1

Lattice QCD:placing a hadron in a computer

Introduction

Current focus: dynamical Nf=2+1 simulation

Nucleon structure functions

Future perspectives

Summary

Akira UkawaCenter for Computational SciencesUniversity of Tsukuba

Nikko Symposium“Creation and development of matter”6 November, 2004

2

Introduction

3

Quantum Chromodynamics

Quantum Chromodynamics (QCD)Fundamental theory of quarks and gluons and their strong interactions

Knowing

1 coupling constant and

6 quark masses

will allow full understanding of strong interactions “Yukawa’s dream(1935) in modern form”

tbcsdu

s

mmmmmm ,,,,,α

( ) ( )( )

( ) ( ) S

ffff

sQCD

eAOddAdZ

AO

miAFFTrS

−∫

∑

=

+−∂⋅+=

ψψψψψψ

ψγψπα µµµµνµν

,,1,,

81

Gross-Wilczek-Politzer 1973

4

QCD on a Lattice

Lattice QCDPowerful mean to calculate the QCD Feyman path integral

From computational point of viewRelatively simple calculation

Uniform meshSingle scale

Requires much computing power due to

4-dim. ProblemFermions essentialPhysics is at lattice spacing a=0

Precision required(<a few % error in many cases)

( ) ( )

( ) ( ) S

ffff

PsQCD

eUOddUdZ

UO

mUUUUUtrS

−∫

∑∑

=

+⋅+=

ψψψψψψ

ψγψα

,,1,,

1

QCDparameterscaleQCDaspacinglattice

Λ

Wilson 1974

5

Development of lattice QCD simulations (I)

msize 15102 −×≈

lattice size lattice spacing

L= 0.8 fm a = 0.1 fm

1981 First lattice QCD simulation

VAX

Mflopsspeed 1≈

４４～ ８４ latticequenched approx (no sea quarks)

Creutz-Jacobs-RebbiCreutzWilsonWeingartenHamber-Parisi

6

L(fm) a(fm)

１９８１ 0.8 0.1１９８５ 1.2 0.1１９８８ 1.6 0.1

1980’s Taking advantage of vector supercomputers

計算機性能の進化

CRAY-1

1 GFLOPS = 毎秒10億回の浮動小数点演算

Development of lattice QCD simulations (II)

7

L(fm) a(fm)１９９３ 2.4 0.07 QCDPAX（JPN） APE（Italy）

1 GFLOPS = 毎秒10億回の浮動小数点演算1 TFLOPS = 毎秒1兆回の浮動小数点演算

Columbia（USA） GF11（USA）

Development of lattice QCD simulations (III)

1990’s QCD dedicated computers

8

L(fm) a(fm)１９９８ 3.0 0.05

Development of lattice QCD simulations (IV)

CP-PACS(JPN) QCDSP(USA)

2000s End of quenching and beginning of full QCD

9

Subject areas of lattice QCD

LQCD

( ) ( ) NONqqxFdxx nn nγ−− =∫ 2

1

0

21 ln,

tbcsdu

s

mmmmmm ,,,,,α

Finite-temperature/density behavior

• eta’ meson mass and U(1) problem• exotic states

glueball, dibaryon, hybrids,…• hadronic matrix elements

proton spin, sigma term, ….• structure functions/form factors

Weak interaction matrix elements

Hadron spectrum and Fundamental constanst of QCD

Hadron physics

• Strong coupling constant• Quark masses

• order of transition• critical temperature/density• equation of state

• K meson amplitudesBKK→ππ decays

• B meson amplitudesfB, BB, form factors

Physics of quark-gluon plasma

CKM matrix and CP violation

10

CP-PACS

Energy density calculated from lattice QCD

Lattice QCD and QGP

11

experiment

Determination of

Comments Davies et al ’97(hep-lat/9703010):Involved extrapolation of Nf=0(quenched) and Nf=2 data to Nf=3

QCDSF-UKQCD ’01(hep-lat/0103023)Continuum estimate with systematic Nf=2 simulations

Davies et al ’02(hep-lat/0209121):Preliminary result based on MILC Nf=3 configurations at a=0.13fm

Systematic Nf=3 full QCD determination expected in a few years

( ) 5=fNz

SMs Mα

Lattice QCD

( ) 5=fNZ

MSs Mα

12

Constraints on the CKM matrix

mixingKK o 0−

mixingBB sdsd0,

0, − ( )

qqq BBB BfM 222 ηρ +∝∆

( ) ( ) 22055 3

811qqq BBBq

oq MBfBqbqbB =−− γγγγ µµ

( )[ ] KK BBA ˆ1 2+−∝ ρηε

( ) ( ) 22055

0

3811 KKK MBfKdsdsK =−− γγγγ µµ

d

s

d

s

B

B

B

B

BB

ff

=ξ

hHh weak'Measured weak amplitude = Known factor

including CKM matrix

QCD matrix element

X

BK

BB fB

13

Summary of lattice results for CKM matrix

240

034

06.013.0

)5(16.1

)37(227.0

87.0ˆ

+−

+−

+−

=

=

=

ξ

GeVBf

B

dd BB

K

Cf. numbers used in the figure left

05.004.021.1

1030228.0

14.006.086.0ˆ

±±=

±±=

±±=

ξ

GeVBf

B

dd BB

K

status 2004 http://www.ckmfitter.in2p3.fr/

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5 2

sin 2β

B → ρρ

B → ρρ

∆md

∆ms & ∆md

εK

εK

|Vub/Vcb|

sin 2β

α

βγ

ρ

η

excluded area has CL < 0.05

C K Mf i t t e r

ICHEP 2004

14

Current focus

15

Current focus: Nf=2+1 dynamical simulation (I)

Quenched approximation : ignore sea quark effects

Acceptable for heavy quarks : c, b, t m>1GeV

Misses important effects with light quarks : u, d, s m<0.1GeV

Dynamical simulation (“unquenching”) is not easy

Need O(10)-O(100) more computer power

Algorithm for odd number of fermions developed only recently

16

Quenching effects on the hadron specturm

Sea quark effects ignored

General pattern reproduced, but clear systematic deviation beyond 10% precision

Calculated quenched spectrum

CP-PACS Collaboration 1998

17

Recent progress due to Development of Tflops-class computersAlgorithms for odd quark flavors (PHMC, RHMC, …)

Two-flavor full QCD (since around 1996)u and d quark dynamical and degenerates quark quenched (no sea quark)

Number of studies: SESAM/UKQCD/MILC/CP-PACS/JLQCD

Two+one-flavor full QCD (since around 2000)u and d quark dynamical and degenerates quark dynamical and heavier

Extensive studies have begun : MILC/CP-PACS-JLQCD

2=fN

12 +=fN

Current focus: Nf=2+1 dynamical simulation (II)

18

Tsukuba/KEK joint effort toward Nf=2+1

1

23

21 2a

β=1.90a ～ 0.10fm20^3 x 408000 trajectory

already finished

β=1.83a ～ 0.12fm16^3 x 325000 trajectory

almost finishedβ=2.05a ～ 0.07fm28^3 x 562000 trajectory

on-going

Fixed physical volume～ (2.0fm)^3 Lattice spacing

Earth simulator@ Jamstec

SR8000/F1@KEK

CP-PACS@Tsukuba

SR8000/G1@Tsukuba

VPP5000@Tsukuba

A three year project 2003-2005

19

Hyperfine splitting at a ～ 0.1fm

Progressively closer agreement with experiment:

Nf=0quenched approxNf=2 u&d quarks dynamicalNf=2+1 u,d,s quarks dynamical

Remaining difference due to finite lattice spacing?-10

-5

0

5

10

(m-m

exp)/

mex

p [%]

mK* (K-input)

Nf=2+1 (RG+clover(NPT))Nf=2 (RG+clover(PT))Nf=0 (RG+clover(PT))

mφ (K-input) mK (φ-input) mK* (φ-input)

a ~ 0.1 fm a ~ 0.1 fma ~ 0.1 fma ~ 0.1 fm

mK* mφ

mK

mK*

examine a dependence

20

Hyperfine splitting toward the continuum limit

0 0.1 0.2 0.3 0.4 0.5

a2 [GeV

-2]

0.85

0.9

0.95

1

1.05

mes

on m

ass

[GeV

]

φ

K*

experiment

K-input

β=1.83

β=1.83

β=1.90

β=1.90

β=2.05

β=2.05

0 0.1 0.2 0.3 0.4 0.5

a2 [GeV

-2]

0.4

0.5

0.6

0.7

0.8

0.9

1

mes

on m

ass

[GeV

]

K

K*

experiment

φ-input

β=1.83β=1.90β=2.05

β=2.05

β=1.90 β=1.83

preliminary

21

Light quark mass results at a ～ 0.1fm

2.0

3.0

4.0

mud

MS(µ

=2G

eV)

Nf=2+1 Nf=2 Nf=0

a ~ 0.1 fm

AWI

a ~ 0.1 fma ~ 0.1 fm

RG + clover(PT) RG + clover(PT)

AWI AWI

K-input

φ-input

RG + clover(NPT)

60

80

100

120

140

160

msM

S(µ

=2G

eV)

Nf=2+1 Nf=2 Nf=0

RG + clover(PT) RG + clover(PT)

a ~ 0.1 fm a ~ 0.1 fmAWI AWI

RG + clover(NPT)

K-input

AWIa ~ 0.1 fm

φ-input

φ-input

K-input

φ-input

K-input

MeVGeVm

MeVmmGeVm

MSs

duMS

385)2(

06.005.32

)2(

±==

±=+

==

µ

µ

Sizably small compared to folklore,

e.g. mud ～ 5MeV, ms ～ 150MeV

22

Nucleon structure functions

23

Nucleon structure functions (I)

Lattice QCD provides normalization of moments in terms of nucleon matrix elements of a set of operators

Progressively difficult for higher moments due to complicated operator structure

Series of calculations by the German group Nf=0 quenched calculationNf=2 u&d dynamical

( ) ( ) NONqqxFdxx nn nγ−− =∫ 2

1

0

21 ln,

qDDqiO nn nn

µµµµµµ γt

Lt

L 2121 1−= etc

QCDSF: M. Goeckeler et al

24

QCDSF quenched results

0.00 0.01 0.02 0.03 0.04(a/r0)

2

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

v3

rgi

v4

rgi

0.00 0.01 0.02 0.03 0.04(a/r0)

2

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

v2a

rgi

v2b

rgi (p=p1)

v2b

rgi (p=0)

3x

x

2x

Flavor non-singlet moments for n=2,3,41st moment shows a sizable deviation from experiment ( a long-standing puzzle)

calculated with 3 types of lattice operators

experiment

hep-ph/0410187

25

QCDSF Nf=2 results

0 1 2 3 4 5 6 7 8

(r0 m

PS)2

0

0.2

0.4

√2 mK

v2b

(u-d) RGI

0 0.02 0.04 0.06

(a / r0)2

0

0.2

0.4

x

2x

3x

Similar trend as the quenched case, i.e., Reasonable agreement for n=3,4Significant deviation for n=2

experiment

hep-lat/0409162

26

Moments of nucleon structure functions (II)

Linear chiralextrapolation misses experimentSmall quark mass behavior is the issuePion not light enough to see curvature?

From D. Dolgov et al hep-lat/0201021

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+Λ+

−= 222

22

2

2

ln4

131 π

πχ

πππ

dmm

mm

fg

ax An

n

W. Detmold et al hep-lat/0103006

dux

−

2πm

27

Future perspectives

28

Next generation of computers

10 TFLOPS class machines for QCD

QCDOC (USA) 2004-2005

Well advanced

1st shipment to UK this month

Large installation in USA early next year

ApeNEXT (Italy) 2005

Some delay in R&D

Large installation in Italy and possibly Germany

CP-PACS II (Japan) 2006

Development to start in fiscal 2005

2/3 installation in early 2006, completion in 2007

29

CP-PACSに至る歴史：20年間の発展

1978 1980 1989 1996

世界最高速を達成したCP-PACS

第１号機PACS-9

22億円 (5年間）競争的資金（科研費創成基礎研究）614億回/秒CP-PACS1996年

3億円 (3年間）競争的資金（科研費特別推進）１４億回/秒QCDPAX1989年

3000万円基盤的研究費＋学内裁量経費3百万回/秒PAX-32J1984年

600万円基盤的研究費4百万回/秒PAX-1281983年

150万円基盤的研究費50万回/秒PAXS-321980年

150万円基盤的研究費7千回/秒PACS-91978年

金額開発・制作費の種類計算速度名称完成年

第２号機PAXS-32

第５号機QCDPAX

30

0.1

1

10

100

1000

104

105

106

1975 1980 1985 1990 1995 2000 2005 20100.0001

0.001

0.01

0.1

1

10

100

1000

GFLOPS

year

TFLOPS

CRAY-1

CP-PACS

Earth Simulator

CP-PACS-II

cluster systemsQCD-PAX

実機設計

試作・検証・確定

実機製作

計算開始

システムソフトウェア開発

計算物理プログラム開発

全システムによる計算の実施

25％ 100％

平成15年4月 平成16年4月 平成17年4月 平成18年4月 平成19年4月

基本仕様検討

テストシステム構築・テスト実施

平成20年4月

CP-PACS-II project (I)

計算科学研究センター発足

CP-PACS稼動10周年

平成18年10月

75％

計画開始

全システム完成

準備研究実施中

高性能化の著しいコモディティ（汎用品）のプ

ロセッサ技術、ネットワーク技術を用いて超

並列システムを短期に開発・製作

CP-PACS-II（仮称） CP-PACS

ピーク性能 24.6TFLOPS 0.61TFLOPS主記憶容量 6.1TByte 0.128TByteディスク容量 1.05PByte 1.05PByte開発期間 2年（H17～18年度） 4.5年（H4年度～8年度）完成時期 平成18年度末 平成8年9月

1TFLOPS=1兆回／秒の計算性能

31

計算物理学研究センターの計算科学研究センターへの

改組・拡充計算物理学研究センター

計算素粒子物理学研究部門

計算宇宙物理学研究部門

計算物性物理学研究部門

計算生命物理学研究部門

並列計算機工学研究部門

計算科学研究センター

素粒子宇宙研究部門

物質生命研究部門

地球生物環境研究部門

超高速計算システム研究部門

計算情報学研究部門

平成4年度～15年度 平成16年度発足

2

2

2

2

3

教員 11名（教授４ 助教授4 講師３）

6

11

3

5

6

教員 31名（教授11 助教授12 講師8）

計算科学研究センター全体の

事業として実施

事業統括責任者（センター長）

物質・生命プロジェクト責任者

素粒子・宇宙プロジェクト責任者

クラスタ計算機プロジェクト責任者

平成16年4月改組・拡充

CP-PACS-II project (II)

32

International Research Network for Computational Particle Physics

UK core institution:University of Edinburgh

Dept. of PhysicsEPCC

Germany core institution:

DESYVon Neumann

Inst.for computing

USA core institution:Fermi National Accelerator

Laboratory(FNAL)

Japan core institution:

University of Tsukuba

Center for Computational

Sciences

ＳｃｉＤＡＣNetworkin USA

Edinburgh

GlasgowLiverpool

Southampton

Swansea DESY/NeumannBerlin/Zeuthen

BielefeldRegensburg LatFor

Networkin Germany

KEK

Hiroshima U

LFT ForumNetworkin Japan

Future expansion to EU NetworkItaly, France, Spain, Denmark,…

Main supercomputer sites

International Lattice Data Grid (ILDG)database of QCD gluon configurations at major supercomputer facilities acceleration of research via mutual usage of QCD gluon configurations via fast internetfuture international sharing of supercomputingand data storage resources

Future expansion to Asia/Oceania

Kyoto U

UKQCDNetworkin United Kingdom

U. Tsukuba

ＦＮＡＬ

Ｗａｓｈｉｎｇｈｏｎ Ｕ

ＵＣＳＢ

MIT/Boston U

BNL/Columbia

JLAB

Arizona

Utah

Indiana

St．Louise

JSPS core-to-core program

QCDOC x 2

QCDOC

APENEXT

CP-PACS II

KEK supercomputer

33

Summary

Definitive prospect toward exact QCD predictions with realistic quark spectrum over the next few years

Firm numbers to our phenomenology/experiment colleaguesQuantitative understanding of the full range of strong interactions

1935 meson theory (Yukawa)1951 strangeness (Gell-Mann-Nishijima) 1961 chiral symmetry and pion(Nambu-Jona-Lasinio)1973 QCD and asymptotic freedom(Gross-Wilczek-Politzer)1974 Lattice QCD(Wilson)1981 Monte Carlo simulation of QCD(Creutz-Jacobs-Rebbi)

Lattice QCD: a prototype of computational research in which theory and instrument making go hand in hand.