Kei Suzuki (KEK) from JLQCD Collaboration: Sinya Aoki (YITP), - - PowerPoint PPT Presentation

The 36th Annual International Symposium on Lattice Field Theory

Kei Suzuki (KEK)

24/Jul/2018

from JLQCD Collaboration: Sinya Aoki (YITP), Yasumichi Aoki (KEK/RIKEN-BNL), Guido Cossu (Edinburgh), Hidenori Fukaya (Osaka U.), Shoji Hashimoto (KEK)

2

QCD phase diagram

(for 𝑣, 𝑒, 𝑡 quarks)

2

ത 𝑟 𝑟 ത 𝑟𝑟 ≠ 0

Phase transition

（crossover）

ത 𝑟𝑟 ~0

Chiral condensate （chiral symmetry breaking）

24/Jul/2018 Lattice 2018

U(1)A symmetry（in vacuum, broken by anomaly） is restored above Tc？

• Above Tc, chiral symmetry breaking by ത

𝑟𝑟 is restored ⇒How about U(1)A symmetry? ത 𝑟 𝑟 ത 𝑟𝑟

𝑈

𝑑

？

U(1)A breaking

∆𝜌−𝜀 = න

∞

𝑒4𝑦 𝜌𝑏(𝑦)𝜌𝑏(𝑦) − 𝜀𝑏(𝑦)𝜀𝑏(𝑦)

24/Jul/2018 3

𝑛𝑣,𝑒 → ∞

If U(1)A is restored…

Colombia plot is modified?

𝑂

𝑔 = 2 world Critical line is shifted?

𝑛𝑑𝑠𝑗(?) 𝑂

𝑔 = 1 world

1st 1st crossover

𝑛𝑣,𝑒,𝑡 → 0 𝑛𝑡 → ∞ 𝑛𝑣,𝑒,𝑡 → ∞

(Pure gauge)

Conventionally, at 𝑛𝑣,𝑒 → 0, 2nd with 𝑃(4) 1st order? 2nd order, not 𝑃(4)?

Cf.) S. Aoki, H. Fukaya, and Y. Taniguchi, PRD86

• Gross-Pisarski-Yaffe (Dilute instanton gas model, 1981) restored at enough high T
• Cohen (1996) w/o zero mode (or instanton)⇒restored
• Aoki-Fukaya-Taniguchi (2012) zero mode suppressed, restored in chiral limit at

𝑂

𝑔 = 2

• HotQCD (DW, 2012) broken
• JLQCD (topology fixed overlap, 2013) restored
• TWQCD (optimal DW, 2013) restored
• LLNL/RBC (DW, 2014) broken (restored at higher T?)
• Dick et al. (overlap on HISQ, 2015) broken
• Sharma et al. (overlap on DW, 2015,2016,2018) broken
• Brandt et al. (Wilson, 2016) restored
• Ishikawa et al. (Wilson, 2017) restored
• JLQCD (reweighted overlap on DW, 2017) restored
• Rohrhofer et al. (DW, 2017) restored

⇒ Many suggestions from lattice QCD (and models)…

U(1)A symmetry above Tc ⇒Long-standing problem in QCD

U(1)A symmetry restoration by JLQCD Collaboration

⇒overlap fermion (exact chiral symmetry on the lattice)

valence/sea quark Setup

• G. Cossu et al. PRD87

(2013) OV on OV (Topology fixed sector)

• A. Tomiya et al. PRD96

(2017) DW on DW OV on DW OV on (reweighted) OV 1/a=1.7GeV (a=0.11fm) In progress OV on DW OV on (reweighted) OV 1/a=2.6GeV (a=0.076fm) (Finer lattice)

24/Jul/2018 Lattice 2018 6

１．Introduction ２．U(1)A susceptibility from Dirac spectra （zero mode and ultraviolet divergence） ３．Results 3-1: U(1)A susceptibility at finite T 3-2: Cutoff and volume dependences

• 4. Summary

Outline

24/Jul/2018 Lattice 2018 7

𝜇ov

𝑅𝑢 = 1

ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟

U(1)A

Chiral condensate and Dirac spectra

𝜇 𝜍 𝜇

ത 𝑟𝑟 = lim

𝑛→0 න ∞

𝑒𝜇 𝜍 𝜇 2𝑛 𝜇2 + 𝑛2

Banks-Casher relation: Chiral condensate induced by low modes

24/Jul/2018 8 Lattice 2018

w/o interaction with interaction

𝜍 𝜇 ~𝜇3 ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟

𝜍 0 = − ത 𝑟𝑟 /𝜌 𝜍 𝜇 ≡ lim

𝑊→∞

1 𝑊 Σ𝜇′ < 𝜀 𝜇 − 𝜇′ >

T-dependence of Dirac spectra

• G. Cossu et al. (JLQCD) PRD87 (2013), 114514

Low T：

ρ(0)≠0 ⇒Spontaneous chiral symmetry breaking

High T：

ρ(0)=0 ⇒Chiral symmetry restoration Critical Temp. Low energy High energy

ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟

U(1)A susceptibility and low modes of Dirac spectra

𝜇 𝜍 𝜇

∆𝜌−𝜀 = න

∞

𝑒𝜇 𝜍 𝜇 2𝑛2 (𝜇2 + 𝑛2)2

ത 𝑟𝑟 = lim

𝑛→0 න ∞

𝑒𝜇 𝜍 𝜇 2𝑛 𝜇2 + 𝑛2 Cf.) Banks-Casher relation: Low mode contribution is enhanced by the factor of 1/𝜇4

24/Jul/2018 10 Lattice 2018

Note １：

U(1)A susc.＝Low modes＋Zero mode？

∆𝜌−𝜀 = න

∞

𝑒𝜇 𝜍 𝜇 2𝑛2 (𝜇2 + 𝑛2)2 ∆𝜌−𝜀

• v

≡ 1 𝑊(1 − 𝑛2)2 ෍

𝑗

2𝑛2(1 − 𝜇ov

(𝑗)2)2

𝜇ov

(𝑗)4

ഥ Δ𝜌−𝜀

• v

≡ ∆𝜌−𝜀

• v

− 2𝑂0 𝑊𝑛2 New order parameter: we subtract zero mode

𝜇ov 𝜍 𝜇ov

The factor of 1/𝜇4 enhances zero-mode contribution?

In 𝑊 → ∞ limit, we know zero- mode contribution is suppressed: Δ0−𝑛𝑝𝑒𝑓

• v

= 2𝑂0 𝑊𝑛2 (∝ 1/ 𝑊)

24/Jul/2018 11

integrated up to λ＝0 subtracted zero mode

Lattice 2018

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

Note ２：

U(1)A susc.＝Physics＋Ultraviolet divergence？

∆𝜌−𝜀 = න

∞

𝑒𝜇 𝜍 𝜇 2𝑛2 (𝜇2 + 𝑛2)2 ∆𝜌−𝜀

• v ∝ 𝑛2 ln Λ + ⋯

We assume valence quark mass dependence of ∆𝜌−𝜀 (for small m):

24/Jul/2018 12 Lattice 2018

∆𝜌−𝜀 (𝑛) = 𝑏 𝑛2 + 𝑐 + 𝑑𝑛2 + 𝑃(𝑛4) ⇒ From 3 eqs. for ∆𝜌−𝜀(𝑛1), ∆𝜌−𝜀(𝑛2), ∆𝜌−𝜀 𝑛3 , 𝑏 and 𝑑 are eliminated ⇒ ∆𝜌−𝜀~ 𝑐 + 𝑃(𝑛4) (, that depends on sea quark mass) 𝜍 𝜇 ~𝜇3 ~1/𝜇4 The term depends on cutoff Λ and valence quark mass 𝑛 Zero-mode

(disappears in 𝑊 → ∞)

𝑛2 ln Λ

(disappears in m → 0)

𝜇ov 𝜍 𝜇ov

≈

Λ

JLQCD, preliminary (2018)

Overlap Dirac spectra at T = 220MeV

Low modes suppressed

JLQCD, preliminary (2018)

Low modes enhanced

𝑛𝑟=2.6MeV 𝑛𝑟=26MeV

24/Jul/2018 13

U(1)A susceptibility at T = 220MeV

JLQCD, preliminary (2018)

14

ഥ Δ𝜌−𝜀 is almost zero ⇒In the chiral limit, U(1)A will be restored

⇒At m=2.6MeV, we found suppression of 10-4GeV2

24/Jul/2018 Lattice 2018

Small mass region

⇒small ഥ Δ𝜌−𝜀 by low mode suppression

Large mass region

⇒large ഥ Δ𝜌−𝜀 by low mode enhancement

U(1)A susceptibility (UV-subt. before/after)

JLQCD, preliminary (2018)

16

⇒Ultraviolet divergence （~𝑛2 ln Λ） is subtracted from ഥ Δ𝜌−𝜀

24/Jul/2018 Lattice 2018

U(1)A susceptibility (UV-subt. before/after)

JLQCD, preliminary (2018)

17

⇒Ultraviolet divergence （~𝑛2 ln Λ） is subtracted from ഥ Δ𝜌−𝜀

24/Jul/2018 Lattice 2018

∆𝜌−𝜀 = න

∞

𝑒𝜇 𝜍 𝜇 2𝑛2 (𝜇2 + 𝑛2)2 ∆𝜌−𝜀

• v

≡ 1 𝑊(1 − 𝑛2)2 ෍

𝑗

2𝑛2(1 − 𝜇ov

𝑗 2)2

𝜇ov

(𝑗)4

To evaluate ഥ Δ𝜌−𝜀, we sum up 40 lowest modes ⇒Cutoff dependence by the number of low modes

𝜇ov 𝜍 𝜇ov

18

40th low modes

Did we really remove the ultraviolet contribution?

Check of cutoff dependence

≈

・・・・・・・・ Lattice cutoff Λ

24/Jul/2018 Lattice 2018

U(1)A susceptibility (cutoff dependence)

JLQCD, preliminary (2018)

19

⇒No cutoff dependence （saturated by a few low modes） 40 modes

3 modes

24/Jul/2018 Lattice 2018

U(1)A susceptibility (volume effect)

JLQCD, preliminary (2018)

20

⇒For small m, V-dependence seems to be small 32 24

48

24/Jul/2018 Lattice 2018

Finite V effect enhanced?

U(1)A susceptibility (T=220, 330MeV)

⇒ With increasing T, U(1)A is more resotored

JLQCD, preliminary (2018)

21

Low T High T

24/Jul/2018

Summary and outlook

• In high-temperature phase

(𝑈 > 𝑈

𝑑) at 𝑂 𝑔 = 2, we

studied U(1)A susceptibility

• Strong suppression in the

chiral limit (for T=220-330MeV)

• Checked volume and

cutoff dependences

• Topological susceptibility ⇒ talk by Y. Aoki
• Parametrization as function of 𝑛𝑟（larger than 𝑛𝑟

2?）

• Near 𝑈

𝑑 (𝑂𝑢 = 14?, chiral transition?)

• 𝑂

𝑔 = 2 + 1 sector

24/Jul/2018 Lattice 2018 22

23

Backup

24/Jul/2018 Lattice 2018

Note １：

U(1)A susc.＝Low modes＋Zero mode？

∆𝜌−𝜀 ≡ න

∞

𝑒𝜇 𝜍 𝜇 2𝑛2 (𝜇2 + 𝑛2)2 ∆zero = න

∞

𝑒𝜇 1 𝑊 ෍

0−𝑛𝑝𝑒𝑓

𝜀(𝜇) 2𝑛2 (𝜇2 + 𝑛2)2 = 1 𝑊 ෍

0−𝑛𝑝𝑒𝑓

2𝑛2 𝑛4 = 1 𝑊 ෍

0−𝑛𝑝𝑒𝑓

2 𝑛2 = 2𝑂0 𝑊𝑛2 Zero mode contributions in ∆𝜌−𝜀 will be suppressed in 𝑊 → ∞ limit

24/Jul/2018 24 Lattice 2018

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

𝜍0−𝑛𝑝𝑒𝑓 𝜇 = 1 𝑊 ෍

0−𝑛𝑝𝑒𝑓

𝜀(𝜇)

𝑂𝑀+𝑆

2

= 𝒫 𝑊 𝑂𝑀+𝑆 = 𝒫 𝑊

lim

𝑊→∞ ∆zero = 0

U(1)A susceptibility (DW/OV reweighting)

JLQCD, preliminary (2018)

25

⇒DW/OV reweighting is crucial in small m region

24/Jul/2018 Lattice 2018

Histogram of topological charge

at T = 220MeV

JLQCD, preliminary (2018)

24/Jul/2018 Lattice 2018 26

Small 𝑛𝑟: all conf. are Q＝0 sector Large 𝑛𝑟: Q≠0 sectors appear

𝑛𝑟=2.6MeV 𝑛𝑟 =10MeV

After reweighting Before reweighting

𝜓𝑢 ≡ 𝑅𝑢

2

𝑊 Using , we plot 𝜓𝑢

After reweighting, Q=1sector survives

𝜇ov 𝜍 𝜇ov

zero mode

Topological susceptibility at T = 220MeV

JLQCD, preliminary (2018)

24/Jul/2018 Lattice 2018 27

⇒In small 𝑛𝑟 region, 𝜓𝑢=0? ⇒Around 𝑛𝑟~10MeV, we found a jump (critical mass?）

Critical mass?

Topological susceptibility

(Temperature dependence)

JLQCD, preliminary (2018)

24/Jul/2018 Lattice 2018 28

⇒With increasing T, 𝜓𝑢 becomes small Low T High T

Topological susceptibility

(Volume dependence)

JLQCD, preliminary (2018)

29

32 24

48

24/Jul/2018 Lattice 2018