QCD: axion - - PowerPoint PPT Presentation

有限温度QCD: 相転移、トポロジー、axion

青木保道 素粒子物理学の進展2018 @ 基研

• Aug. 9, 2018

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

1st order transition ? JLQCD: Lattice 2017

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

1st order transition ? make sense al a Pisarski & Wilczek JLQCD: Lattice 2017 U(1)A restoration suggested from y.a. measurement

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

JLQCD: Lattice 2017

physical ud

where is the physical ud mass point ? ud quark のみの世界の話です

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

JLQCD: Lattice 2017

physical ud

where is the physical ud mass point ? ud quark のみの世界の話です もし現実世界がこうであったら 宇宙初期に一次相転移 axion window が閉じる…

JLQCD members involved in recent finite temperature study

Sinya Aoki YA Guido Cossu Hidenori Fukaya
 Shoji Hashimoto Takashi Kaneko Kei Suzuki …

もくじ

• QCD相図: 理解の現状 (μ=0: zero chemical potential)
• 格子作用のいろいろ
• axion との関係
• Nf=2 JLQCD の結果を中心に
• topological susceptibility
• fate of the UA(1) symmetry
• Nf=2+1
• review of topological susceptibility

現在でも: Columbia Plot = 大方の人の理解 || 期待 mud ms ∞ ∞

クロスオーバー 一次転移 二次転移

[original Columbia plot: Brown et al 1990]

現在でも: Columbia Plot = 大方の人の理解 || 期待 mud ms ∞ ∞

physical pt. [original Columbia plot: Brown et al 1990]

Nf=2+1相図

• 連続極限で分かっていること
• Nf=0: 一次転移
• 右上隅はよく分かっている
• Nf=2+1 物理点: cross-over
• staggered (YA, Endrodi,

Fodor, Katz, Szabo: Nature 2006)

• 他の正則化でも反証なし
• 厳密なカイラル対称性を持つ

アプローチでは未踏

• その他の領域は未確定

mud ms ∞ ∞

physical pt.

QCD 有限温度相転移の理論: Nf=2+1 Lattice

• Nf=2+1 相図が完成すれば
• QCD の理解
• 物理点の相転移の存在、次数が分かる。
• 遠回りだが確実な方法
• 相境界(μ=0)の μ>0 への伸び方を調べる→(T,μ)臨界終点の研究へつなげる
• 大変重要／有用である → ポスト京 重点課題9 のプロジェクトのひとつ

mud ms ∞ ∞

physical pt.

まずは Nf=2

• Nf=2+1 physical pt. から遠い？
• ms ~100 MeV → ∞
• T=0 では s のあるなしは微細効果
• boundary の情報としては有用
• Nf=2
• Wilson, staggered: 未確定
• 厳密な格子カイラル対称性

➡U(1)A 回復を示唆[JLQCD16] ➡一次転移の可能性 → χt(m)に飛び? [Pisarski&Wilczek]

mud ms ∞ ∞

physical pt.

一次転移だとどうなるか？

• 0 ≤ mf < mc : 一次転移
• 一つの可能性として: 左下(Nf=3)の一次転移領域と繋がる
• 物理点への影響も考えられる
• 現状では staggered → 連続極限の結果のみ

mud ms ∞ ∞

physical pt.

mud ms ∞ ∞ ?

Columbia plot: direct search of PT / scaling

• 2nd order
• improved Wilson
• WHOT-QCD Lat2016 (O(4) scaling)
• Ejiri et al PRD 2016 [heavy many flavor]
• 1st oder
• imaginary μ → 0
• staggered Bonati et al PRD 2014
• Wilson Phillipsen et al PRD 2016

external parameter → phase boundary → point of interest

➡ detour the demanding region

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Columbia plot: direct search of PT / scaling

• 2nd order
• improved Wilson
• WHOT-QCD Lat2016 (O(4) scaling)
• Ejiri et al PRD 2016 [heavy many flavor]
• 1st oder
• imaginary μ → 0
• staggered Bonati et al PRD 2014
• Wilson Phillipsen et al PRD 2016

0.05 0.1 0.15 0.2 0.25

(am u,d)2/5

• 1
• 0.75
• 0.5
• 0.25

(µ/T)

2

first order second order B region region

Bonati et al

external parameter → phase boundary → point of interest

➡ detour the demanding region

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Columbia plot: direct search of PT / scaling

• 2nd order
• improved Wilson
• WHOT-QCD Lat2016 (O(4) scaling)
• Ejiri et al PRD 2016 [heavy many flavor]
• 1st oder
• imaginary μ → 0
• staggered Bonati et al PRD 2014
• Wilson Phillipsen et al PRD 2016

0.05 0.1 0.15 0.2 0.25

(am u,d)2/5

• 1
• 0.75
• 0.5
• 0.25

(µ/T)

2

first order second order B region region

Bonati et al

external parameter → phase boundary → point of interest

➡ detour the demanding region

problem not settled yet

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Nt = 1/(aT) =4 or 6 so far → far from continuum limit

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Nf=2+1 or 3

• either
• no PT found
• 1st order region
• shrinks as a→0

with both staggered and Wilson

• r even disappear ?
• for more information see eg
• Meyer Lattice 2015
• Ding Lattice 2016
• de Forcrand

“Surprises in the Columbia plot” (Lapland talk 2018)

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Nf=2+1 or 3

• either
• no PT found
• 1st order region
• shrinks as a→0

with both staggered and Wilson

• r even disappear ?
• for more information see eg
• Meyer Lattice 2015
• Ding Lattice 2016
• de Forcrand

“Surprises in the Columbia plot” (Lapland talk 2018)

Understanding of the diagram being changed a lot

格子作用いろいろ

現状良く行われる改良

• Wilson → improved version
• staggered → improved version
• domain wall fermion → “reweighting” to overlap [JLQCD]

U(1)B SU(Nf)V SU(Nf)A simulation cost Wilson ✓ ✓ × moderate staggered ✓ × U(1) cheep domain wall ✓ ✓ almost exact expensive

• verlap

✓ ✓ ✓ almost impossible

QCD and Lattice QCD

• Lattice QCD = QCD defined on discretized Euclidian space-time
• discreteness: lattice spacing = a ( ~0.1 fm ~ (2 GeV)-1 )
• eventually continuum limit: a → 0 needed
• put the system in finite 4d box : V = Ls3 x Lt
• eventually: V → ∞ needed
• able to put on the computer as a statistical system
• Z = Σ exp( −S ) → Monte Carlo simulation
• some symmetry is lost
• infinitesimal translation and rotation
• chiral: partially or completely lost
• expected to recover in the continuum lim. a → 0
• exact symmetry
• gauge !
• “chiral” for special discretization
• (close to) exact chiral symmetry crucial for some applications

ψ(n + ψ(n) Uµ(n) a

QCD and Lattice QCD

• Lattice QCD = QCD defined on discretized Euclidian space-time
• discreteness: lattice spacing = a ( ~0.1 fm ~ (2 GeV)-1 )
• continuum limit is needed: a → 0
• near the continuum limit
• lattice operators can be expanded in powers of a

ψ(n + ˆ µ) ψ(n) Uµ(n) a

O|LQCD = O|QCD + ac1O1 + a2c2O2 . . .

• for some operators in some lattice discretizations
• c1 = 0 automatically → effectively close to cont. lim.
• c1 = 0 by engineering = “improvements”
• most of the lattice actions used now → c1 = 0 or c1 ≃ 0
• However, the size of c2 term wildly varies among different actions

Nf=4: stout improved staggered [LatKMI collab.]

• t0 from Symanzik flow:
• a2(β=3.7)/a2(β=3.8) ≃ 1.3
• taste symmetry violation

0.01 0.02 0.03 0.04

mf a

1 1.2 1.4 1.6

t0 a

2

~30% difference in a

2

β=3.8 β=3.7

0.02 0.04 0.06 mf 0.1 0.2 Mπ

2

π5 π05 πi5 πi0 πij π0 πi πI 0.02 0.04 0.06 mf 0.1 0.2 0.3 Mπ

2

ξI ξi ξ4 ξiξj ξiξ4 ξ4ξ5 ξiξ5 ξ5

Nf=4, β=3.7

改良した作用でも SU(4)V の破れが大きい

T=0

HISQ

Nf=4 topological susceptibility [LatKMI collab.]

• normalized with t0
• x-axis: NG pion → taste singlet

0.05 0.1 0.15 0.2 0.25 0.3 Mπ

2 t0

0.0001 0.0002 0.0003 χtop t0

2

β=3.7 β=3.8

does not vanish in Mπ→0 ? better chiral behavior, and better a-scaling

0.05 0.1 0.15 0.2 0.25 0.3 MπI

2 t0

0.0001 0.0002 0.0003 χtop t0

2

β=3.7 β=3.8

(a la sChPT: see Billeter, DeTar, Osborn, PRD2004)

χt: O(a2)が巨大 EFT (sChPT)に頼れば改善するが、 第一原理計算の意義は？

T=0

Nf=2+1 domain wall fermion

[JLQCD: S.Aoki et al 2017, Nf=2+1 DWF]

T=0

2x108 4x108 6x108 8x108 1x109 10 20 30 40 50 χt(MeV4) mMSbar

ud (MeV)

β=4.17, ms=0.04 β=4.17, ms=0.03 β=4.17, ms=0.04 (L=48) β=4.35, ms=0.025 β=4.35, ms=0.018 β=4.47, ms=0.015 ChPT fit (β=4.17) ChPT fit (β>=4.35)

O(a2) error 制御可 ChPT matching 良好 DWF を使うべき!

転移はともかく、U(1)A 回復すると…

• ( U(1)A broken case: χt(T) ∝ m2 : m=u,d quark mass )

➡ χt ~ O(m4) Cohen ➡ χt |m=0 = 0 & dnχt / dmn|m=0 = 0 Aoki-Fukaya-Tanigchi ➡ χt = 0 for 0 ≤ m < mc → 実現？ →

• physical u,d で χt = 0 の可能性

mud ms ∞ ∞

physical pt.

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

r e fi n e d

topological susceptibility and axion mass

• Peccei Quinn mechanism for a solution to strong CP problem
• new complex pseudo scalar field to remedy the fine tuning problem of θ
• U(1) symmetry is spontaneously broken → axion
• effective potential tilted by chiral anomaly
• → gets mass through χ = topological susceptibility at θ=0
• Axion is a candidate of dark matter
• axion mass as a function of temperature mA(T) is a crucial information
• χ(T) of QCD @ θ=0 is the target quantity!

U(1)A 回復すると…

• axion cosmology scenario may fail for U(1)A restoration

due to vanishing / suppressed topological susceptivility

• χt |m=0 = 0 & dnχt / dmn|m=0 = 0 Aoki-Fukaya-Tanigchi

➡ χt = 0 for small non-zero m OR ➡ exponential decay for T>Tc

• axion mass and decay constant:

➡ axion window can possibly be closed

Kitano-Yamada JHEP [1506.00370]

χt(T) ∼

• mqΛ3

QCD,

T < Tc, m2

qΛ2 QCDe−2c(mq)T 2/T 2

c , T > Tc,

h c(mq) → ∞ as mq → 0,

s χt = m2

af2 a.

topological susceptibility, U(1)A:

• QCD相図の理解
• axion の可能性

に重要。 それらを順に見ていく。 得に断りの無いものは JLQCD による仕事 (最新結果はpreliminary)

topological susceptibility

Method

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry: mres = 0.05(3) MeV (β=4.3, Ls=16)
• Overlap: exact chiral symmetry
• Qt measurements
• global sum of the gluonic charge density (clover) after Wilson Flow (t≃t0)
• Overlap Index
• reweighting: before / after and above 2 meas. yield 4 χt values
• current main focus: 1/a = 2.6 GeV *** PRELIMINARY ***

χt = hQ2i V

susceptibility

Method

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry: mres = 0.05(3) MeV (β=4.3, Ls=16)
• Overlap: exact chiral symmetry
• Qt measurements
• global sum of the gluonic charge density (clover) after Wilson Flow (t≃t0)
• Overlap Index
• reweighting: before / after and above 2 meas. yield 4 χt values
• current main focus: 1/a = 2.6 GeV *** PRELIMINARY ***

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV-DW OV index on DW ensemble OV-OV OV index on OV ensemble

χt = hQ2i V

susceptibility

χt(mf) for Nf=2 T=220 MeV, 323

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV-DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

increased stat. from Lattice 2017 : ~30k traj.

χt(m) T=~220 MeV discretization effect

comparing 1/a=1.7 GeV and 1/a=2.6 GeV; ( (3.6fm)3 and (2.4fm)3 )

• OV-OV: better scaling
• GL-DW: large scaling violation for smaller m
• OV-OV: χt = 0 (within error) for 0 ≤ m ≲ 10 MeV
• GL-DW: χt > 0, but, may well decrease as a

➡ (consistent with OV-OV with large error of OV-OV)

5 10 15 20 25 30 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

2.0×10

8

χt [MeV

4]

β=4.1 GL-DW β=4.3 GL-DW β=4.1 OV-OV β=4.3 OV-OV

compare Nt=8(β=4.1) and 12(4.3) at simlar temperature (217 and 220 MeV)

5 10 15 m [MeV] 0.0 2.0×10

7

4.0×10

7

χt [MeV

4]

β=4.1 GL-DW β=4.3 GL-DW β=4.1 OV-OV β=4.3 OV-OV

compare Nt=8(β=4.1) and 12(4.3) at simlar temperature (217 and 220 MeV)

χt(m) T=220 MeV a2 scaling: m=6.6 MeV

continuum scaling in 1st region

• m=6.6 MeV
• vanishing towards continuum limit
• caveat: physical volume is different → needs further invest.

( V=(3.6fm)3 and (2.4fm)3 )

0.005 0.01 0.015

a

2 [fm 2] 1e+07 2e+07 3e+07 4e+07 5e+07

χt [MeV

4] GL-DW OV-OV

m=6.6MeV

χt(m) T=~220 MeV, 323x12

1/a=2.6 GeV

suggesting 2 regions 1: χt is very small (may vanish in a→0): 0 ≤ m ≲ 10 MeV (→ consistent w/ Aoki-Fukaya-Tanigchi for U(1)A symm.) 2: sudden growth of χt : 10 MeV ≲ m

• physical ud mass point: m≃4 MeV

physical ud

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

5 10 15 mf [MeV] 2e+07 4e+07 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV-DW OV index on DW ensemble OV-OV OV index on OV ensemble

χt(m) T=~220 MeV, 323x12

1/a=2.6 GeV

suggesting 2 regions 1: χt is very small (may vanish in a→0): 0 ≤ m ≲ 10 MeV (→ consistent w/ Aoki-Fukaya-Tanigchi for U(1)A symm.) 2: sudden growth of χt : 10 MeV ≲ m

• physical ud mass point: m≃4 MeV

physical ud

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

5 10 15 mf [MeV] 2e+07 4e+07 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV-DW OV index on DW ensemble OV-OV OV index on OV ensemble

a→0

χt(m) T=~220 MeV, 323x12

1/a=2.6 GeV

suggesting 2 regions 1: χt is very small (may vanish in a→0): 0 ≤ m ≲ 10 MeV (→ consistent w/ Aoki-Fukaya-Tanigchi for U(1)A symm.) 2: sudden growth of χt : 10 MeV ≲ m

• physical ud mass point: m≃4 MeV

physical ud

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

5 10 15 mf [MeV] 2e+07 4e+07 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

transition suggested ?

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV-DW OV index on DW ensemble OV-OV OV index on OV ensemble

from both GL-DW & OV-OV a→0

χt(m) T=~220 MeV, 323x12

1/a=2.6 GeV

suggesting 2 regions 1: χt is very small (may vanish in a→0): 0 ≤ m ≲ 10 MeV (→ consistent w/ Aoki-Fukaya-Tanigchi for U(1)A symm.) 2: sudden growth of χt : 10 MeV ≲ m

• physical ud mass point: m≃4 MeV

physical ud

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

5 10 15 mf [MeV] 2e+07 4e+07 χ [MeV

4]

GL-DW OV-OV

32

3x12, β=4.3

transition suggested ?

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV-DW OV index on DW ensemble OV-OV OV index on OV ensemble

Next step: Volume Study In addition to 323, 243 & 483 are studied from both GL-DW & OV-OV a→0 GL-DW is precise, maybe useful

323 m=2.6 MeV history and histogram

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32

5000 10000 15000 20000 25000 trajectory

• 2
• 1

1 2 Qt GL-DW OV-DW

β=4.3, 32

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram GL-DW OV-DW

32

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram GL-DW GL-OV

gluonic

32

resolution of susceptibility (ex: m=2.6 MeV)

null measurement of topological excitation after reweighting

• does not readily mean χt=0:

(this case <Q2>=4(4) x10-6 ↔ 6(3) x10-3 @m=13MeV)

• there must be a resolution of χt under given statistics
• [resolution of <Q2>] = 1/Neff
• shall take the “statistical error” of <Q2> = max(Δ<Q2>, 1/Neff)

Effective number of statistics

• decreases with reweighting
• Neff=Nconf <R>/Rmax
• Nconf=1326 → Neff = 32
• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32

Results of χt(m) at T=220 MeV; multiple volume

• Statistics in trajectory

~30k, 30k, 10k

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

OV-OV

Results of χt(m) at T=220 MeV; multiple volume

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

OV-OV

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

GL-DW

Statistics in trajectory ~30k, 30k, 10k

Results of χt(m) at T=220 MeV; multiple volume

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

OV-OV

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

GL-DW

Statistics in trajectory ~30k, 30k, 10k

• V dependence at m=10 MeV is strange
• non-monotonic: cannot take thermodynamic limit
• important region, where a phase boundary was suggested w/ 323
• Let’s look at the histogram of Q

summary of histogram: T=220 MeV,

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram GL-DW GL-OV

gluonic

32

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram GL-DW GL-OV

gluonic

24

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram GL-DW GL-OV

gluonic

48

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

24

• 3
• 2
• 1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

48

323x12 243x12 483x12

m=10 MeV

# trajectory: sample rate: ~30k ~30k ~10k 100 20 100

low stat. low stat.

competing scenarios with multiple volumes for χt given Δπ-δ (UA(1) oder parameter) @ T=220 MeV

• AFK scenario: χt= 0 for 0<m<mc
• KY scenario: χt= 2 fA m2
• There are no strong tensions
• Neither scenario is excluded

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

KY scenario from ∆π−δ (m=3MeV) AFT scenario

OV-OV

5 10 15 20 25 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

24

3

32

3

48

3

KY scenario from ∆π−δ (m=3MeV) AFT scenario

GL-DW

Kanazawa-Yamamoto

• assume fA≠ 0 (breaking param)
• expansing free energy in m
• discussing
• finite m and V effect
• in each topological sector

U(1)A

U(1) axial

• violated by quantum anomaly

up to contact terms

• at T=0, responsible for η’ mass
• non-trivial topology of gauge field
• at high T, this Ward-Takahashi identity is still valid
• however, if configurations that contribute to RHS is suppressed………

➡ the symmetry effectively recovers ๏ here Nf=2 (including Nf=2+1 with “2” driven to chiral limit)

∂µJµ

5 = Nf

32π2 F ˜ F

h∂µJµ

5 (x) · O(0)i = Nf

32π2 hF ˜ F(x) · O(0)i

Why bother ?

• Because it is unsettled problem !
• fate of U(1)A - analytic
• Gross-Pisarski-Yaffe (1981) restores in high temperature limit
• Dilute instanton gas
• Cohen (1996)
• measure zero instanton effect → restores
• Lee-Hatsuda (1996)
• zero mode does contributes → broken
• Aoki-Fukaya-Tanigchi (2012)
• QCD analysis (overlap) → restores w/ assumption (lattice)
• Kanazawa-Yamamoto (2015)
• EFT case study how restore / break
• Azcoiti (2017)
• case study how restore / break

Why bother ?

• Because it is unsettled problem !
• fate of U(1)A lattice
• HotQCD (DW, 2012) broken
• JLQCD (topology fixed overlap, 2013) restores
• TWQCD (optimal DW, 2013) restores ?
• LLNL/RBC (DW, 2014) broken
• HotQCD (DW, 2014) broken
• Dick et al. (overlap on HISQ, 2015) broken
• Brandt et al. (O(a) improved Wilson 2016) restores
• JLQCD (reweighted overlap from DW, 2016) restores
• JLQCD (current: see Suzuki et al Lattice 2017) restores
• Ishikawa et al (Wilson, 2017) at least Z4 restores

U(1)A restoration or not

• need to make sure if not comparing apples and oranges…
• key points
• systematics effects of lattice discretization under control ?
• finite V, a, m…
• ud chiral limit of
• Nf=2 QCD or
• Nf=2+1 QCD → strange quark mass effect !
• discussing mud→0 or just around physical ud mass
• discussing X = 0 ? or X ≃ 0 ?

a U(1)A order parameter

• symmetry in switching flavor non-singlet pseudoscalar and

scalar

• rder parameter:

➡ 0 for U(1)A restoration

• as a result, screening masses for these channel will degenerate
• not a sufficient condition for U(1)A restoration

Δπ−δ ¼ Z d4x½hπaðxÞπað0Þi − hδaðxÞδað0Þi;

π δ

τ 2

: qγ5 q : q τ

2 q

: q : qγ5 σ η

L R

SU(2) x SU(2) SU(2) x SU(2)

L R

U(1)A U(1)A

q q

5,con

screening mass from O(a) improved Wilson f Nf=2

• mass difference between π and δ
• 1200
• 1000
• 800
• 600
• 400
• 200

200 10 20 30 40 50 ∆MP S [MeV] mud [MeV] T = TC linear chiral extrapolation T = 0 physical point T = 0 chiral limit

• Nt = 1/(aT) = 16 - quite fine lattice
• T=Tc - on top of transition temperature
• nly one existing study for Nf=2
• ΔMPS = 0 (with a sizable error) → consistent with U(1)A restoration

Brandt et al JHEP [1608.06882]

relation with Dirac eigenmode spectrum ρ(λ)

• chiral condensate : order parameter of SU(2)A : Banks-Casher rel.
• U(1)A:

very roughly speaking

• very sensitive to the spectrum near λ=0
• overlap fermion, able to distinguish zero/nonzero modes, is ideal

hqqi = lim

m→0

Z ∞ dλρ(λ) 2m λ2 + m2 = πρ(0)

∆πδ = Z 1 dλρ(λ) 2m2 (λ2 + m2)2 →∼ ρ0(0)

JLQCD 16: Hov, HDW spectrum: above Tc Nf=2

[JLQCD 2016 Tomiya et al]

• DW: Domain wall fermion sea
• OV: Overlap valence
• exact “chiral symmetry”
• reweighting to OV

JLQCD 16: Hov, HDW spectrum: above Tc Nf=2

Lowest bin→0 consistent with SUA(2) restoration

[JLQCD 2016 Tomiya et al]

Δ~0 OV on DW OV on OV DW on DW

• DW: Domain wall fermion sea
• OV: Overlap valence
• exact “chiral symmetry”
• reweighting to OV

Comparison: unitary <-> partially quench

Δ>0 Δ~0

range of JLQCD

OV on DW OV on HISQ OV on OV

Dick et al PRD [1502.06190]

Partially quench effect needs to be investigated

JLQCD 16: UA(1) susceptibility: T=190-220 MeV

¯ Δov

π−δ ≡ Δov π−δ − 2N0

Vm2 : ð

zero mode effect

ð Þ

seemingly Δ→0

HotQCD 2014: DWF Nf=2+1

50 100 150 200 250 130 140 150 160 170 180 190 200 T [MeV] (χMS

π - χMS σ )/T2

mπ=135 MeV mπ=200 MeV 50 100 150 200 250 130 140 150 160 170 180 190 200 T [MeV] (χMS

π - χMS δ )/T2

mπ=135 MeV mπ=200 MeV

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

χπ − χσ χη− χδ χπ − χδ χσ− χη

SU(2)A U(1)A

[figures from Ding Lattice 2016]

体積研究は？

π δ

τ 2

: qγ5 q : q τ

2 q

: q : qγ5 σ η

L R

SU(2) x SU(2) SU(2) x SU(2)

L R

U(1)A U(1)A

q q

5,con

JLQCD 16: UA(1) susceptibility

is this showing really, exactly Δ→0 ?

ð Þ

seemingly Δ→0 update available closer to continuum limit

チェック事項: V→∞ 極限を取れるか

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Δ V

[JLQCD 2016 Tomiya et al]

JLQCD: zero-mode subtracted

チェック事項: V→∞ 極限を取れるか

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Δ V

[JLQCD 2016 Tomiya et al]

JLQCD: zero-mode subtracted 2 1 6 J L Q C D : V → ∞ 取 れ な い U ( 1 )

A

r e s t

• r

e : 結 論 で き な い

チェック事項: V→∞ 極限を取れるか

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Δ V

[JLQCD 2016 Tomiya et al]

JLQCD: zero-mode subtracted 2 1 6 J L Q C D : V → ∞ 取 れ な い U ( 1 )

A

r e s t

• r

e : 結 論 で き な い r e w e i g h t i n g に 近 似 : 信 頼 度 ?

チェック事項: V→∞ 極限を取れるか

x ) )

• x = 2V4fAm2

fix V: Δ→0 as m2 for m→0 even for U(1)A br. case fix m : Δ∝V

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Δ V

[JLQCD 2016 Tomiya et al]

KY: without |Q|>0 sector JLQCD: zero-mode subtracted 必ずしも一致する必要ない しかし x→0 の極限では似たような物 2 1 6 J L Q C D : V → ∞ 取 れ な い U ( 1 )

A

r e s t

• r

e : 結 論 で き な い r e w e i g h t i n g に 近 似 : 信 頼 度 ?

チェック事項: V→∞ 極限を取れるか

x ) )

• x = 2V4fAm2

fix V: Δ→0 as m2 for m→0 even for U(1)A br. case fix m : Δ∝V

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Δ V

[JLQCD 2016 Tomiya et al]

KY: without |Q|>0 sector JLQCD: zero-mode subtracted 必ずしも一致する必要ない しかし x→0 の極限では似たような物 2 1 6 J L Q C D : V → ∞ 取 れ な い U ( 1 )

A

r e s t

• r

e : 結 論 で き な い r e w e i g h t i n g に 近 似 : 信 頼 度 ? 最新のJLQCD では reweighting は exact

Note

U(1)A susc.Low modesZero mode

∆"#$ = ' () * ) 2,- ()- + ,-)-

1 2

∆"#$

34 ≡

1 7(1 − ,-)- 9 2,-(1 − )34

(:)-)-

)34

(:);

• :

Δ >"#$

34

≡ ∆"#$

34

− 2?2 7,- New order parameter: we subtract zero mode

)34 * )34

The factor of 1/); enhances zero-mode contribution?

In 7 → ∞ limit, we know zero- mode contribution is suppressed: Δ2#CDEF

34

= 2?2 7,- (∝ 1/ 7

• )

24/Jul/2018 11

integrated up to λ0 subtracted zero mode

Lattice 2018

• S. Aoki, H. Fukaya, and Y. Taniguchi PRD86 (2012), 114512
• A. Tomiya et al. (JLQCD) PRD96 (2017), 034509

Note

U(1)A susc.PhysicsUltraviolet divergence

∆"#$ = ' () * ) 2,- ()- + ,-)-

1 2

∆"#$

34 ∝ ,- ln Λ + ⋯

We assume valence quark mass dependence of ∆"#$ (for small m):

24/Jul/2018 12 Lattice 2018

∆"#$ (,) = : ,- + ; + <,- + =(,>) From 3 eqs. for ∆"#$(,?), ∆"#$(,-), ∆"#$ ,@ , : and < are eliminated ∆"#$~ ; + =(,>) (, that depends on sea quark mass) * ) ~)@ ~1/)> The term depends on cutoff Λ and valence quark mass , Zero-mode

(disappears in D → ∞)

,- ln Λ

(disappears in m → 0)

)34 * )34

≈

Λ

JLQCD, preliminary (2018)

U(1)A susceptibility Nf=2 [JLQCD preliminary]

48

32 24

seemingly vanishing as m→0

Suzuki, Lattice 2018 physical ud

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 50 100 150 200 250 ρ(|λ|) [GeV3] Overlap Dirac eigenvalue |λ| [MeV] 323x12, β=4.3, T=220MeV, m=0.001(2.64MeV)

• n DW
• n OV

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 50 100 150 200 250 ρ(|λ|) [GeV3] Overlap Dirac eigenvalue |λ| [MeV] 323x12, β=4.3, T=220MeV, m=0.01(26.4MeV)

• n DW
• n OV

U(1)A susceptibility Nf=2 [JLQCD preliminary]

seemingly vanishing as m→0, more evident in log-log prot

U(1)A susceptibility Nf=2 [JLQCD preliminary]

seemingly vanishing as m→0, more evident in log-log prot ただし UV subtraction が finite V effect も引いていないかは 精査する必要あり to be continued … その精査を通過したら この結果は U(1)A 回復を示している

もう一つの見方？

• Because it is unsettled problem !
• fate of U(1)A lattice
• HotQCD (DW, 2012) broken
• JLQCD (topology fixed overlap, 2013) restores
• TWQCD (optimal DW, 2013) restores ?
• LLNL/RBC (DW, 2014) broken
• HotQCD (DW, 2014) broken
• Dick et al. (overlap on HISQ, 2015) broken
• Brandt et al. (O(a) improved Wilson 2016) restores
• JLQCD (reweighted overlap from DW, 2016) restores
• JLQCD (current: see Suzuki et al Lattice 2018) restores ?
• Ishikawa et al (Wilson, 2017) at least Z4 restores

2+1 2+1 2+1 2+1

Nf

2 2 2 2 2 2

ここまでのまとめ

• topological susceptibility
• T>Tc でゼロの可能性: 結論出ず
• 相転移の有無: 結論出ず
• fate of U(1)A
• T>Tc で回復するか: 結論出ず
• しかし、より連続極限に近い格子で、より精密な手法を開発
• 更なる研究が必要: そもそも簡単な問題ではない
• 今後
• 現状の統計で 様々な解析手法を使い調査継続
• subtraction の理解 (得に個人的)
• parameter の変更により、より見やすい所を追跡: Tc 近傍など
• T=220 MeV → 180 MeV (> Tc chiral transition)

U(1)A @ Nf=2+1 (+1) その他のグループ

references

• topological susceptibility for axion mass
• 1606.07494, S. Borsanyi et al, (Budapest-Wuppertal), Nature
• “Calculation of the axion mass based on high-temperature lattice

quantum chromodynamics”

• 1606.07175, J.Frison, R.Kitano, H.Matsufuru, S.Mori, N.Yamada
• “Topological susceptibility at high temperature on the lattice”
• crucial technique of above

simulation parameters and integral path

• T direction: integrate
• m-direction:

10 20 30 100 200 500 1000 2000 5000 nf=3+1 nf=2+1+1 phys mud/mud

phys

T[MeV] staggered staggered fix Q

• verlap fix Q
• Z1

Z0

• 2+1+1

= exp Z mphys

s

mphys

ud

d log mud mudhψψudi1−0 ! · Z1 Z0

• 3+1
• bQ ⌘ d log ZQ/Z0

d log T = dβ d log ahSgiQ−0 + X

f

d log mf d log a mfhψψfiQ−0.

• starting from 3+1 at T=300 MeV

↓ use Q=1

possible (our choice is given ion = −

−

O O O

Q Q

• bservable between sectors Q

• ther key methodologies
• “reweighting” of staggered simulations to better ones with small O(a2)
• lowest modes engineering :
• would induce non-local term in the action → similar to 4th root ?
• isospin breaking effect
• finite volume effect
• charm quark effect
• line of constant physics: mud(β), ms(β), mc(β), a(β); β=6/g2
• systematic error associated with the ambiguous definition of Q from Gluonic
• …

↖この操作の正当性が大きな問題

the result and comparison with other works

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 200 500 χ[fm-4] T[MeV] 1512.06746 1606.03145 χt 1606.03145 m2χdisc this work

the result and comparison with other works

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 200 500 χ[fm-4] T[MeV] 1512.06746 1606.03145 χt 1606.03145 m2χdisc this work

• he direct determination of χ(T) all the way

up to 3 GeV means that we do not have to rely on the dilute instanton gas approximation (DIGA). Note that a posteriori the exponent predicted by DIGA turned out to be compatible with

• ur finding, but its prefactor is off by an
• rder of magnitude, similar to the quenched

case. Though some of our simulations (see Supplementary Fig. 18) are already carried

• ut with chiral (overlap) fermions, where

large cut-off effects are a priori absent, it is an important task for the future to crosscheck these results with a calculation using chiral fermions only.

the result and comparison with other works

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 200 500 χ[fm-4] T[MeV] 1512.06746 1606.03145 χt 1606.03145 m2χdisc this work

• Bonati et al 2015: outlier ?

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

a

2 [fm 2]

10 20 30 40 50

χ

1/4[MeV]

Borsanyi et al. Bonati et al.

Bonati et al 2018 new now, consistent T=430 MeV Note: these are all based on staggered DW or overlap での検証必要

Thank you very much for your attention !

Lattice framework

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

• Overlap: exact chiral symmetry
• DW→OV reweighting

Lattice framework

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

• Overlap: exact chiral symmetry
• DW→OV reweighting

hOiov ¼ hORiDW hRiDW ;

Lattice framework

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

• Overlap: exact chiral symmetry
• DW→OV reweighting

hOiov ¼ hORiDW hRiDW ;

R ≡ det½HovðmÞ2 det½H4D

DWðmÞ2 × det½H4D DWð1=4aÞ2

det½Hovð1=4aÞ2 :

Lattice framework

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

• Overlap: exact chiral symmetry
• DW→OV reweighting

Dov = 1 2

• λi<λth

(1 + γ5sgnλi) |λi⇤⇥λi|

• Exact low modes

+D4D

DW

• 1
• λi<λth

|λi⇤⇥λi|

• High modes

,

hOiov ¼ hORiDW hRiDW ;

R ≡ det½HovðmÞ2 det½H4D

DWðmÞ2 × det½H4D DWð1=4aÞ2

det½Hovð1=4aÞ2 :

Lattice framework

• DWF ensemble → reweighted to overlap
• Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

• Overlap: exact chiral symmetry
• DW→OV reweighting

Dov = 1 2

• λi<λth

(1 + γ5sgnλi) |λi⇤⇥λi|

• Exact low modes

+D4D

DW

• 1
• λi<λth

|λi⇤⇥λi|

• High modes

,

hOiov ¼ hORiDW hRiDW ;

R ≡ det½HovðmÞ2 det½H4D

DWðmÞ2 × det½H4D DWð1=4aÞ2

det½Hovð1=4aÞ2 :

HM ¼ γ5 αDW 2 þ DW ;

λ for

simply speaking, in the m→0 limit

• U(1)A restores if
• and not if

ρ(λ)

with ρ(0)→0 and ρ’(0)→0

ρ(λ)

with ρ(0)→0 and ρ’(0)≠ 0 non-analyticity at λ→0 required

Analytic works

• Aoki-Fukaya-Taniguchi
• QCD with OV regulator
• assuming analyticity of ρ(0)
• fA → 0 : U(1)A br. parameter
• χtop= 0 for 0<m<mc
• Kanazawa-Yamamoto
• assuming fA≠ 0
• expansing free energy in m
• discussing
• finite m and V effect
• contributions of topological

sectors

Kanazawa - Yamamoto

• assuming fA≠ 0
• expansing free energy in m

Z(T, V3, M) = exp

• −V3

T f(T, V3, M)

• ,

f(T, V3, M) = f0 − f2 tr M†M − fA(det M + det M†) + O(M4) ,

represents the effect of axial as det M → e4iθA det M

breaks U(1)A

• ther terms are invariant under U(1)A

all invariant under SU(2)LxR

M → e−2iθAVLMV †

R

• logical sectors. As is w

via M → M eiθ/Nf

• to study topological sectors

ZQ(T, V3, M) ≡ dθ 2π e−iQθ Z(T, V3, Meiθ/2). = e−V4[f0−f2(m2

u+m2 d)]

dθ 2π e−iQθ e2V4fAmumd cos θ = e−V4[f0−f2(m2

u+m2 d)] IQ(2V4fAmumd) ,

∆π−δ =

∞

X

Q=−∞

ZQ Z PQ PQ = 8fA

I′

Q(2V4fAm2)

IQ(2V4fAm2)

Kanazawa - Yamamoto: U(1)A br. scenario

• logical sectors. As is w

via M → M eiθ/Nf

• to study topological sectors

ZQ(T, V3, M) ≡ dθ 2π e−iQθ Z(T, V3, Meiθ/2). = e−V4[f0−f2(m2

u+m2 d)]

dθ 2π e−iQθ e2V4fAmumd cos θ = e−V4[f0−f2(m2

u+m2 d)] IQ(2V4fAmumd) ,

∆π−δ =

∞

X

Q=−∞

ZQ Z PQ PQ = 8fA

I′

Q(2V4fAm2)

IQ(2V4fAm2)

x = 2V4fAm2

relative contribution of modes

x ) )

• ∆|Q=0

∆|full

Kanazawa - Yamamoto: U(1)A br. scenario

• logical sectors. As is w

via M → M eiθ/Nf

• to study topological sectors

ZQ(T, V3, M) ≡ dθ 2π e−iQθ Z(T, V3, Meiθ/2). = e−V4[f0−f2(m2

u+m2 d)]

dθ 2π e−iQθ e2V4fAmumd cos θ = e−V4[f0−f2(m2

u+m2 d)] IQ(2V4fAmumd) ,

∆π−δ =

∞

X

Q=−∞

ZQ Z PQ PQ = 8fA

I′

Q(2V4fAm2)

IQ(2V4fAm2)

x = 2V4fAm2

relative contribution of modes

x ) )

• ∆|Q=0

∆|full

KY tells

• fixed topology gives wrong result at small V
• adding all Q sector or large enough volume necessary

Kanazawa - Yamamoto: U(1)A br. scenario

• logical sectors. As is w

via M → M eiθ/Nf

• to study topological sectors

ZQ(T, V3, M) ≡ dθ 2π e−iQθ Z(T, V3, Meiθ/2). = e−V4[f0−f2(m2

u+m2 d)]

dθ 2π e−iQθ e2V4fAmumd cos θ = e−V4[f0−f2(m2

u+m2 d)] IQ(2V4fAmumd) ,

∆π−δ =

∞

X

Q=−∞

ZQ Z PQ PQ = 8fA

I′

Q(2V4fAm2)

IQ(2V4fAm2)

x = 2V4fAm2

relative contribution of modes

x ) )

• ∆|Q=0

∆|full

KY tells

• fixed topology gives wrong result at small V
• adding all Q sector or large enough volume necessary

JLQCD

• does not fix topology (DW)
• zero-mode subtraction may have similar effect to fix Q=0
• for smallest m: actually effectively fixed to Q=0