Nf=3 QCD 相図 中村 宜文 理化学研究所 計算科学研究機構 ２０１５年９月５日 「有限温度密度系の物理と格子ＱＣＤシミュレーション」研究会 1 / 40
共同研究者 • 金 暁勇（アルゴンヌ国立研究所） • 藏増 嘉伸（筑波大学 / 理化学研究所） • 武田 真滋（金沢大学 / 理化学研究所） • 宇川 彰（理化学研究所） 2 / 40
内容 • QCD の相図の概観 • phase diagram for N f = 3 , µ = 0 • Critical endpoint of finite temperature phase transition for three flavor QCD, Phys. Rev. D 91, 014508 (2015) • phase diagram for N f = 3 , µ � 0 • Curvature of the critical line on the plane of quark chemical potential and pseudo scalar meson mass for three-flavor QCD, arXiv:1504.00113[hep-lat] 3 / 40
QCD の相図の概観 4 / 40
有限温度有限密度（物理点） 初期宇宙 クォークグルーオンプラズマ相 温度 重イオン衝突実験 臨界終点 ハドロン相 カラー超伝導相 中性子星 陽子・中性子・・ 密度 5 / 40
3D φ 4 theory g g O(4) O(4) T T O(4) 1st order U(2)xU(2)/U(2) O(4) strong U ( 1 ) A anomaly weak U ( 1 ) A anomaly Basile et al. (hep-lat/0509018) 6 / 40
Columbia plot （ possible ） standard scenario alternative scenario (weak U ( 1 ) A anomaly) Kanaya (arXiv:1012.4247[hep-lat]) 7 / 40
phase diagram for N f = 3 , µ = 0 8 / 40
Motivation • Critical endpoint (CEP) obtained with staggered and Wilson type fermions is inconsistent. → Results in the continuum limit is necessary m π at the endpoint at µ = 0 (bottom-left corner of Columbia plot) m E N t action π [MeV] 4 unimproved staggered 260 de Forcrand, 6 unimproved staggered 150 Philipsen ’07 4 p4-improved staggered 70 Karsch et al. ’03 6 stout-improved staggered ≲ 50 Endr˝ odi et al. ’07 6 HISQ ≲ 50 Ding et al. ’11,...’15 4 unimproved Wilson ∼ 1100 Iwasaki et al. ’96 • N f = 3 study is a stepping stone • curvature of critical surface • to the physical point We determine CEP on m l = m s line with clover fermions in the continuum limit 9 / 40
crossover physical surface world 2nd order line temperature chemical potential quark mass s i g n p r o s e b 1st order r l e i m o u s transition surface 10 / 40
crossover physical surface world 2nd order line temperature chemical potential quark mass s i g n p r o s e b 1st order r l e i m o u s transition surface 10 / 40
crossover physical surface world 2nd order line temperature chemical potential quark mass s i g n p r o s e b 1st order r l e i m o u s transition surface 10 / 40
Distinguishing between 1st, 2nd and crossover criterion first order second order crossover distribution double peak single peak singe peak ∝ N γ/ν ∝ N d χ peak - l l ∝ N − 1 /ν ∝ N − d β ( χ peak ) − β c - l l kurtosis at N l → ∞ K= -2 -2 < K < 0 - • scaling might work with wrong exponents near CEP • peaks in histgram might emerge only at large N l on weak 1st order • K does not depend on volume at 2nd order phase transition point M = N − β/ν f M ( tN 1 /ν ) l l N − 4 β/ν f M 4 ( tN 1 /ν ) l l ] 2 = f B ( tN 1 /ν K + 3 = B 4 ( M ) = ) l [ N − 2 β/ν f M 2 ( tN 1 /ν ) l l 11 / 40
Method to determine CEP • determine the transition point (peak position of susceptibility) • determine kurtosis at transition point at each spatial lattice size • find intersection point of kurtosis by fit, K E + aN 1 /ν ( β − β E ) l light heavy 0 x V>V>V K t 1st order K x E K crossover -2 • interpolate/extrapolate √ t 0 m PS , t measured at transition point to β E • extrapolate √ t 0 m PS , E to the continuum limit • use scale determined from Wilson flow 1 / √ t 0 = 1 . 347 ( 30 ) GeV [Borsanyi et al. ’12] 12 / 40
Higher moments i -th derivative of ln Z with respect to control parameter c : E = ∂ ln Z ∂ c • Variance V = ∂ 2 ln Z = σ 2 ∂ c 2 • Skewness ( e.g. right-skewed → S > 0 , left-skewed → S < 0 ) ∂ 3 ln Z S = 1 σ 3 ∂ c 3 • Kurtosis( e.g. Gaussian → K = 0 , 2 δ func. → K = − 2 ) ∂ 4 ln Z K = 1 = B 4 − 3 σ 4 ∂ c 4 13 / 40
Simulations • action: Iwasaki gluon + N f = 3 clover (non perturbative c SW , degenerate) • observables • gauge action density, G • plaquette, P • Polyakov loop, L • chiral condensate, Σ • and their higher moments • temporal lattice size N t = 4 , 6 , 8 • statistics: O(100K) traj • preliminary N t = 10 • statistics: O(1K) traj 14 / 40
0.1434 -0.4 -1 -1.2 -1.4 -1.6 -1.8 0.1437 0.1436 0.1435 0 0.1433 0.1432 -0.2 -0.6 0 0.2 kurtosis of 𝑄 𝜆 4 3.5 3 2.5 2 1.5 1 0.5 -0.8 plaquette at β = 1 . 60 , N t = 4 𝑂 𝑡 = 6 𝑂 𝑡 = 8 𝑂 𝑡 = 10 𝑂 𝑡 = 12 𝑂 𝑡 = 16 susceptibility of 𝑄 15 / 40
0.2 -0.2 -1 0.1415 0.1414 0.1413 0.1412 0.1411 0.141 0.1 kurtosis of 𝑄 -0.4 0 -0.6 𝜆 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 -0.8 plaquette at β = 1 . 65 , N t = 4 𝑂 𝑡 = 6 𝑂 𝑡 = 8 𝑂 𝑡 = 10 𝑂 𝑡 = 12 𝑂 𝑡 = 16 susceptibility of 𝑄 16 / 40
0.75 1.600 0.15 0.18 0.85 0.80 -2 0.70 0.65 0.60 0.55 0.19 1.610 0.17 1.620 1.630 1.640 1.650 1.660 1.670 𝛾 0 -0.5 -1 -1.5 0.16 Kurtosis intersection at N t = 4 𝑂 𝑚 = 10 , 𝐻 𝑂 𝑚 = 12 , 𝐻 𝐿 𝑢 𝑂 𝑚 = 16 , 𝐻 𝑂 𝑚 = 10 , 𝑄 𝑂 𝑚 = 12 , 𝑄 𝑂 𝑚 = 16 , 𝑄 𝑂 𝑚 = 10 , 𝑀 𝑂 𝑚 = 12 , 𝑀 𝑂 𝑚 = 16 , 𝑀 𝑢 0 𝑛 PS,t 𝑢 0 𝑈 t 17 / 40
1.670 0.85 1.660 1.650 1.640 1.630 1.620 1.610 1.600 0.19 0.18 0.17 0.16 0.15 -2 0.80 𝛾 0.75 0.70 0.65 0.60 0.55 0 -0.5 -1 -1.5 Kurtosis intersection at N t = 4 𝑂 𝑚 = 10 , Σ 𝑂 𝑚 = 12 , Σ 𝐿 𝑢 𝑂 𝑚 = 16 , Σ 𝑢 0 𝑛 PS,t 𝑢 0 𝑈 t 18 / 40
1.75 1.68 3D 𝑃(4) 𝛾 𝑐(= 𝛿⁄𝜈) 1.77 1.76 0 1.74 1.73 1.72 1.71 1.7 1.69 1.67 3D 𝑎 2 1.66 1.65 1.64 1.63 1.62 1.61 1.6 3 2.5 2 1.5 1 0.5 3D 𝑃(2) γ/ν v.s. β 𝛾 𝐹 at 𝑂 𝑢 = 4 𝛾 𝐹 at 𝑂 𝑢 = 6 𝛾 𝐹 at 𝑂 𝑢 = 8 𝑂 𝑢 = 4, 𝐻 𝑂 𝑢 = 4, 𝑀 𝑂 𝑢 = 4, Σ 𝑂 𝑢 = 6, 𝐻 𝑂 𝑢 = 6, 𝑀 𝑂 𝑢 = 6, Σ 𝑂 𝑢 = 8, 𝐻 𝑂 𝑢 = 8, 𝑀 𝑂 𝑢 = 8, Σ χ max = aN γ/ν l 19 / 40
continuum extrapolation for √ 0.01 G, P, L Σ SU(3) sysmmetric point 1st order crossover 𝑢 0.06 0.05 0.04 0.03 0.02 0.07 0 0.45 0.2 0.7 0.3 0.35 0.4 0.25 0.5 0.55 0.6 0.65 t 0 m PS , E 𝑢 0 𝑛 PS, E 1⁄𝑂 2 ▲ : √ = √ √ t 0 m phy ; sym ( m 2 π + 2 m 2 t 0 K ) / 3 ∼ 0 . 305 PS m PS , E = 304 ( 7 )( 14 )( 7 ) MeV 20 / 40
continuum extrapolation for √ 0.01 G, P, L Σ 𝑢 0.07 0.06 0.05 0.04 0.03 0.02 0 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 t 0 T E 𝑢 0 𝑈 E 1⁄𝑂 2 T E = 131 ( 2 )( 1 )( 3 ) MeV 21 / 40
Summary at N t = 4 , 6 , 8 • kurtosis intersection analysis is consistent with χ max analysis • results at N t = 4 is out of scaling region • √ t 0 m PS , E in the continuum limit is smaller than the SU(3) sysmmetric point, m PS , E / m phys , sym = 0 . 739 ( 17 )( 34 )( 17 ) PS • further studies at larger temporal sizes to obtain conclusive results are needed Phys. Rev. D 91, 014508 (2015) 22 / 40
Σ at N t = 10 (1/3, preliminary) 14000 β =1.76 N s =16 β =1.77 N s =16 β =1.77 N s =20 β =1.78 N s =16 12000 β =1.78 N s =20 β =1.78 N s =24 β =1.79 N s =16 β =1.79 N s =20 10000 β =1.79 N s =24 β =1.80 N s =16 β =1.80 N s =20 8000 pbpz sus 6000 4000 2000 0 0.1386 0.1388 0.139 0.1392 0.1394 0.1396 0.1398 0.14 0.1402 κ 23 / 40
Σ at N t = 10 (2/3, preliminary) 2 Z 2 β =1.76 N s =16 β =1.77 N s =16 β =1.77 N s =20 1.5 β =1.78 N s =16 β =1.78 N s =20 β =1.78 N s =24 β =1.79 N s =16 1 β =1.79 N s =20 β =1.79 N s =24 β =1.80 N s =16 0.5 β =1.80 N s =20 pbpz krt 0 -0.5 -1 -1.5 -2 0.1386 0.1388 0.139 0.1392 0.1394 0.1396 0.1398 0.14 0.1402 κ 24 / 40
0.13 1.800 0.30 0.35 𝛾 0.09 0.10 0.11 0.12 0.25 1.760 1.770 1.780 1.790 Σ at N t = 10 (3/3, preliminary) 𝑢 0 𝑛 PS,t 𝑢 0 𝑈 t assuming β E = 1 . 78 ( 1 ) 25 / 40
G, P, L 0.13 1st order SU(3) sysmmetric point Σ 𝑢 0.09 0.1 0.11 0.12 0.14 𝑢 0.15 0.16 0.17 0 0.01 0.02 0.03 0.04 0.05 crossover Σ 0.07 0.6 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.65 G, P, L 0.7 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 continuum extrapolation and results at N t = 10 (preliminary) 𝑢 0 𝑈 E 𝑢 0 𝑛 PS, E 1⁄𝑂 2 1⁄𝑂 2 • assuming β E = 1 . 78 ( 1 ) at N t = 10 • excluding results at N t = 10 from continuum extrapolation • T E would not change very much • m PS , E may become smaller than results at smaller N t 26 / 40
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