LΛ - 名古屋大学 素粒子宇宙起源研究機構（KMI）

LatKMI collaboration• Y.Aoki, T.Aoyama, M.Kurachi, T.Maskawa, K.-i.Nagai, H.Ohki,E. Rinaldi, K.Yamawaki, T.YamazakiKofor the O• K. Hasebe, A.Shibata

β ≡ 6/g 2 = m π ,f π ,m ρ

m_pi0.90.80.70.60.50.40.30.20.1m_pi vs mfL^3x(4L/3)00 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mfm_rho1.110.90.80.70.60.50.40.30.20.1L=12L=18L=24L=30L=36f_pim_rho vs mfL^3x(4L/3)0.20.180.160.140.120.10.080.060.040.0200 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mff_pi vs mfL^3x(4L/3)L=12L=18L=24L=30L=3600 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mfL=12L=18L=24L=30L=36

f π /M πfpi(mf)/mpi(mf)0.30.280.260.240.22fpi/mpi vs mpiL^3x(4L/3)L=12L=18L=24L=30L=360.20.180 0.2 0.4 0.6 0.8 1mpi

f π = 0 or = 0? as m f → 0f_pi vs mff_pi/mf vs mf0.2L^3x(4L/3)4L^3x(4L/3)0.180.163.5f_pi0.140.120.10.080.060.040.02L=12L=18L=24L=30L=36f_pi/mf32.521.5L=12L=18L=24L=30L=3600 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf10 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2mf

f π = 0 as m f → 0f π = F + Am f + Bm 2 f0.20.180.16f_pi vs mfL^3x(4L/3), Quadratic fit: y=c0+c1*mf+c2*mf^20.140.12f_pi0.10.080.060.040.020data (on largest L at mf)0.0-0.16 (c2d=34.2)0.0-0.1 (c2d=7.84)0.0-0.08 (c2d=4.06)0.0-0.06 (c2d=4.09)0.0-0.05 (c2d=5.56)0.0-0.04 (c2d=0.46)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf

f π = F + Am α f0.20.18f_pi vs mfL^3x(4L/3), Power fit: y=c0+c1*(mf^c2)α = 11+γf_pi0.160.140.120.10.080.060.040.020data (on largest L at mf)0.0-0.16 (c2d=3.43)0.0-0.1 (c2d=4.62)0.0-0.08 (c2d=4.72)0.0-0.06 (c2d=6.71)0.0-0.05 (c2d=9.43)0.0-0.04 (c2d=0.28)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf

0.2f_pi vs mfL^3x(4L/3)f_pi0.180.160.140.120.10.080.060.040.02L=12L=18L=24L=30L=3600 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf

0.2f_pi vs mfL^3x(4L/3)0.180.16f_pi0.140.120.10.080.060.040.02L=12L=18L=24L=30L=3600 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf

m 2 π ∝ m f ?m_pi^2 vs mfL^3x(4L/3)4.5m_pi^2/mf vs mfL^3x(4L/3)0.64m_pi^20.50.40.30.20.1L=12L=18L=24L=30L=36m_pi^2/mf3.532.521.510.5L=12L=18L=24L=30L=3600 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf00 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mf

LM H = F H (x),LF H = G F (x)x ≡ Lm 1/1+γ

4Lf vs. X(=0.6)Nf=8, =3.8, L^3x(4L/3)4Lf vs. X(=1.0)Nf=8, =3.8, L^3x(4L/3)4Lf vs. X(=1.4)Nf=8, =3.8, L^3x(4L/3)333Lf 21L=12L=18L=24L=30L=36Lf 21L=12L=18L=24L=30L=36Lf 21L=12L=18L=24L=30L=3600 2 4 6 8 10X()00 2 4 6 8 10X()00 2 4 6 8 10X()LM H = F H (x),LF H = G F (x)x ≡ Lm 1/1+γ

4Lf vs. X(=0.6)Nf=8, =3.8, L^3x(4L/3)4Lf vs. X(=1.0)Nf=8, =3.8, L^3x(4L/3)4Lf vs. X(=1.4)Nf=8, =3.8, L^3x(4L/3)333Lf 21L=12L=18L=24L=30L=36Lf 21L=12L=18L=24L=30L=36Lf 21L=12L=18L=24L=30L=3600 2 4 6 8 10X()00 2 4 6 8 10X()00 2 4 6 8 10X()LM H = F H (x),LF H = G F (x)x ≡ Lm 1/1+γ

4Lf vs. X(=0.6)Nf=8, =3.8, L^3x(4L/3)4Lf vs. X(=1.0)Nf=8, =3.8, L^3x(4L/3)4Lf vs. X(=1.4)Nf=8, =3.8, L^3x(4L/3)333Lf 21L=12L=18L=24L=30L=36Lf 21L=12L=18L=24L=30L=36Lf 21L=12L=18L=24L=30L=3600 2 4 6 8 10X()00 2 4 6 8 10X()00 2 4 6 8 10X()LM H = F H (x),LF H = G F (x)x ≡ Lm 1/1+γ

P (γ) 20151052 4 6 8

P (γ) 100001000γP100M πf π10M M ρF M 10 0.2 0.4 0.6 0.8 1 1.2 1.4

LM H = F H (x),LF H = G F (x)x ≡ Lm 1/1+γLM H (LF H )=C 0 + C 1 x.

L*m_pi201510L=12L=18L=24L=30L=36linear fithyperscaling (m_pi vs x)(m_pi)=0.5925(17), ^2/dof=12.4L*f_pi432L=12L=18L=24L=30L=36linear fithyperscaling (f_pi vs x)(f_pi)=0.9528(38), ^2/dof=3.375100 1 2 3 4 5 6 7γ(Mx=L*mf^(1/(1+))π ) = γ(f π )00 1 2 3 4 5 6 7 8 9x=L*mf^(1/(1+))

hyperscaling (m_pv vs x)(m_pv)=0.8421(65), ^2/dof=1.8930hyperscaling (m_sc-p vs x)(m_sc-p)=0.6541(195), ^2/dof=1.92L*m_pv201510L=12L=18L=24L=30L=36linear fitL*m_sc-partner25201510L=12L=18L=24L=30L=36linear fit5500 1 2 3 4 5 6 7 8x=L*mf^(1/(1+))00 1 2 3 4 5 6 7x=L*mf^(1/(1+))

γγM πM πf πf πM ρM ργ(M π ) = γ(f π )

γ(M π ) = γ(f π ) N f =4, 12

γ N f = 12N f = 12 (β =3.7) N f =8γγM πM πf πf πM ρM ργ(M π ) = γ(f π )

N f =4=3.7, = 0.0=3.7, = 1.0=3.7, = 2.03332.52.52.5222L f 1.5L f 1.5L f 1.51110.5L=12, T=18L=16, T=24L=20, T=300.5L=12, T=18L=16, T=24L=20, T=300.5L=12, T=18L=16, T=24L=20, T=3000 1 2 3 4 5 6 7L m 1/(1+)00 1 2 3 4 5 6 7L m 1/(1+)00 1 2 3 4 5 6 7L m 1/(1+)f π 0 < γ < 2

LatKMI collaboration, Phys.Rev. D85 (2012) 074502Nf cr 11.9

(Nf=11), α*~αcr , just below conformal windowRaw data(Mock data)Finite size Hyperscaling test = 1.022m 0/ 0.70.60.50.4L = 12L = 16L = 20L = 25L = 30m PL2018161412L = 12L = 16L = 20L = 25L = 300.31080.260.1 0.2 0.3 0.4 0.5 0.6m P/ 10 15 20 25x

(Nf=11), α*~αcr , just below conformal windowRaw data(Mock data)Finite size Hyperscaling test = 1.022m 0/ 0.70.60.50.4L = 12L = 16L = 20L = 25L = 30m PL2018161412L = 12L = 16L = 20L = 25L = 300.31080.260.1 0.2 0.3 0.4 0.5 0.6m P/ 10 15 20 25x

Summary (Preliminary)SU(3) gauge theory with 8 HISQ quarks.Preliminary result of spectrumSχSB from ChPT analysis (Fπ≠0, & similar Fπ/Mπ to Nf=4)Remnant of conformal (not ordinary QCD) , γ(Fπ)~0.95from hyperscaling test and from comparison with Nf=4 and 12.From SD-eq analysis,indicates near conformal.→ Candidate of Walking dynamicsFuture planγ 1Simulation on larger volumes at lighter massFinite Size Scaling (due to the difficulty to take V=∞ and mf→0)Lattice spacing dependence (UV cutoff dep.) ←many βsSpectroscopy (Mglueball, M”dilaton”, Mbaryon, Mmeson, Fπ, S-para. etc.)M”dilaton”, Mglueball, MH 126GeV?Mglueball and Mscalar(singlet) → E. Rinaldi’s talk (Nf=12)

m_pi vs L at each mff_pi vs L at each mfm_pi0.90.80.70.60.50.40.3L^3x(4L/3)mf=0.10mf=0.08mf=0.07mf=0.06mf=0.05mf=0.04mf=0.03mf=0.02f_pi0.20.180.160.140.120.10.080.06L^3x(4L/3)mf=0.10mf=0.08mf=0.07mf=0.06mf=0.05mf=0.04mf=0.03mf=0.020.20.040.10.0200 4 8 12 16 20 24 28 32 36 40L00 4 8 12 16 20 24 28 32 36 40L

m_pv vs mfL^3x(4L/3), Quadratic fit: y=c0+c1*mf+c2*mf^2m_pv vs mfL^3x(4L/3), Power fit: y=c0+c1*(mf^c2)110.80.8m_pv0.60.40.20data (on largest L at mf)0.0-0.16 (c2d=5.03)0.0-0.1 (c2d=1.59)0.0-0.08 (c2d=0.37)0.0-0.06 (c2d=0.59)0.0-0.05 (c2d=0.84)0.0-0.04(c2d=0.18)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mfm_pv0.60.40.20data (on largest L at mf)0.0-0.16 (c2d=0.78)0.0-0.1 (c2d=0.99)0.0-0.08 (c2d=1.19)0.0-0.06 (c2d=0.91)0.0-0.05 (c2d=1.29)0.0-0.04 (c2d=0.20)0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18mfM ρ = 0 at m f → 0.

Nf=9, α*>αcr , broken phasem 0/ 0.70.8 L = 12L = 160.6 L = 20L = 25L = 300.5 0.70.40.60.3Raw data(Mock data)0.20.50 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6m P/ m PLFinite size Hyperscaling test25201510 = 2.0L = 12L = 16L = 20L = 25L = 3050 5 10 15 20 25x

Nf=12, α*