Physical Point Simulation in 2+1 Flavor

Lattice QCD

Y.Kuramashi for PACS-CS Collaboration

(Univ. of Tsukuba)

July 29, 2009

1

Plan of talk

§1. The PACS-CS project

§2. Reweighting method

§3. Parameters

§4. Preliminary Results

§5. Summary

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§1. The PACS-CS projectParallel Array Computer System for Computational Sciences

operation started on 1 July 2006 at CCS in U.Tsukuba

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collaboration membersphysicists:S.Aoki, N.Ishizuka, D. Kadoh, K.Kanaya, TsukubaY.Kuramashi, Y.Namekawa, Y.Taniguchi, A.Ukawa,N.Ukita, T.YoshieK.-I.Ishikawa, M.Okawa HiroshimaT.Izubuchi BNL

computer scientists:T.Boku, M.Sato, D.Takahashi, O.Tatebe TsukubaT.Sakurai, H.Tadano

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Physics plan

aim: 2+1 flavor QCD simulation at the physical point

PACS-CS CP-PACS/JLQCDgauge action Iwasaki Iwasakiquark action clover with cNP

SW clover with cNPSW

a[fm] 0.07,0.1,0.122 0.07,0.1,0.122volume ∼> (3fm)3 ∼ (2fm)3

mAWIud physical point 64MeV

algorithm for ud DDHMC with improvements HMCalgorithm for s UV-filtered exact PHMC exact PHMC

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Why is physical point simulation necessary?

• difficult to trace chiral logs for chiral extrapolation

• ChPT is not always a good guiding principle

• need a proper treatment of resonances

• simulations with different up and down quark masses

⇒ there are two types of problems

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(1) Computaional cost

0 0.2 0.4 0.6 0.8 1mπ/mρ

0.0

2.0

4.0

6.0

8.0

10.0

Tfl

ops

year

s

HMCDDHMCMPDDHMC

Nf=2+1

10000 MDa=0.1fmL=3fm

pysi

cal p

t

successfully solved by (Mass-Preconditioned) DDHMC

PRD79(2009)034503

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(2) Fine tuning on physical point

need 3 simulation points within a few MeV differenes around the

physical point in 2+1 flavor case

⇒ demanding computational cost

try reweighting method both for ud and s quarks

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§2. Reweighting method

original:(κud, κs) ⇒ target:(κ′ud, κ′

s) assuming ρq ≡ κ′q/κq ≈ 1

⟨O[U ](κ′ud, κ′

s)⟩(κ′ud,κ′

s)=

⟨O[U ](κ′ud, κ′

s)Rud[U ]Rs[U ]⟩(κud,κs)

⟨Rud[U ]Rs[U ]⟩(κud,κs)

reweighting factors

Rud[U ] = |det [W [U ](ρud)]|2 , Rs[U ] = det [W [U ](ρs)]

where W [U ](ρq) ≡Dκ′q

[U ]

Dκq[U ]

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Evaluation of Rud[U ]

introduce a complex bosonic field η

Rud[U ] = |det [W [U ](ρud)]|2

= ⟨e−|W−1[U ](ρud)η|2+|η|2⟩η

given a set of η(i) (i = 1, . . . , Nη) with the Gaussian distribution

Rud[U ] = limNη→∞

1

Nη

Nη∑i=1

e−|W−1[U ](ρud)η|2+|η|2

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Evaluation of Rs[U ]

assume detW [U ](ρs) is positive

Rs[U ] = det [W [U ](ρs)]

= ⟨e−|W−1/2[U ](ρs)η|2+|η|2⟩ηTaylor expansion for W−1/2[U ](ρs)η

W−1[U ](ρs) =Dκs[U ]

Dκ′s[U ]

= 1 − (1 − ρs)(1 − (Dκ′

s[U ])−1

)= 1 − X[U ](ρs)

where |1 − ρs| ≪ 1

⇒ expansion of W−1/2[U ](ρs)η in terms of X[U ](ρs)

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Additional technique Hasenfratz-Hoffmann-Schaefer

determinant breakup: divide (κ′q − κq) into NB subintervals

κq ⇒ κq + ∆q ⇒. . . ⇒ κq + (NB − 1)∆q ⇒ κ′q

with ∆q = (κ′q − κq)/NB

det[W−1[U ](ρq)

]= det

[W−1[U ]

(κq + ∆q

κq

)]× det

[W−1[U ]

(κq + 2∆q

κq + ∆q

)]

× . . . × det

[W−1[U ]

(κ′

q

κq + (NB − 1)∆q

)],

reduce fluctuations of the reweighting factors

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§3. Parameters

simulation parameters

– original: (κud,κs)=(0.137785,0.136600)– 1000 MD time, still increasing– MP2DDHMC for ud quark with 84 block, ρ1 = 0.9995, ρ2 = 0.99– UV-filtered PHMC for s quark with Npoly = 220

reweighting parameters

– target: (κ′ud,κ

′s)=(0.137800,0.136645), (0.137800,0.136690)

– breakup intervals: ∆ud = (0.137800 − 0.137785)/2,– breakup intervals: ∆s = (0.136690 − 0.136600)/4– Nη = 10 for stochastic estimation of Rud,s

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§4. Preliminary Results

concentrate on (κ′ud, κ′

s) = (0.137800,0.136645)

• results for Rud,s

• Reweighting for plaquette

• Reweighting for mπ, mK, mΩ

(physical inputs for mud, ms, a−1)

• hadron spectrum

• locate the physical point

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Reweighting factors on each configuration

0 50 100 150 200

Conf.

0.01

0.1

1

10

100

Rud

Rs

Rud

Rs

Reweighting factors

normalized with ⟨Rud,s⟩ = 1

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Reweighting factors vs. plaquette value

0.5708 0.5709 0.571 0.5711 0.5712 0.5713

PLQ

0.01

0.1

1

10

100

Rud

Rs

Rud

Rs

Reweighting factors

clear dependence

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Plaquette histgram w/ and w/o Rud

-0.0002 -0.0001 0 0.0001 0.0002PLQ − <PLQ>

0

5

10

15

20

25 originalreweighted w/ R

ud

distribution is slightly moved toward larger values

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Plaquette histgram w/ and w/o Rs

-0.0002 -0.0001 0 0.0001 0.0002PLQ − <PLQ>

0

5

10

15

20

25 originalreweighted w/ R

s

distribution is slightly moved toward larger values

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π effective mass

0 10 20 30

t0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

originalPQPQ+RW

π effective mass

reweighting effects are observed

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K effective mass

0 10 20 30

t0.22

0.23

0.24

0.25

originalPQPQ+RW

K effective mass

similar to π case

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Ω effective mass

0 5 10 15 20

t0.72

0.74

0.76

0.78

0.80

0.82

0.84

originalPQPQ+RW

Ω effective mass

mΩ is slightly decreased

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Hadron spectrum in comparison with experiment

0

-0.2

-0.1

0.0

0.1

0.2

0.3

π K ρ K* φ N Λ Σ Ξ ∆ Σ* Ξ* Ωη

ss

originaltarget

(mh/mΩ)

lat / (m

h/mΩ)

exp - 1

mπ/mΩ, mK/mΩ are properly tuned

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Locate the physical point

7.256 7.257 7.258 7.2581/κ

ud

7.3150

7.3175

7.3200

7.3225

1/κ s

originaltarget (0.137800,0.136645)target (0.137800,0.136690)estimated physical point

physical point

confirmed with three data point analysis

∆mud ∼ 1MeV, ∆ms∼<3MeV

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§5. Summary

– Chiral extrapolation is not necessary anymore

– (6fm)3 box simulation is under way

– starting point for precision measurements

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