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

The 36th Annual International Symposium on Lattice Field Theory Kei Suzuki (KEK) from JLQCD Collaboration: Sinya Aoki (YITP), Yasumichi Aoki (KEK/RIKEN-BNL), Guido Cossu (Edinburgh), Hidenori Fukaya (Osaka U.), Shoji Hashimoto (KEK) 24/Jul/2018

𝑟𝑟 ~0 ത QCD phase diagram (for 𝑣, 𝑒, 𝑡 quarks) Phase transition （ crossover ） 𝑟 𝑟 ത 𝑟𝑟 ≠ 0 ത Chiral condensate （ chiral symmetry breaking ） 24/Jul/2018 Lattice 2018 2 2

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? ∞ 𝑒 4 𝑦 𝜌 𝑏 (𝑦)𝜌 𝑏 (𝑦) − 𝜀 𝑏 (𝑦)𝜀 𝑏 (𝑦) ∆ 𝜌−𝜀 = න 0 U(1) A breaking ？ 𝑟 𝑟 ത 𝑟𝑟 ത 𝑈 𝑑 24/Jul/2018 3

Cf.) S. Aoki, H. Fukaya, and Y. Taniguchi, PRD86 If U(1) A is restored… Colombia plot is modified? 𝑛 𝑑𝑠𝑗 (?) 𝑂 𝑔 = 2 world 𝑛 𝑡 → ∞ 𝑛 𝑣,𝑒,𝑡 → ∞ (Pure gauge) Conventionally, at 𝑛 𝑣,𝑒 → 0 , 2 nd with 𝑃(4) 1st 𝑔 = 1 world Critical line is shifted? 1 st order? crossover 2 nd order, not 𝑃(4) ? 𝑂 1st 𝑛 𝑣,𝑒,𝑡 → 0 𝑛 𝑣,𝑒 → ∞

U(1) A symmetry above Tc ⇒ Long-standing problem in QCD • 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 restoration by JLQCD Collaboration ⇒ overlap fermion (exact chiral symmetry on the lattice) valence/sea quark Setup G. Cossu et al. PRD87 OV on OV (2013) (Topology fixed sector) DW on DW A. Tomiya et al. PRD96 1/a=1.7GeV OV on DW (2017) (a=0.11fm) OV on (reweighted) OV 1/a=2.6GeV OV on DW In progress (a=0.076fm) OV on (reweighted) OV (Finer lattice) 24/Jul/2018 Lattice 2018 6

Outline 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 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത U(1) A 𝑟𝑟 ത 𝑅 𝑢 = 1 𝜇 ov 24/Jul/2018 Lattice 2018 7

Chiral condensate and Dirac spectra Banks-Casher relation: ∞ 2𝑛 𝑟𝑟 = lim ത 𝑛→0 න 𝑒𝜇 𝜍 𝜇 𝜇 2 + 𝑛 2 0 1 𝑊 Σ 𝜇 ′ < 𝜀 𝜇 − 𝜇 ′ > 𝜍 𝜇 ≡ lim 𝑊→∞ 𝜍 𝜇 with interaction 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത Chiral condensate induced by low modes 𝜍 𝜇 ~𝜇 3 𝜇 w/o interaction 𝜍 0 = − ത 𝑟𝑟 /𝜌 24/Jul/2018 Lattice 2018 8

G. Cossu et al. (JLQCD) PRD87 (2013), 114514 T-dependence of Dirac spectra Low T ： ρ(0)≠ 0 ⇒ Spontaneous chiral symmetry breaking 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത 𝑟𝑟 ത Critical Temp. High T ： ρ(0)=0 ⇒ Chiral symmetry restoration Low energy High energy

U(1) A susceptibility and low modes of Dirac spectra ∞ 2𝑛 2 ∆ 𝜌−𝜀 = න 𝑒𝜇 𝜍 𝜇 (𝜇 2 + 𝑛 2 ) 2 0 𝜍 𝜇 Low mode contribution is enhanced by the factor of 1/𝜇 4 𝜇 ∞ 2𝑛 𝑟𝑟 = lim ത 𝑛→0 න 𝑒𝜇 𝜍 𝜇 Cf.) Banks-Casher relation: 𝜇 2 + 𝑛 2 0 24/Jul/2018 Lattice 2018 10

S. Aoki, H. Fukaya, and Y. Taniguchi PRD86 (2012), 114512 A. Tomiya et al. (JLQCD) PRD96 (2017), 034509 Note １： U(1) A susc. ＝ Low modes ＋ Zero mode ？ ∞ 2𝑛 2 (𝑗)2 ) 2 2𝑛 2 (1 − 𝜇 ov 1 ∆ 𝜌−𝜀 = න 𝑒𝜇 𝜍 𝜇 ov ∆ 𝜌−𝜀 ≡ 𝑊(1 − 𝑛 2 ) 2 ෍ (𝜇 2 + 𝑛 2 ) 2 (𝑗)4 0 𝜇 ov 𝑗 𝜍 𝜇 ov integrated up to λ ＝ 0 The factor of 1/𝜇 4 enhances zero-mode contribution? subtracted zero mode In 𝑊 → ∞ limit, we know zero- 𝜇 ov mode contribution is suppressed: = 2𝑂 0 ov Δ 0−𝑛𝑝𝑒𝑓 𝑊𝑛 2 (∝ 1/ 𝑊) − 2𝑂 0 New order parameter: ov ov ഥ Δ 𝜌−𝜀 ≡ ∆ 𝜌−𝜀 𝑊𝑛 2 we subtract zero mode 24/Jul/2018 Lattice 2018 11

JLQCD, preliminary (2018) Note ２： U(1) A susc. ＝ Physics ＋ Ultraviolet divergence ？ ∞ 2𝑛 2 ov ∝ 𝑛 2 ln Λ + ⋯ ∆ 𝜌−𝜀 = න 𝑒𝜇 𝜍 𝜇 ∆ 𝜌−𝜀 (𝜇 2 + 𝑛 2 ) 2 0 The term depends on cutoff Λ and 𝜍 𝜇 ~𝜇 3 ~1/𝜇 4 valence quark mass 𝑛 We assume valence quark mass 𝜍 𝜇 ov dependence of ∆ 𝜌−𝜀 (for small m) : ∆ 𝜌−𝜀 (𝑛) = 𝑏 𝑛 2 + 𝑐 + 𝑑𝑛 2 + 𝑃(𝑛 4 ) 𝜇 ov 𝑛 2 ln Λ Zero-mode ≈ (disappears in 𝑊 → ∞ ) (disappears in m → 0 ) Λ ⇒ From 3 eqs. for ∆ 𝜌−𝜀 (𝑛 1 ) , ∆ 𝜌−𝜀 (𝑛 2 ) , ∆ 𝜌−𝜀 𝑛 3 , 𝑏 and 𝑑 are eliminated ⇒ ∆ 𝜌−𝜀 ~ 𝑐 + 𝑃(𝑛 4 ) (, that depends on sea quark mass) 24/Jul/2018 Lattice 2018 12

JLQCD, preliminary (2018) Overlap Dirac spectra at T = 220MeV 𝑛 𝑟 =2.6MeV Low modes suppressed 𝑛 𝑟 =26MeV Low modes enhanced 24/Jul/2018 13

JLQCD, preliminary (2018) U(1) A susceptibility at T = 220MeV ഥ Δ 𝜌−𝜀 is almost zero ⇒ In the chiral limit, U(1) A will be restored ⇒ At m=2.6MeV, we found suppression of 10 -4 GeV 2 24/Jul/2018 Lattice 2018 14

Large mass region ⇒ large ഥ Δ 𝜌−𝜀 by low mode enhancement Small mass region ⇒ small ഥ Δ 𝜌−𝜀 by low mode suppression

JLQCD, preliminary (2018) U(1) A susceptibility (UV-subt. before/after) ⇒ Ultraviolet divergence （ ~ 𝑛 2 ln Λ ） is subtracted from ഥ Δ 𝜌−𝜀 24/Jul/2018 Lattice 2018 16

JLQCD, preliminary (2018) U(1) A susceptibility (UV-subt. before/after) ⇒ Ultraviolet divergence （ ~ 𝑛 2 ln Λ ） is subtracted from ഥ Δ 𝜌−𝜀 24/Jul/2018 Lattice 2018 17

Did we really remove the ultraviolet contribution? Check of cutoff dependence ∞ 2𝑛 2 𝑗 2 ) 2 2𝑛 2 (1 − 𝜇 ov 1 ∆ 𝜌−𝜀 = න 𝑒𝜇 𝜍 𝜇 ov ∆ 𝜌−𝜀 ≡ 𝑊(1 − 𝑛 2 ) 2 ෍ (𝜇 2 + 𝑛 2 ) 2 (𝑗)4 0 𝜇 ov 𝑗 40th low modes 𝜍 𝜇 ov ・・・・・・・・ Lattice cutoff Λ ≈ 𝜇 ov To evaluate ഥ Δ 𝜌−𝜀 , we sum up 40 lowest modes ⇒ Cutoff dependence by the number of low modes 24/Jul/2018 Lattice 2018 18

JLQCD, preliminary (2018) U(1) A susceptibility (cutoff dependence) 40 modes 3 modes ⇒ No cutoff dependence （ saturated by a few low modes ） 24/Jul/2018 Lattice 2018 19

JLQCD, preliminary (2018) U(1) A susceptibility (volume effect) Finite V effect enhanced? 48 32 24 ⇒ For small m, V-dependence seems to be small 24/Jul/2018 Lattice 2018 20

JLQCD, preliminary (2018) U(1) A susceptibility (T=220, 330MeV) Low T High T ⇒ With increasing T, U(1) A is more resotored 24/Jul/2018 21

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 𝑛 𝑟 （ larger than 𝑛 𝑟 2 ? ） • Near 𝑈 𝑑 ( 𝑂 𝑢 = 14? , chiral transition?) • 𝑂 𝑔 = 2 + 1 sector 24/Jul/2018 Lattice 2018 22

Backup 24/Jul/2018 Lattice 2018 23

S. Aoki, H. Fukaya, and Y. Taniguchi PRD86 (2012), 114512 A. Tomiya et al. (JLQCD) PRD96 (2017), 034509 Note １： U(1) A susc. ＝ Low modes ＋ Zero mode ？ ∞ 2𝑛 2 ∆ 𝜌−𝜀 ≡ න 𝑒𝜇 𝜍 𝜇 (𝜇 2 + 𝑛 2 ) 2 0 𝜍 0−𝑛𝑝𝑒𝑓 𝜇 = 1 ෍ 𝜀(𝜇) 𝑊 0−𝑛𝑝𝑒𝑓 ∞ 2𝑛 2 𝑒𝜇 1 ∆ zero = න ෍ 𝜀(𝜇) (𝜇 2 + 𝑛 2 ) 2 𝑊 0 0−𝑛𝑝𝑒𝑓 2𝑛 2 = 1 ෍ 𝑛 4 𝑊 0−𝑛𝑝𝑒𝑓 = 1 𝑛 2 = 2𝑂 0 2 2 𝑂 𝑀+𝑆 = 𝒫 𝑊 𝑊→∞ ∆ zero = 0 lim ෍ 𝑊𝑛 2 𝑂 𝑀+𝑆 = 𝒫 𝑊 𝑊 0−𝑛𝑝𝑒𝑓 Zero mode contributions in ∆ 𝜌−𝜀 will be suppressed in 𝑊 → ∞ limit 24/Jul/2018 Lattice 2018 24

JLQCD, preliminary (2018) U(1) A susceptibility (DW/OV reweighting) ⇒ DW/OV reweighting is crucial in small m region 24/Jul/2018 Lattice 2018 25

JLQCD, preliminary (2018) 𝜍 𝜇 ov Histogram of topological charge zero mode at T = 220MeV 𝜇 ov 𝑛 𝑟 =2.6MeV 𝑛 𝑟 =10MeV After reweighting, Before After Q=1sector reweighting reweighting survives Large 𝑛 𝑟 : Q ≠ 0 sectors appear Small 𝑛 𝑟 : all conf. are Q ＝ 0 sector 2 𝜓 𝑢 ≡ 𝑅 𝑢 Using , we plot 𝜓 𝑢 𝑊 24/Jul/2018 Lattice 2018 26

JLQCD, preliminary (2018) Topological susceptibility at T = 220MeV Critical mass? ⇒ In small 𝑛 𝑟 region, 𝜓 𝑢 =0? ⇒ Around 𝑛 𝑟 ~10MeV, we found a jump (critical mass? ） 24/Jul/2018 Lattice 2018 27