Study of high temperature QCD with chiral fermions Hidenori Fukaya - - PowerPoint PPT Presentation

Study of high temperature QCD with chiral fermions

Hidenori Fukaya (Osaka U.)

for JLQCD collaboration

• S. Aoki, Y. Aoki, G. Cossu, HF,
• S. Hashimoto, T. Kaneko, C. Rohrhofer,
• K. Suzuki, in preparation.

JLQCD’s finite T project (2012~)

• 1. Nf=2 QCD with overlap fermions at fixed topology [
• G. Cossu ] 2012-2013, on IBM BG/L, Hitachi SR11000
• 2. Nf=2 QCD with Mobius domain-wall fermions [G.

Cossu, A.Tomiya ] 2013-2015, on IBM BG/Q.

• 3. Nf=2 QCD with MDW, finer and larger lattices [Y. Aoki,
• K. Suzuki, C. Rohrhofer] 2016-2020, on IBM BG/Q and

Oakforest-PACS [today’s topic]

• 4. Nf=2+1 QCD with MDW started ! [I. Kanamori, Y.

Nakamura joined.] 2020- Oakforest-PACS, Fugaku?

Results 2016-2019 (phase 3)

Symanzik gauge action Nf=2 Mobius domain-wall fermion action m = [1-10] mphys 1/a = 0.075 fm (0.1fm in phase 2) Lt = 8,10,12,14 [T=190-330MeV] L=24,32,40,48 [1.8-3.6fm] 15000-30000trj. Checking overlap/domain-wall consistency with reweighting.

Results 2016-2019 (phase 3) Target observables are

• Dirac spectrum
• Topological charge,
• axial U(1) susceptibility,
• meson/baryon correlators,
• chiral susceptibility.

Special focus = axial U(1) anomaly

Anomalous WTI looks non-zero: but the real question is if to which only lattice QCD can answer.

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup
• 3. Numerical results
• Dirac spectrum
• Topology
• U(1) susceptibility
• Meson correlators
• Chiral susceptibility
• 4. Summary

Simulation setup

Nf=2 flavor QCD 1/a = 2.6 GeV (0.075fm) Symanzik gauge action L=24,32,40,48 [1.8-3.6fm] Mobius domain-wall fermions with mres<1MeV Quark mass from 3MeV (< phys. pt. ~4MeV) to 30MeV. T=190, 220, 260, 330 MeV and higher. (Lt=8,10,12,14) Tc is estimated to be around 175MeV.

t z

Overlap vs. Mobius domain-wall

with L5=16. OV is obtained by exactly computing the sgn function for low-modes of HM.

perfect chiral sym. good chiral sym.

numerically mres ~ 1keV numerically mres ~ 1MeV

Violation of chiral symmetry enhanced at finite T

Checking chiral sym. for EACH eigenmode Bad modes appear above Tc for a~0.1fm.

[JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

Note: residual mass is (weighted) average

• f them.

For T=0, gi are consistent with residual mass.

Overlap/domain-wall reweighting

Essential for a > 0.1fm. [our previous work] DW and OV are consistent for a~0.08fm. (for Meson/Baryon study, we use DW) [this work].

Bonus = topology tunnelings

For dynamical overlap fermion, we needed to fix the topology. But DW + OV reweighting, we do not.

• 3
• 2
• 1

1 2 3 2000 4000 6000 8000 10000 12000 14000 trj Q (L=48) m=0.001 m=0.0025 m=0.00375 m=0.005

Data at T=220MeV

The use of overlap only in valence sector is dangerous !

In our work, reweighted OV and DW are

• consistent. But partially quenched OV

is NOT. Fake chiral zero modes appear.

0.0005 0.001 0.0015 0.002 0.0025 0.003 DW OV PQOV ρ(λ=0) (GeV3) m=0.001 m=0.0025

density of zero modes

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup

Nf=2 QCD w/ MDWF and rewegihteg overlap. at

T=190-330MeV near physical m~4MeV.

• 3. Numerical results
• Dirac spectrum
• Topology
• U(1) susceptibility
• Meson correlators
• Chiral susceptibility
• 4. Summary

✔

Dirac spectrum

Zero eigenvalues are related to SU(2)xSU(2) breaking, through the Banks-Casher relation, and axial U(1) anomaly through the index theorem,

lim

m→0 lim V →∞h¯

qqi = πρ(0)

ρ(λ) = 1 V X

i

hδ(λ λi(A))i

λi(A) :

i-th eigenvalue of Dirac op. with gauge background A.

n+ − n− = 1 32⇡2 Z d4x ✏µνρσtrcF µνF ρσ

Dirac spectrum at T=220MeV

* A remarkable peak at zero but disappears as m→0. U(1)A? * Strong supression of non-zero near zero modes. SU(2)? * DW and OV are consistent.

0.01 0.02 0.03 0.04 0.05 0.06 50 100 150 200 ρ(λ) (GeV3) λ(MeV) β=4.30, T=220MeV, L=32(2.4fm)

• verlap, m=0.01
• verlap, m=0.005
• verlap, m=0.00375
• verlap, m=0.0025
• verlap, m=0.001

domain-wall <- qq>/π at T=0, m=0

Different volumes

* 3 different volumes show consistent results * except for L=24 m=0.01 (heaviest data, L/Lt=2)

2x10-5 4x10-5 6x10-5 8x10-5 0.0001 0.00012 0.00014 0.00016 0.00018 20 40 60 80 100 A(λ) (GeV4) λ(MeV) β=4.30, T=220MeV L=24 (1.8fm), m=0.01 L=32 (2.4fm), m=0.01 L=40 (3.0fm), m=0.01 m=0.005 m=0.00375 m=0.0025

0.01 0.02 0.03 0.04 0.05 0.06 50 100 150 200 250 300 ρ(λ) (GeV3) λ(MeV) β=4.30, T=260MeV, L=32(2.4fm)

• verlap, m=0.015
• verlap, m=0.01
• verlap, m=0.008
• verlap, m=0.005
• verlap, m=0.003 (reweighted from 0.005)

The larger T, the larger the pseudo-gap.

0.01 0.02 0.03 0.04 0.05 0.06 20 40 60 80 100 120 140 ρ(λ) (GeV3) λ(MeV) β=4.24, T=195MeV, L=32(2.7fm)

• verlap, m=0.01
• verlap, m=0.005
• verlap, m=0.0025

domain-wall 0.01 0.02 0.03 0.04 0.05 0.06 50 100 150 200 ρ(λ) (GeV3) λ(MeV) β=4.30, T=220MeV, L=32(2.4fm)

• verlap, m=0.01
• verlap, m=0.005
• verlap, m=0.00375
• verlap, m=0.0025
• verlap, m=0.001

domain-wall <- qq>/π at T=0, m=0 0.01 0.02 0.03 0.04 0.05 0.06 100 200 300 400 500 ρ(λ) (GeV3) λ(MeV) β=4.30, T=330MeV, L=32(2.4fm)

• verlap, m=0.04
• verlap, m=0.02
• verlap, m=0.015
• verlap, m=0.01
• verlap, m=0.005

T=195MeV T=220MeV T=260MeV T=330MeV

(Near)zero mode peaks

Consistent with zero BEFORE the chiral limit.

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 20 40 60 80 100 120 ρ(λ=0) (GeV3) m (MeV) T=195 MeV (β=4.24) T=190 MeV (β=4.30) T=220 MeV T=260 MeV T=330 MeV

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup

Nf=2 QCD w/ MDWF and rewegihteg overlap. at

T=190-330MeV near physical m~4MeV.

• 3. Numerical results
• Dirac spectrum has a peak but vanishes in the m→0 limit.
• Topology
• U(1) susceptibility
• Meson correlators
• Chiral susceptibility
• 4. Summary

✔

Topological susceptibility

Overlap Dirac index Gluonic definition

χt = hQ2i V

(using clover term) (w/ OV/DW reweghting) We try both definitions.

Topological susceptibility at T=220MeV

5x107 1x108 1.5x108 2x108 5 10 15 20 25 30 χt [MeV4] m [MeV] OV index, T=220MeV L=24 (1.8fm) OV index, T=220MeV L=32 (2.4fm) OV index, T=220MeV L=40 (3.0fm) OV index, T=220MeV L=48 (3.6fm) Gluonic on DW

* Strong supression around m=10MeV. * Data for L=1.8-3.6 fm are consistent. * Gluonic def. on DW and reweighted OV index agree.

Different temperatures

* Sharp drop at FINITE quark mass * Gluonic def. on DW and reweighted OV index agree.

5x107 1x108 1.5x108 2x108 2.5x108 3x108 3.5x108 20 40 60 80 100 120 χt [MeV4] m [MeV] OV index, T=195MeV OV index, T=190MeV OV index, T=220MeV OV index, T=260MeV OV index, T=330MeV OV index, T=330MeV (L=3.6fm) Gluonic on DW

Taking 4th root

* Topology fluctuation is suppressed to ~ m4. * goes down to a few MeV, at most.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 20 40 60 80 100 120 χt

1/4/T

m [MeV] OV index, T=195MeV OV index, T=190MeV OV index, T=220MeV OV index, T=260MeV T=330MeV Gluonic on DW

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup

Nf=2 QCD w/ MDWF and rewegihteg overlap. at

T=190-330MeV near physical m~4MeV.

• 3. Numerical results
• Dirac spectrum has a peak but vanishes in the m→0 limit.
• Topology fluctuation is suppressed by ~m4.
• U(1) susceptibility
• Meson correlators
• Chiral susceptibility
• 4. Summary

✔

Axial U(1) susceptibility

Definition: Difference between S and PS triplet correlator Spectral decomposition (with overlap D)

∆(m) = X

x

[hπ(x)π(0)i hδ(x)δ(0)i] ,

∆(m) = 1 V (1 − m2)2 *X

λm

2m2(1 − λ2

m)2

λ4

m

+ ,

λm : eigenvalues of Hov(m) = γ5[(1 − m)Dov + m]

→ DW w/ noise method → OV w/ reweighting We try 2ways:

Chiral zero-mode subtraction

We find chiral zero modes are noisy.

• > subtract by hand:

This is justified since

∆(m) = h|Q|i m2V (1 m2)2 + 1 V (1 m2)2 * X

λm6=0

2m2(1 λ2

m)2

λ4

m

+ ,

¯ ∆(m) ⌘ ∆(m) 2h|Q|i m2(1 m2)2V

h|Q|i / V 1/2 (/ L3/2)

• 1

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 0.6 <|Q|>/(mL)3/2 1/(TL) m=0.01 m=0.005 m=0.00375 m=0.0025 m=0.001

UV subtraction

In the expansion in valence quark mass, c has a UV divergence, while we are interested in the IR part, a and b.

• > Let us remove c, so that

by linear combinations w/ 3 mv’s.

¯ ∆(mv) = a m2

v

+ b + m2

vc + O(m4 v)

¯ ∆UV subt.(m) = a m2 + b + O(m4)

UV subtraction

More concretely, where we choose

¯ ∆UV subt.(m) = m2

2m2 3

m2

2 − m2 3

 ¯ ∆(m1) − ¯ ∆(m2) m2

1 − m2 2

− ¯ ∆(m1) − ¯ ∆(m3) m2

1 − m2 3

• + (m2

1 + m2 2)(m2 1 + m2 3)

m2

3 − m2 2

m2

1 ¯

∆(m1) − m2

2 ¯

∆(m2) m4

1 − m4 2

− m2

1 ¯

∆(m1) − m2

3 ¯

∆(m3) m4

1 − m4 3

• ,

m1 = m, m2 = 0.95m m3 = 1.05m.

Low-mode saturation

10 20 30 40 50 60 70 80 100 200 300 400 500 ΔUVsubt.

1/2 [MeV]

λth (MeV) T=260MeV, m=0.015 T=330MeV, m=0.04

100 200 300 400 500 600 700 50 100 150 200 ΔUVsubt.

1/2 [MeV]

λth (MeV) T=190MeV, m=0.005 T=220MeV, m=0.01

It looks O.K. for lower T

But for higher T, we need higher modes contribution using stochastic method.

* Different volumes show consistent results, except for L=24 at heavier masses (L/Lt=2) * anomaly goes down to a few MeV, at most.

50 100 150 200 250 300 350 400 450 5 10 15 20 25 30 ΔUVsubt.

1/2 [MeV]

m (MeV) New definition T=220MeV, L=1.8fm T=220MeV, L=2.4fm T=220MeV, L=3.0fm T=220MeV, L=3.6fm

Axial U(1) susceptibility at T=220MeV

Different temperatures

* Axial U(1) anomaly goes down to a few MeV, at most. * Results here are with low-modes only. K. Suzuki is (re)analyzing the stochastic measurements.

100 200 300 400 500 600 20 40 60 80 100 120 ΔUVsubt.

1/2 [MeV]

m (MeV) T=190MeV T=220MeV T=260MeV T=330MeV

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup

Nf=2 QCD w/ MDWF and rewegihting overlap. at

T=190-330MeV near physical m~4MeV.

• 3. Numerical results
• Dirac spectrum has a peak but vanishes in the m→0 limit.
• Topology fluctuation is suppressed by ~m4.
• U(1) susceptibility goes down to (a few MeV)2.
• Meson correlators
• Chiral susceptibility
• 4. Summary

✔

“Meson” correlator

We consider spacial correlator in z direction, where

* We find that the chiral symmetry is good enough with MDW. * Rotationally symmetric average taken. * Low-mode averaging is performed for noisy ensembles.

CΓ(z) = X

x,y,t

h¯ uΓd(x, y, z, t) ¯ dΓu(0, 0, 0, 0)i,

Γ = γ5(PS), 1(S), γ1,2(V ), γ5γ1,2(A), γ4γ3(Tt) and γ5γ4γ3(Xt).

Tensor channels

We find that comparison is difficult. Instead, we investigate

For the reference, we also study SU(2)LxSU(2)R pair in vector channel,

too noisy.

¯ qτ aγ4γ3q(x)¯ qτ aγ4γ3q(0) ↔ ¯ qτ aγ5γ4γ3q(x)¯ qτ aγ5γ4γ3q(0)

¯ qτ aq(x)¯ qτ aq(0) ↔ ¯ qτ aγ5q(x)¯ qτ aγ5q(0)

¯ qτ aγ1q(x)¯ qτ aγ1q(0) ↔ ¯ qτ aγ1γ5γ1q(x)¯ qτ aγ5γ1q(0)

Is it really “meson”?

In general form (of bosonic correlator), it depends on the details of spacial spectral function . If it has

• 1. an isolated pole →
• 2. a cut (like 2-quark states) from

, →

ρΓ(M)

mth

CΓ(z) = Z dMρΓ(M) Z dpz 2π 2Meipzz p2

z + M 2 =

Z dMρΓ(M)e−Mz.

∝ δ(M − mg)

ρΓ(M) = θ(M − mth) (c0 + c1M + · · · )

CΓ(z) ∼ e−mgz

CΓ(z) ∼ e−mthz 1/z + O(1/z2)

• <latexit sha1_base64="2ODUrRDb128Xl6w6M/KU703HT0=">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</latexit>

Pole vs. cut

2 4 6 8 10 12 14 16 nz 300 600 900 1200 1500 1800 2100 2400 2700 3000 meff

32 × 12 β = 4.30 mud = 0.001 PS cosh: 701 ± 31 MeV 2-q LO: 409 ± 27 MeV

cosh 2-q LO 2 4 6 8 10 12 14 16 nz 300 600 900 1200 1500 1800 2100 2400 2700 3000 meff

32 × 12 β = 4.30 mud = 0.001 Ax cosh: 1349 ± 18 MeV 2-q LO: 1035 ± 15 MeV

cosh 2-q LO

Effective mass plots favor 2-quark picture. T=220MeV

Screening mass difference

We fit our data to and see their difference. Anomaly effect disappears to a few MeV (~1% of temperature), at most. Note: mscreen itself ~ 1GeV ～ 2πT.

Ae−mz/z

• 100
• 50

50 100 5 10 15 20 25 30 Δmscreen (MeV) m (MeV) T=220MeV U(1)A SU(2)xSU(2)

Volume dependence

• 150
• 100
• 50

50 100 150 5 10 15 20 25 30 Δmscreen (MeV) m (MeV) T=220MeV L=24 L=32 L=48

• 150
• 100
• 50

50 100 150 5 10 15 20 25 30 Δmscreen (MeV) m (MeV) T=220MeV L=24 L=32 L=48

axial U(1) SU(2)xSU(2)

The data are consistent with each other, except for noisy L=24.

Higher temperatures

No big difference from SU(2)xSU(2) restoration.

• 30
• 20
• 10

10 20 30 10 20 30 40 50 Δmscreen (MeV) m (MeV) T=260MeV U(1)A SU(2)xSU(2)

• 4
• 3
• 2
• 1

1 2 3 4 10 20 30 40 50 60 Δmscreen (MeV) m (MeV) T=330MeV U(1)A SU(2)xSU(2)

Suppression is stronger for higher T.

• C. Rohrhofer is now working

hard on baryons.

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup

Nf=2 QCD w/ MDWF and rewegihting overlap. at

T=190-330MeV near physical m~4MeV.

• 3. Numerical results
• Dirac spectrum has a peak but vanishes in the m→0 limit.
• Topology fluctuation is suppressed by ~m4.
• U(1) susceptibility goes down to (a few MeV)2.
• Meson 2quark correlators show a good U(1)A symmetry.
• Chiral susceptibility
• 4. Summary

✔

What is chiral susceptibility?

Z(m) = Z [dA] det(D(A) + m)Nf e−SG(A) = Z [dA] Y

λ

(iλ(A) + m)Nf e−SG(A)

Partition function chiral condensate chiral susceptibility

h¯ qqi = 1 NfV ∂ ∂m ln Z(m) = 1 V *X

λ

1 iλ(A) + m + ,

empirically small

χ(m) = 1 NfV ∂2 ∂m2 ln Z(m) = χcon.(m) + χdis.(m),

What is chiral susceptibility?

Broken phase Symmetric phase

Z(m)

h¯ qqi(m)

χ(m)

m

SU(2) or U(1)?

Chiral condensate and chiral susceptibility is used for the probe of SU(2)xSU(2) chiral symmetry breaking. But they also break axial U(1). In this work we show that chiral susceptibility is dominated by axial U(1) anomaly, rather than SU(2)xSU(2).

Dirac eigenmode decomposition

h¯ qqi = 1 NfV ∂ ∂m ln Z(m) = 1 V *X

λ

1 iλ(A) + m + ,

Z(m) = Z [dA] det(D(A) + m)Nf e−SG(A) = Z [dA] Y

λ

(iλ(A) + m)Nf e−SG(A)

χ(m) = 1 NfV ∂2 ∂m2 ln Z(m) = χcon.(m) + χdis.(m),

χcon.(m) = ∂ ∂mvalence h¯ qqi

• mvalence=m

χdis.(m) = ∂ ∂msea h¯ qqi

• msea=m

Connected part

χcon.(m) = − 1 V *X

λ

1 (iλ(A) + m)2 + = − 1 V *X

λ

2m2 (λ(A)2 + m2)2 + + 1 m " 1 V *X

λ

m λ(A)2 + m2 +#

= ∆(m) + h¯ qqi m ,

axial U(1) susceptibility! Namely, connected part includes pure U(1) anomaly effect.

Disconnected part

It is interesting to compare with topological susceptibility

χdis.(m) = Nf V " hN 2

0 i hN0i2

m2 + 2 m * N0 X

λ>0

2m λ(A)2 + m2 + hN0i *X

λ>0

2m λ(A)2 + m2 +! + * X

λ>0

2m λ(A)2 + m2 !2+

• *X

λ>0

2m λ(A)2 + m2 +23 5 .

Nf m2 χt = Nf m2 hQ2i hQi2 V

N0 = n+ + n− : number of zero modes

Q = n+ − n−

UV divergence

The chiral condensate has quadratic divergence at most. quadratic divergence appears only in connected part. Disconnected part is logarighmically divergent.

h¯ qqi = sgn(m)

• Σ + α|m|Λ2 + β|m|2Λ + γ|m|3 + · · ·
• <latexit sha1_base64="zH/rzyVedSE4mYrz2ErEtYBTRo=">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</latexit>

Λ : cut off

d dmh¯ qqi = χcon.(m) + χdis.(m) = 2δ(m)Σ αΛ2 2β|m|Λ 3γm2 + · · · .

Lattice formulas

With the overlap Dirac operator, we have where

λm = eigenvalues of Hm = γ5[(1 − m)Dov + m]

χdis.lat(m) = Nf V 2 4 1 (1 m2)2 * X

allλm

m(1 λ2

m)

λ2

m

!2+ |h¯ qqilat|2V 2 3 5 .

χcon.lat(m) = ∆lat(m) + h¯ qqilat m , ∆lat(m) = 1 V (1 m2)2 * X

allλm

2m2(1 λ2

m)2

λ4

m

+ , h¯ qqilat = 1 V (1 m2) * X

allλm

m(1 λ2

m)

λ2

m

+ .

Low mode approximation

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.02 0.04 0.06 0.08 0.1 0.12 chiral sus. connected λthre L=24, m=0.01 L=24, m=0.005 L=24, m=0.00375 L=24, m=0.0025 L=24, m=0.001 L=32 L=40

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.02 0.04 0.06 0.08 0.1 0.12 chiral sus. disconnected λthre L=24, m=0.01 L=24, m=0.005 L=24, m=0.00375 L=24, m=0.0025 L=24, m=0.001 L=32 L=40

connected disconnected part is well described by 40 lowest modes.

Disconnected part

χdis.(m) = Nf V 2 4 * X

λ

m λ(A)2 + m2 !2+ − *X

λ

m λ(A)2 + m2 +23 5 .

⇠ Nf V hN 2

0 i hN0i2

m2

N0 :

number of zero modes ⇠ Nf V hQ2i hQi2 m2

Q: topological charge

dominated by topological susceptibility. (vol. dependence is small.)

0.02 0.04 0.06 0.08 0.1 0.002 0.004 0.006 0.008 0.01 0.012 chiral susceptibility disconnected m beta=4.30(T=220MeV) threshold=0.07 w/ nonzeromodes, L=24 w/ nonzeromodes, L=32 w/ nonzeromodes, L=40 zeromodes only 2*top.sus./m2

Disconnected part at different T

The topological susceptibility (or U(1) anomaly) dominance is seen at 4 different temperatures.

200 400 600 800 1000 1200 5 10 15 20 25 30 35 40 chiral susceptibility1/2(MeV) m(MeV) chiral susceptibility1/2 (disconnected) T=200 MeV T=220 MeV T=260 MeV T=330 MeV 2*top.sus./m2

Contents

✔ 1. Introduction

We study Nf=2 QCD with chiral fermions at ~mphys, focusing on U(1) anomaly.

• 2. Lattice setup

Nf=2 QCD w/ MDWF and rewegihting overlap. at

T=190-330MeV near physical m~4MeV.

• 3. Numerical results
• Dirac spectrum has a peak but vanishes in the m→0 limit.
• Topology fluctuation is suppressed by ~m4.
• U(1) susceptibility goes down to (a few MeV)2.
• Meson 2quark correlators show a good U(1)A symmetry.
• Chiral susceptibility is dominated by axial U(1) anomaly.
• 4. Summary

✔ ✔

Summary

We study Nf=2 QCD with Mobius domain-wall and rewegihting overlap fermions at T=190-330MeV near physical m~4MeV. We observe disappearance of axial U(1) anomaly (~ a few MeV)

• Dirac spectrum has a peak but vanishes in the

m→0 limit.

• Topology fluctuation is suppressed by ~m4.
• U(1) susceptibility goes down to (a few MeV)2.
• Meson 2quark correlators show a good U(1)A

symmetry (mscreen difference ~ a few MeV).

• Chiral susceptibility is dominated by axial U(1)

anomaly (anomaly controls the phase transition?).

Outlook

Enhancement of symmetry to SU(4)? Polyakov loop Comparison with pQCD + instantons Axion dark matter Nf=2+1 QCD started! beta=4.17, L=32, Lt=12, T=204MeV m=0.002(almost physical),0.0035,0.007,0.012 ms =0.04(almost physical)