H.S., Takaura, arXiv:1812.06444

Same lattice data as arXiv:1610.07810, but with the higher order coefficients! (Harlander, Kluth, Lange, arXiv:1808.09837). For the entropy density

0.000 0.005 0.010 0.015 0.020 0.025 0.030

tT2

4.8 5.0 5.2 5.4 5.6

( + p)/T4

T/TC = 1.68 (NLO) =

d

643 × 12 963 × 16 1283 × 20 Range-1 Range-2 Range-3 0.000 0.005 0.010 0.015 0.020 0.025 0.030

tT2

4.8 5.0 5.2 5.4 5.6

( + p)/T4

T/TC = 1.68 (N2LO) =

d

643 × 12 963 × 16 1283 × 20 Range-1 Range-2 Range-3

The higher order coefficient renders the behavior more stable ⇒ Less sensitive to the method of the t → 0 extrapolation.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 24 / 42

Iritani, Kitazawa, H.S., Takaura, arXiv:1812.06444

For the trace anomaly:

0.000 0.005 0.010 0.015 0.020 0.025 0.030

tT2

1.0 1.1 1.2 1.3 1.4

( 3p)/T4

T/TC = 1.68 (N2LO) =

d

643 × 12 963 × 16 1283 × 20 Range-1 Range-2 Range-3 0.000 0.005 0.010 0.015 0.020 0.025 0.030

tT2

1.0 1.1 1.2 1.3 1.4

( 3p)/T4

T/TC = 1.68 (N3LO) =

d

643 × 12 963 × 16 1283 × 20 Range-1 Range-2 Range-3

The two-loop coefficient already gives a well-stable behavior.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 25 / 42

Iritani, Kitazawa, H.S., Takaura, arXiv:1812.06444

Already the field of a precise determination:

1.0 1.5 2.0 2.5

T/Tc

1 2 3 4 5 6

( + p)/T4

Boyd et al. Borsanyi et al. Giusti-Pepe Caselle et al. FlowQCD 2016 NLO (this work) N2LO (this work)

1.0 1.5 2.0 2.5

T/Tc

4.5 5.0 5.5 6.0 6.5

( + p)/T4

Boyd et al. Borsanyi et al. Giusti-Pepe Caselle et al. FlowQCD 2016 NLO (this work) N2LO (this work)

1.0 1.5 2.0 2.5

T/Tc

0.0 0.5 1.0 1.5 2.0 2.5

( 3p)/T4

Boyd et al. Borsanyi et al. Giusti-Pepe Caselle et al. FlowQCD 2016 N2LO (this work) N3LO (this work)

1.0 1.2 1.4 1.6 1.8 2.0

T/Tc

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

( 3p)/T4

Boyd et al. Borsanyi et al. Giusti-Pepe Caselle et al. FlowQCD 2016 N2LO (this work) N3LO (this work)

Figure: Boyd et al., Borsanyi et al.: Integral method, Giusti, Pepe: Moving frame method, Caselle et al.: Jarzynski’s equality.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 26 / 42

The two point functions (Kitazawa, Iritani, Asakawa, Hatsuda, arXiv:1708.01415)

The connected part Cµν;ρσ(τ) ≡ 1 T 5 ˆ

V

d3x ⟨δTµν(x)δTρσ(0)⟩ , where δTµν(x) ≡ Tµν(x) − ⟨Tµν(x)⟩.

0.0 0.1 0.2 0.3 0.4 0.5 τT 10 15 20 25 30 35 C44; 44(τ)

T/Tc = 1. 68

tT 2 = 0. 0024 tT 2 = 0. 0035 tT 2 = 0. 0052 tT 2 = 0. 0069

0.0 0.1 0.2 0.3 0.4 0.5 τT 15 10 5 5 10 −C44; 11(τ)

T/Tc = 1. 68

s/T 3 tT 2 = 0. 0024 tT 2 = 0. 0035 tT 2 = 0. 0052 tT 2 = 0. 0069

3 5 7 0.0 0.1 0.2 0.3 0.4 0.5 τT 15 10 5 5 10 −C41; 41(τ)

T/Tc = 1. 68

s/T 3 tT 2 = 0. 0024 tT 2 = 0. 0035 tT 2 = 0. 0052

3 5 7

Indicating the conservation law of the EMT, ∂τCµν;ρσ(τ) = 0!!! Confirms the linear response relations, s.t, ε + p T 4 = 1 T 3 dp dT = −C44,;11(τ).

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 27 / 42

Stress tensor distribution around the static quark–anti-quark pair (Yanagihara, Iritani, Kitazawa, Asakawa, Hatsuda, arXiv:1803.05656)

The EMT around the static quark–anti-quark pair: Tµν(x) ≡ ⟨Tµν(x)⟩Q ¯

Q = lim T→∞

⟨Tµν(x)W(R, T)⟩ ⟨W(R, T)⟩ . Eigenvectors: Tijn(k)

j

= λkn(k)

j

0.4 0.2 0.0 0.2 0.4

z[fm]

0.3 0.2 0.1 0.0 0.1 0.2 0.3

y[fm]

λk < 0 λk > 0 (a) SU(3) Yang-Mills 0.4 0.2 0.0 0.2 0.4

z[fm]

0.3 0.2 0.1 0.0 0.1 0.2 0.3

y[fm]

λk < 0 λk > 0 (b) Maxwell

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 28 / 42

WHOT-QCD Collaboration: Baba, Ejiri, Iwami, Kanaya, Kitazawa, Shimojo, Shirogane, A. Suzuki, H.S., Taniguchi, Umeda

For Full QCD? We are studying the Nf = 2 + 1 QCD by using the NP O(a)-improved Wilson quark action and the RG improved Iwasaki gauge action. Somewhat heavy ud quarks (mπ/mρ ≃ 0.63, mηss/mφ ≃ 0.74)

a = 0.0701(29) fm, 283 × 56 (JLQCD), 323 × Nt (Nt = 6, 8, . . . , 16) a = 0.0970(26) fm, 323 × 40, 323 × Nt (Nt = 8, 10, 11, 12) [a = 0.04976 fm, 403 × 80]

Aiming at the test of the methodology, the continuum limit. Physical mass quarks

a = 0.08995(40) fm, 323 × 64 (PACS-CS), 323 × Nt (Nt = 4, 5, 6, . . . , 14, 16, [18])

Physical prediction on the EoS etc.. . .

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 29 / 42

Somewhat heavy ud quarks, a ≃ 0.07 fm, arXiv:1609.01417

Typical t → 0 extrapolation (Nt = 12)

5 10 15 20 25 30 35 0.5 1 1.5 2 (e+p)/T4 t/a2 T=232 MeV (Nt=12) linear fit nonlinear fit linear+log fit

• 15
• 10
• 5

5 10 15 20 0.5 1 1.5 2 (e-3p)/T4 t/a2 T=232 MeV (Nt=12) linear fit nonlinear fit linear+log fit

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 30 / 42

Somewhat heavy ud quarks, a ≃ 0.07 fm, arXiv:1609.01417

5 10 15 20 25 30 100 200 300 400 500 600 (e+p)/T4 T (MeV) gradient flow T-integration

• 4
• 2

2 4 6 8 10 12 100 200 300 400 500 600 (e-3p)/T4 T (MeV) gradient flow T-integration

Comparison to Umeda et al. (WHOT-QCD), arXiv:1202.4719. Indicating a ≃ 0.07 fm is fine enough for T ≲ 300 MeV. Disagreement for T ≳ 350 MeV (Nt ≤ 8) may be attributed to O((aT)2 = 1/N2

t ) error.

It appears that the method is basically working.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 31 / 42

Somewhat heavy ud quarks, a ≃ 0.097 fm (Preliminary)

Typical t → 0 extrapolation (Nt = 10). The “linear region” becomes smaller, as expected.

5 10 15 20 25 30 35 0.2 0.4 0.6 0.8 1 1.2 1.4 (e+p)/T4 t/a2 T=203 MeV (Nt=10) linear fit non-linear fit linear+log fit

• 10

10 20 30 40 50 0.2 0.4 0.6 0.8 1 1.2 1.4 (e-3p)/T4 t/a2 T=203 MeV (Nt=10) linear fit non-linear fit linear+log fit

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 32 / 42

Somewhat heavy ud quarks, a ≃ 0.07 fm and a ≃ 0.097 fm (Preliminary)

5 10 15 20 25 30 35 50 100 150 200 250 300 350 400 450 (e+p)/T4 T [MeV] b=2.05 b=1.9 linear fit b=1.9 non-linear fit

• 5

5 10 15 50 100 150 200 250 300 350 400 450 (e-3p)/T4 T [MeV] b=2.05 b=1.9 linear fit b=1.9 non-linear fit

It appears that the a dependence is fairly small. Systematic fit is ongoing

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 33 / 42

Physical mass ud, a ≃ 0.09 fm, arXiv:1710.10015, plus new Nt = 16 (Preliminary)

Typical t → 0 extrapolation (Nt = 12)

5 10 15 20 25 30 35 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 (e+p)/T4 t/a2 T=183 MeV (Nt=12) linear fit non-linear fit linear+log fit

• 10

10 20 30 40 50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 (e-3p)/T4 t/a2 T=183 MeV (Nt=12) linear fit non-linear fit linear+log fit

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 34 / 42

Physical mass ud, a ≃ 0.09 fm, arXiv:1710.10015, plus new Nt = 16 (Preliminary)

5 10 15 20 25 30 35 100 200 300 400 500 600 (e+p)/T4 T (MeV) linear fit

• 5

5 10 15 20 25 30 35 40 100 200 300 400 500 600 (e-3p)/T4 T (MeV) linear fit

Entropy seems to be consistent with that by the staggered fermion. Trace anomaly is much larger compared with the staggered. Increasing the statistics and a lower temperature are ongoing. Finer lattices, the continuum limit are future problem.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 35 / 42

Two point functions, the somewhat heavy ud case, a ≃ 0.07 fm, arXiv:1711.02262

The connected part (δTµν(x) ≡ Tµν(x) − ⟨Tµν(x)⟩): ) Cµν;ρσ(τ) ≡ 1 T 5 ˆ

V

d3x ⟨δTµν(x)δTρσ(0)⟩ .

• 100
• 50

50 100 2 4 6 8 10 C00;00 time flow time=0.5 flow time=1.0 flow time=1.5 flow time=2.0

• 40
• 20

20 40 60 2 4 6 8 10 C20;20 time flow time=0.5 flow time=1.0 flow time=1.5 flow time=2.0

• 40
• 20

20 40 2 4 6 8 10 C00;22 time flow time=0.5 flow time=1.0 flow time=1.5 flow time=2.0

Indicating the conservation law, restoration of the rotational symmetry, and the linear response relations. Shear viscosity from C12;12, η/s = 0.145(51) @ T = 232 MeV (Preliminary), (JPS meeting @ Shinshu).

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 36 / 42

Chiral condensate

Gradient flow can be employed also to construct the (renormalized) scalar operator to compute the chiral condensate and (disconnected) chiral susceptibility. For the somewhat heavy ud quarks, a ≃ 0.07 fm, arXiv:1609.01417. Tpc ≃ 190 MeV?

0.05 0.1 0.15 0.2 100 200 300 400 500 600 chiral condensate T (MeV) u quark s quark u quark s quark 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 100 200 300 400 500 600 chiral susceptibility T (MeV) u quark s quark

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 37 / 42

Chiral condensate

For the physical mass ud, a ≃ 0.09 fm, arXiv:1710.10015, plus new Nt = 16 (Preliminary). VEV extracted chiral condensate. It appears that sharper for ud quarks

• 0.2
• 0.15
• 0.1
• 0.05

100 200 300 400 500 600 subtracted chiral condensate T (MeV) u quark

• 0.2
• 0.15
• 0.1
• 0.05

100 200 300 400 500 600 subtracted chiral condensate T (MeV) s quark

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 38 / 42

3D scalar theory (Morikawa, Sonoda, H.S., work in progress)

3D N-component scalar theory S = ˆ dDx [1 2∂µϕI∂µϕI + m2 2 ϕIϕI + λ0 8N ( ϕIϕI)2] The flow equation ∂tφI(t, x) = ∂µ∂µφI(t, x), φI(t = 0, x) = ϕI(x). A universal formula for EMT (C = 3.844365111074):

Tµν = ∂µϕI∂νϕI − δµν [ 1 2 ∂ρϕI∂ρϕI + m2 2 ϕIϕI + λ 8N ( ϕIϕI)2 ] − δµν ( λ 4π ( 1 + 2 N ) ( − 1 3 ) (8πt)−1/2 + λ2 (4π)2 {( 1 + 2 N )2 ( − 1 4π ) + 1 N ( 1 + 2 N ) ( − 1 8 ) [ ln(8πµ2t) − 1 3 + C ]}) ϕIϕI.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 39 / 42

3D scalar theory (Morikawa, Sonoda, H.S., work in progress)

The theory around the Wilson–Fisher fixed point can be realized as the long-distance limit, ⟨ϕ(x1) . . . ϕ(xn)⟩gE = lim

τ→∞ enxhτ ⟨ϕ(eτx1) . . . ϕ(eτxn)⟩m2,λ ,

where m2 = m2

cr(λ) + gEe−yEτ.

(m2 = m2

cr(λ) is the critical line).

The theory with gE = 0 flows to a CFT in IR. It can be interesting to explore the GF fixed point and the critical exponents by using the universal formula.

• cf. in the large N limit,

xh = 1 2, yE = 1, and m2

cr(λ) = 0.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 40 / 42

Summary and prospects

We wrote down a universal formula for the EMT in vector-like gauge theories by employing the gradient flow. The formula can be used in nonperturbative lattice simulations. The window problem: a ≪ √ 8t ≪ 1 Λ. Numerical experiments so far show encouraging results; the method appears usable practically.

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 41 / 42

Summary and prospects

Asymptotic form in t → 0? (work in progress). Push applications further: EoS of QCD, viscosities in gauge theory, momentum/spin structure of baryons, critical exponents in low-energy conformal field theory, dilaton physics, . . . Further theoretical study, including the equal-point correction. The axial U(1)A anomaly in gravitational field is not automatically reproduced (Morikawa, H.S., arXiv:1803.04132), ∂x

α ⟨j5α(x)Tµν(y)Tρσ(z)⟩

̸= ˆ

p,q

eip(x−y)eiq(x−z) 1 (4π)2 1 6ϵµρβγpβqγ(qνpσ − δνσpq) + (µ ↔ ν, ρ ↔ σ), but requires a correction by a “local counterterm” ∝ δ(x − y)δ(x − z). Other Noether currents, such as the axial and super currents (partially already done).

鈴木 博 Hiroshi Suzuki (九州大学) Gradient flow and the. . . 2019/04/18 @ FLQCD2019 42 / 42