PACS-CS Collaboration Member S. Aoki, N. Ishii, N. Ishizuka, D. - - PowerPoint PPT Presentation

PACS-CSの取り組み

Ref: PRD79.034503 (2009)

浮田 尚哉 (筑波大学計算科学研究センター)

for PACS-CS Collaboration

「ストレンジを含むクォーク多体系分野の理論的将来を考える」研究会 ２００９年２月２７－２８日 @ KKRホテル熱海（熱海市）

1

PACS-CS Collaboration Member

• S. Aoki, N. Ishii, N. Ishizuka, D. Kadoh, K. Kanaya,
• Y. Kuramashi, Y. Namekawa, Y. Taniguchi,
• A. Ukawa, N. Ukita, T. Yoshie, T. Yamazaki

(University of Tsukuba) K.-I. Ishikawa, M. Okawa, Y. Shimizu (Hiroshima Univ.)

• T. Izubuchi (BNL, Kanazawa Univ.)

2

Introduction

PACS-CS Project: Nf = 2 + 1 Simulations at the Physical Point on large enough lattices

• O(a) improved Wilson quark action with nonperturbative cSW

(CP-PACS and JLQCD Collaborations, 2006)

• Iwasaki gauge action

(Iwasaki, 1983)

• β = 1.90 (a = 0.0907(13)fm)
• 323 × 64 lattice (V ≈ (2.9fm)3)
• mπ = 156 ∼ 702 MeV
• mπL 2.3

u-d quarks : Domain-Decomposed HMC (DDHMC) algorithm (L¨

uscher, 2003)

+ Hasenbusch trick (Hasenbusch, 2001; Hasenbusch, Jansen, 2003) +· · · s quark : UV-filterd Polynomial HMC (UVPHMC) algorithm (JLQCD Collaborations, 2002)

• n the PACS-CS (2560nodes, 14.3TFLOPS) and the T2K (95+140+61TFLOPS).

3

Introduction

Comparison with other groups for Nf = 2 + 1 simulations: Collaboration Dirac op. mπ(MeV) mmin

π

L a(fm) PACS-CS clover 156 2.3 0.09 [MΩ] BMW(’08) stout clover 190 4 0.065-0.125 [MΞ] MILC(’04,’07) staggered 240 4 0.06-0.18 [Fπ] NPLQCD(’07) staggered+DW 290 3.7 0.13 [r0] RBC/UKQCD(’08) DW 330 4.6 0.1 [MΩ] JLQCD/TWQCD(’08) Overlap 310 2.8 0.108 [r0]

4

Introduction The light hadron spectrum of QCD at a = 0.09fm

0.5 1 1.5 2 m[GeV] ρ K

* φ

N Λ Σ Ξ ∆ Σ

∗ Ξ ∗ Ω

meson

• ctet baryon

decuplet baryon π K chpt fse ( mπ,mK,mΩ-input )

5

Introduction The light hadron spectrum of QCD : PACS-CS (a=0.09fm) and BMW (a → 0)

0.5 1 1.5 2 m[GeV] ρ K

* φ

N Λ Σ Ξ ∆ Σ

∗ Ξ ∗ Ω

meson

• ctet baryon

decuplet baryon π K PACS-CS ( mπ,mK,mΩ-input ) BMW (mπ,mK,mΞ-input)

6

Plan to this talk :

• Introduction
• Algorithm for Nf = 2 part : DDHMC, Hasenbusch trick, Solver
• Simulation Parameters and Data Set
• Run Status
• ChPT Analysis : SU(2) and SU(3) ChPT up to NLO
• The Physical Point Simulation
• Conclusion

Algorithm for Nf = 2 + 1 part

DDHMC, Hasenbusch trick, Solver

7

DDHMC Algorithm

(L¨ uscher, 2003)

Key to reduce the cost : Domain Decomposition & Multi Time Step Integrator

• Domain Decomposition splitting lattice sites into even & odd domains

as a preconditionor for Nf = 2 O(a)-improved Wilson-Dirac op. Even Odd Even domain size we use is 84 D = „ DEE DEO DOE DOO « = „ DEE DOO « „ 1 D−1

EEDEO

D−1

OODOE

1 « | det D|2 = | det DEE|2 | det DOO|2 | {z }

UV part

| det(1 − D−1

EEDEOD−1 OODOE

| {z }

≡ DIR : IR part

)|2

8 After this preconditioning, we have a partition function, Z = Z

U

e−SG | det(1 + T )|2 | {z }

Gauge part

| det ¯ DEE|2 | det ¯ DOO|2 | {z }

UV part

| det DIR | {z }

IR part

|2 = Z

P,U,φ,χ

exp  −1 2Tr(P 2) − S′

G[U] − SUV [U, φ] − SIR[U, χ]

ff . Molecular dynamics equation : set random φ, χ ˙ U = P , ˙ P = FG[U] + FUV [U, φ] + FIR[U, χ]

• Multi time step integrator for Gauge, UV and IR parts (Sexton and Weingarten, 1992)

Relative magnitudes of force terms, FG, FIR, FUV : ||FG|| : ||FUV || : ||FIR|| ≈ 16 : 4 : 1. We choose the associated step sizes, δτG, δτUV , δτIR such that δτG||FG|| ≈ δτUV ||FUV || ≈ δτIR||FIR||, δτG = τ/N0N1N2, δτUV = τ/N1N2, δτIR = τ/N2, N0 = N1 = 4.

9 For strange quark, we employ UVPHMC algorithm (CP-PACS and JLQCD Collaborations, 2006) where the domain decomposition is not used. ∥Fs∥ ≈ ∥FIR∥ ⇒ δτs = δτIR. ⇓ ⃝ mπ 296MeV : DDHMC + UVPHMC algorithm works stable. × mπ = 156MeV : large fluctuation of ∥FIR∥, slow to keep simulation stable.

• Hasenbusch trick (Hasenbusch, 2001; Hasenbusch, Jansen, 2003)

D′

IR = DIR(κ → κ′ = ρκ), eg. ρ = 0.9995 to shift to the heavier mass

| det DIR|2 = | det D′

IR|

˛ ˛ ˛ ˛det „DIR D′

IR

«˛ ˛ ˛ ˛

2

⇒ FIR′, FIR/IR′ Step sizes, δτG, δτUV , δτIR′, δIR/IR′, are controlled by (N0, N1, N2, N3). N0 = N1 = 4, N2 and N3 are chosen to reduce the fluctuation of ∥FIR′∥, ∥FIR/IR′∥.

10

Solver for mπ 296MeV : Dx = b

For DDHMC algorithm (mπ 296MeV),

• IR solver : SAP(single prec.) preconditioned GCR(double prec.) (L¨

uscher, 2004)

• UV solver : SSOR(single prec.) preconditioned GCR(double prec.)
• Stopping condition : |Dx − b|/|b| ≤ 10−14 for H, 10−9 for F

11

Solver for mπ = 156MeV : Dx = b

For MPDDHMC algorithm (mπ =156MeV), ⋆ Chronological guess for IR part (Brower, Ivanenko, Levi, Orginos, 1997) ⋆ nested BiCGStab solver for IR and UV part :

• Outer solver(double prec.) : Solve Dx = b with preconditioner M ≈ D−1

with strict stopping condition 10−14 for F

• Inner solver(single prec.) : Solve M ≈ D−1 with appropriate precoditioner

with automatic tolerance control tolinner = min “ max “

errouter tolouter , 10−6”

, 10−3” ⋆ Deflation technique (Morgan, Wilcox, 2002; L¨

uscher, 2007)

• inner BiCGStab stagnant → GCRO-DR (Parks et al, 2006)

(Generalized Conjugate Residual with implicit inner Orthogonalization and Deflated Restarting)

12

Simulation Cost

0.2 0.4 0.6 0.8 1

mπ/mρ

0.0 2.0 4.0 6.0 8.0 10.0

Tflops years

HMC κud=0.13770 κud=0.13781

Nf=2+1 100 configs a=0.1fm L=3fm

physical pt

Cost ∝ (mπ/mρ)−3 for HMC, ∝ (mπ/mρ)−2 for MPDDHMC.

(Ukawa, 2002; PACS-CS Collaboration, 2007)

Physical Point simulations require HMC : O(100) Tflops computer , MPDDHMC : O(10) Tflops computer.

Simulation Parameters and Data Set

13

Simulation Parameters and Data Set

κud 0.13700 0.13727 0.13754 0.13754 0.13770 0.13781 0.137785 κs 0.13640 0.13640 0.13640 0.13660 0.13640 0.13640 0.13660 Algorithm

DDHMC DDHMC DDHMC DDHMC DDHMC MPDDHMC MPDDHMC

τ 0.5 0.5 0.5 0.5 0.25 0.25 0.25

(N0,N1,N2,N3,N4)

(4,4,10) (4,4,14) (4,4,20) (4,4,28) (4,4,16) (4,4,4,6) (4,4,2,4,4) (4,4,6,6) ρ1 − − − − − 0.9995 0.9995 ρ2 − − − − − − 0.9990 Npoly 180 180 180 220 180 200 220 Replay

• n
• n
• n
• n
• n
• ff
• ff

MD time 2000 2000 2250 2000 2000 1400 1000 mud[MeV ] 67 45 24 21 12 3.5 3.6 mπ[MeV ] 702 570 411 385 296 156 162 mπ/mρ 0.64 0.57 0.46 0.45 0.35 0.20 0.23 CPU time [h]/τ 0.29 0.44 1.3 1.1 2.7 7.1 6.0 shifted hopping parameter κ′

ud = ρ1κud ∼ 0.1377

Run Status

dH and Force histories, Effective masses

14

dH History

mπ = 570MeV mπ = 296MeV mπ = 156MeV

1000 1500 2000 2500

τ

• 2
• 1

1 2 3 4 5 dH 1000 1500 2000 2500 traj.

• 2
• 1

1 2 3 4 5 dH 500 550 600 650 700 750 800 850 900

τ

• 2
• 1

1 2 3 4 5 dH

acc(HMC)=0.87 acc(HMC)=0.84 acc(HMC)=0.88 replay trick∼ 0.1% replay trick∼ 3% rate(|dH| > 2) ∼ 3%

15

Force History

mπ = 570MeV mπ = 296MeV mπ = 156MeV

5000 10000 15000 20000 0.1 1 10

F0 F1 F2

4e+05 4.5e+05 5e+05 5.5e+05 6e+05 6.5e+05 7e+05 0.1 1 10 1e+06 1.2e+06 1.4e+06 1.6e+06 0.001 0.01 0.1 1 10

F0 F1 F2 F3

F0 : Gauge + clv F1 : UV F2 : IR F0 : Gauge + clv F1 : UV F2 : IR′ F3 : IR/IR′

16

Effective mass : Meson kud = 0.13727 mπ = 570MeV

10 20 30

Nt

0.2 0.3 0.4 0.5

mMeson

π K η

s

ρ K

*

φ

kud = 0.13770 mπ = 296MeV

10 20 30

Nt

0.1 0.2 0.3 0.4 0.5

mMeson

π K η

s

ρ K

*

φ

kud = 0.13781 mπ = 156MeV

10 20 30

Nt

0.1 0.2 0.3 0.4 0.5

mMeson

π K η

s

ρ K

*

φ

Fit Range [tmin, tmax] : Pseudoscalar [13 − 30], Vector [10 − 20]

17

Effective mass : Baryon kud = 0.13727 mπ = 570MeV

5 10 15 20 25

Nt

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

mB

∆ Σ

*

Ξ

*

Ω N Λ Σ Ξ

kud = 0.13770 mπ = 296MeV

5 10 15 20 25

Nt

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

mB

∆ Σ

*

Ξ

*

Ω N Λ Σ Ξ

kud = 0.13781 mπ = 156MeV

5 10 15 20 25

Nt

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

mB

∆ Σ

*

Ξ

*

Ω N Λ Σ Ξ

Fit Range [tmin, tmax] : Decuplet [13 − 30], Octet [10 − 20]

18

Edinburgh Plot

0.1 0.2 0.3 0.4 0.5 0.6 0.7

mπ / mρ

1 1.1 1.2 1.3 1.4 1.5

mN / mρ

κs=0.13640 κs=0.13660 Phisical Point

19

m2

π/mud, fK/fπ vs mud

0.02 0.04 0.06 0.08

m

AWI

3.4 3.6 3.8 4.0 4.2

CP-PACS/JLQCD κs=0.13640 PACS-CS κs=0.13640 PACS-CS κs=0.13660

ud

mπ

2/m AWI

ud 0.02 0.04 0.06 0.08

m

AWI

0.9 1.0 1.1 1.2 1.3

CP-PACS/JLQCD κs=0.13640 PACS-CS κs=0.13640 PACS-CS κs=0.13660 Experiment

ud

fK/fπ

SU(3) ChPT analysis

19

SU(3) ChPT analysis on the PS meson sector

Continuum SU(3) ChPT up to NLO = WChPT up to NLO : m2

π

2mud = B0  1 + µπ − 1 3µη + 2B0 f 2 (16mudL85 + 16(2mud + ms)L64) ff , m2

K

(mud + ms) = B0  1 + 2 3µη + 2B0 f 2 (8(mud + ms)L85 + 16(2mud + ms)L64) ff , fπ = f0  1 − 2µπ − µK + 2B0 f 2 (8mudL5 + 8(2mud + ms)L4) ff , fK = f0  1 − 3 4µπ − 3 2µK − 3 4µη + 2B0 f 2 (4(mud + ms)L5 + 8(2mud + ms)L4) ff , where we have six unknown LECs B0, f0, L4,5,6,8. µPS denotes the chiral logarithm defined by µPS = 1 16π2 ˜ m2

PS

f 2 ln ˜ m2

PS

µ2 ! , where ˜ m2

π = 2mudB0,

˜ m2

K = (mud + ms)B0,

˜ m2

η = 2

3(mud + 2ms)B0

20

Result for the LEC’s in the SU(3) ChPT

PACS-CS phenomenology RBC/UKQCD MILC w/o FSE w/ FSE f0[GeV] 0.1160(88) 0.1185(90) 0.115 0.0935(73) − fπ/f0 1.159(57) 1.145(56) 1.139 1.33(7) 1.21(5) `+13

−3

´ L4 −0.04(10) −0.06(10) 0.00(80) 0.139(80) 0.1(3) `+3

−1

´ L5 1.43(7) 1.45(7) 1.46(10) 0.872(99) 1.4(2) `+2

−1

´ 2L6 − L4 0.10(2) 0.10(2) 0.0(1.0) −0.001(42) 0.3(1) `+2

−3

´ 2L8 − L5 −0.21(3) −0.21(3) 0.54(43) 0.243(45) 0.3(1)(1) χ2/dof 4.2(2.7) 4.4(2.8) − 0.7 −

• 1

1 2 3 4 5 6 7 8

l3-bar

GL CERN ETM MILC(SU3) PACS-CS(SU2) JLQCD RBC/UKQCD(SU2) RBC/UKQCD(SU3) PACS-CS(SU2-FSE) PACS-CS(SU3) PACS-CS(SU3-FSE)

2 3 4 5 6 7

l4-bar

GL CGL ETM MILC(SU3) PACS-CS(SU2) JLQCD RBC/UKQCD(SU2) RBC/UKQCD(SU3) PACS-CS(SU2-FSE) PACS-CS(SU3) PACS-CS(SU3-FSE)

21

SU(3) ChPT Fit up to NLO

0.01 0.02 0.03

m

AWI

3.4 3.6 3.8 4.0 4.2

κs=0.13640 κs=0.13660 SU(3) SU(3)-FSE SU(3)@ph SU(3)-FSE@ph

ud

mπ

2/m AWI

ud 0.02 0.03

(m

AWI+m AWI)/2

3.2 3.4 3.6 3.8

κs=0.13640 κs=0.13660 SU(3) SU(3)-FSE SU(3)@ph SU(3)-FSE@ph

ud

2mK

2/(m AWI+m AWI)

ud ud s s 0.01 0.02 0.03

m

AWI

0.05 0.06 0.07 0.08 0.09 0.10

κs=0.13640 κs=0.13660 SU(3) SU(3)-FSE SU(3)@ph SU(3)-FSE@ph

ud

fπ

0.01 0.02 0.03

m

AWI

0.070 0.075 0.080 0.085 0.090 0.095 0.100

κs=0.13640 κs=0.13660 SU(3) SU(3)-FSE SU(3)@ph SU(3)-FSE@ph

ud

fK

22

Ratio of NLO contribution to LO one

0.005 0.01 0.015 0.02 0.025 0.03

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

fπ (su(2)-fit) fπ (su(3)-fit) fK (su(2)-fit) fK (su(3)-fit)

NLO/LO

mud

AWI

ms is not small enough to be treated in the SU(3) ChPT up to NLO. ⇓ The SU(2) ChPT with the analytical expansion of ms about the physical value.

SU(2) ChPT analysis

23

SU(2) ChPT + ms linear dependence

Continuum SU(2) ChPT up to NLO : m2

π

2mud = B ( 1 + 1 16π2 ¯ m2

π

f 2 ln ¯ m2

π

µ2 ! + 4 ¯ m2

π

f 2 l3 ) , fπ = f ( 1 − 1 8π2 ¯ m2

π

f 2 ln ¯ m2

π

µ2 ! + 2 ¯ m2

π

f 2 l4 ) . B and f are linearly expanded in terms of ms : B = B(0)

s

+ msB(1)

s ,

f = f (0)

s

+ msf (1)

s .

For mK and fK we employ the following fit formulae: m2

K

= αm + γmms + βmmud, fK = ¯ f  1 + βfmud − 3 4 2Bmud 16π2f ln „2Bmud µ2 «ff , where ¯ f = ¯ f (0)

s

+ ms ¯ f (1)

s .

24

SU(2) ChPT Fit up to NLO

0.01 0.02 0.03

m

AWI

3.4 3.6 3.8 4.0 4.2

κs=0.13640 κs=0.13660 SU(2) SU(2)-FSE SU(2)@ph SU(2)-FSE@ph

ud

mπ

2/m AWI

ud 0.01 0.02 0.03

m

AWI

0.05 0.06 0.07 0.08 0.09 0.10

κs=0.13640 κs=0.13660 SU(2) SU(2)-FSE SU(2)@ph SU(2)-FSE@ph

ud

fπ

0.01 0.02 0.03

m

AWI

0.070 0.075 0.080 0.085 0.090 0.095 0.100

κs=0.13640 κs=0.13660 SU(2) SU(2)-FSE SU(2)@ph SU(2)-FSE@ph

ud

fK

χ2/dof = 0.43(77) with FSE, = 0.33(68) without FSE,

25

Ratio of NLO contribution to LO one

0.005 0.01 0.015 0.02 0.025 0.03

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

fπ (su(2)-fit) fπ (su(3)-fit) fK (su(2)-fit) fK (su(3)-fit)

NLO/LO

mud

AWI

The ratio NLO/LO in the SU(2) ChPT is reduced from the SU(3) ChPT case.

26

Result in the SU(2) ChPT

0.5 1 1.5 2 m[GeV] ρ K

* φ

N Λ Σ Ξ ∆ Σ

∗ Ξ ∗ Ω

meson

• ctet baryon

decuplet baryon π K chpt fse ( mπ,mK,mΩ-input )

PACS-CS experiment RBC/UKQCD MILC a−1 [GeV] 2.176(31) − 1.729(28) continum mMS

ud [MeV]

2.527(47) − 3.72(16)(33)(18) 3.2(0)(1)(2)(0) mMS

s

[MeV] 72.72(78) − 107.3(4.4)(9.7)(4.9) 88(0)(3)(4)(0) ms/mud 28.8(4) − 28.8(0.4)(1.6) 27.2(1)(3)(0)(0) fπ [MeV] 134.0(4.2) 130.7 ± 0.1 ± 0.36 124.1(3.6)(6.9) input fK [MeV] 159.4(3.1) 159.8 ± 1.4 ± 0.44 149.6(3.6)(6.3) 156.5(0.4)(+1.0

−2.7)

fK/fπ 1.189(20) 1.223(12) 1.205(18)(62) 1.197(3)(+6

−13)

κud = 0.137785, κs = 0.13660

is estimated as the physical point from our ChPT analysis.

27

κud = 0.137785, κs = 0.13660

0.5 1 1.5 2 m[GeV] ρ K

* φ

N Λ Σ Ξ ∆ Σ

∗ Ξ ∗ Ω

meson

• ctet baryon

decuplet baryon π K

chpt fse ( mπ,mK,mΩ-input ) κ ud=0.137785

ChPT experiment κud = 0.137785 mMS

ud

[MeV] 2.53(5) − 3.5(3) mMS

s

[MeV] 72.7(8) − 73.4(2) fπ [MeV] 134.0(4.2) 130.7 ± 0.1 ± 0.36 129.0(5.4) fK [MeV] 159.4(3.1) 159.8 ± 1.4 ± 0.44 160.6(1.4)

28

Conclusion

• Nf = 2 + 1 full QCD with O(a) improved Wilson quarks on (2.9fm)3
• domain-decomposed HMC + Hasenbusch trick
• mπ = 156 ∼ 702[MeV]
• mπ/mρ = 0.20 ∼ 0.64
• a = 0.0907(13) fm
• mπL 2.3

We need ⋆ Nonperturbative renormalization factors to remove perturbative uncertainties, ⋆ To investigate the finite size effects and the discritization errors.

29

Linear fit for the vector meson masses

0.01 0.02 0.03

m

AWI

0.3 0.4 0.5 0.6

κs=0.13640 κs=0.13660 linear physical pt Experiment

ud

mρ

0.01 0.02 0.03

m

AWI

0.4 0.5 0.6

κs=0.13640 κs=0.13660 linear physical pt Experiment

ud

mΚ

∗

0.01 0.02 0.03

m

AWI

0.3 0.4 0.5 0.6

κs=0.13640 κs=0.13660 linear physical pt Experiment

ud

mρ

30

Linear fit for the octet baryon masses

0.005 0.01 0.015 0.02 0.025 0.4 0.5 0.6 0.7 0.8 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mN

0.005 0.01 0.015 0.02 0.025 0.5 0.55 0.6 0.65 0.7 0.75 0.8 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mΛ

0.005 0.01 0.015 0.02 0.025 0.5 0.55 0.6 0.65 0.7 0.75 0.8 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mΣ

0.005 0.01 0.015 0.02 0.025 0.6 0.65 0.7 0.75 0.8 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mΞ

31

Linear fit for the decuplet baryon masses

0.005 0.01 0.015 0.02 0.025 0.5 0.6 0.7 0.8 0.9 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

m∆

0.005 0.01 0.015 0.02 0.025 0.65 0.7 0.75 0.8 0.85 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mΣ∗

0.005 0.01 0.015 0.02 0.025 0.7 0.75 0.8 0.85 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mΞ∗

0.005 0.01 0.015 0.02 0.025 0.75 0.8 0.85 0.9 ks=0.13640 ks=0.13660 PP chpt (13640) chpt (13660) PP (ronbun) Exp.

mΩ