素粒子宇宙起源研究機構

for the Origin of Particles and the Universe

格子量子色力学を用いた軽い原子核の計算

素粒子宇宙起源研究機構

Calculation of small nuclei from lattice QCD

Takeshi Yamazaki
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe
Nagoya University
Y. Kuramashi and A. Ukawa for the PACS-CS Collaboration
PRD84:054506(2011)
素核宇融合による計算基礎物理学の進展 - ミクロとマクロのかけ橋の構築 @合歓の郷, Dec. 3–5

1. Introduction
Nuclei from first principle of strong interaction→ lattice QCD
Recent studies of lattice QCD for bound state of multi-baryon systems
1. Three- and four-nucleon systems’10 PACS-CS Nf = 0 mπ = 0.8 GeV
2. H dibaryon in ΛΛ system (S=−2, I=0)(’88 Iwasaki et al. Nf = 0 mπ > 0.7 GeV)
’11 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV’11 HALQCD Nf = 3 mπ = 0.67–1.02 GeV’11 Luo et al. Nf = 0 mπ = 0.5–1.3 GeVBound state in simplest multi-nucleon system, NN system?
1

Expoloratory study of three- and four-nucleon systems
PACS-CS Collaboration, PRD81:111504(R)(2010)

Identification of bound state from volume dependence of ∆E00-0.01

ΔEL[GeV]-0.02

4He-0.03

-0.04

-0.04
S1
S2exp.

-0.06
-0.07

02e-054e-05
33He-0.02

-0.03

-0.05

ΔEL[GeV]-0.01

S1S2exp.

-0.05
-0.06
6e-05

8e-05

-0.07

02e-054e-05
36e-05

8e-05

1/L

1/L

∆E4He = 27.7(7.8)(5.5) MeV∆E3He = 18.2(3.5)(2.9) MeV

1. Observe bound state in both channels
2. Same order of ∆E to experiment

However, several systematic errors, e.g., mπ = 0.8 GeV
1-a

1. Introduction
Nuclei from first principle of strong interaction→ lattice QCD
Recent studies of lattice QCD for bound state of multi-baryon systems
1. Three- and four-nucleon systems’10 PACS-CS Nf = 0 mπ = 0.8 GeV
2. H dibaryon in ΛΛ system (S=−2, I=0)(’88 Iwasaki et al. Nf = 0 mπ > 0.7 GeV)
’11 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV’11 HALQCD Nf = 3 mπ = 0.67–1.02 GeV’11 Luo et al. Nf = 0 mπ = 0.5–1.3 GeVBound state in simplest multi-nucleon system, NN system?
1-b

Scattering length a0 in NN system from lattice QCD at ∼’0925220

1a0( S0)[fm]150

3a0( S1)[fm]experiment a0=23.7[fm]Fukugita et al.(1995) (L=2.7 fm, Nf=0)

105-2

NPLQCD(2006) (L=2.5 fm, Nf=2+1)

Fukugita et al.(1995) (L=2.7fm, Nf=0)

Ishii et al.(2008) (L=4.4 fm, Nf=0)

NPLQCD(2006) (L=2.5 fm, Nf=2+1)

NPLQCD(2009) (L=2.5 fm, Nf=2+1)

Ishii et al.(2008) (L=4.4 fm, Nf=0)-4NPLQCD(2009) (L=2.5 fm, Nf=2+1)
experiment a0=-5.47[fm]000.511.522mπ[GeV ]22.5-600.5

11.522mπ[GeV ]22.5a0: Far from experiments due to mπ >∼0.3 GeV
Deuteron: 3S1 channel ∆Ed = 2.2 MeV

Assumption: unbound due to mπ >∼0.3 GeVAim of this work : check assumption with simpler methodc.f. using nuclear force ’09 HALQCD
2

Existence of bound state for a0System0th
1sta0w/ bound statebound state
scattering state< 0 from 1stw/o bound state
scattering state
scattering state> 0 from 0th

Bound state exists → a0 never obtained from 0th state
by Lüscher’s finite volume method
4πa00∆EL = EN N − 2M = −+ ···(’86, ’91 Lüsher)
M L3

Need to check existence of bound state to calculate a0
Two methods
1. Volume dependence of 0th state
2. Properties of 1st state energy3Outline
1. Introduction
2. Methods
1. Identification of bound state
2. Properties of 1st excited state
3. Simulation parameters
4. Results
1. Single state analysis
2. Two-state analysis
5. Summary and future work4

2. Methods

1. Identification of bound state in finite volume
observe small ∆EL = E − 2m < 0 at one L is not enough2mbound0

2e-05

4e-05

attractive

6e-05

38e-050.0001

1/L!
"−CLBound state : ∆EL = ∆E∞ + O e<0’04# Beane et$ al., ’06 Sasaki & TY
aAttractive scattering state : ∆EL = O − 0 3 < 0 (a0 > 0)
ML’86, ’91 Lüscher
52. Methods

1. Identification of bound state in finite volume
observe small ∆EL = E − 2m < 0 at several L2mbound0

2e-05

4e-05

attractive

6e-05

38e-050.0001

1/L!
"−CL
Bound state : ∆EL = ∆E∞ + O e<0’04# Beane et$ al., ’06 Sasaki & TY
aAttractive scattering state : ∆EL = O − 0 3 < 0 (a0 > 0)
ML’86, ’91 Lüscher
5-a2. Methods
2. Properties of 1st excited state in finite volume’06 Sasaki & TY

2(m 2+(2π/L 2) ) 1/2

1st excited state ∆EL = E − 2m > 0 at finite L2mring
scattebound
02e-05

4e-05

6e-05

3

#1/L

8e-05

0.0001$a0Scattering state : ∆EL = O −> 0 (a0 < 0)’86, ’91 Lüscher
M L3
1st excited state ← diagonalization method ’90 Lüscher & Wolff
6

3. Simulation parameters• Quenched Iwasaki gauge action at β = 2.416
a−1 = 1.54 GeV with r0 = 0.49 fm• Tad-pole improved Wilson fermion actionmπ = 0.8 GeV and mN = 1.62 GeV
reduce large statistical fluctuation

1. Finite volume dependepce of 0th state (Single state analysis)• Three volumes: L = 3.1, 6.1, 12.3 fm• Two smearing sources: for consistency check2. Property of 1st excited state (Two-state analysis)• Two volumes: L = 3.1, 6.1 fm• Wavefunction smearing sink operators
assumption: 0th energy = one of single state analysis

Simulations:
PACS-CS, T2K-Tsukuba at Univ. of Tsukuba, HA8000 at Univ. of Tokyo
7

4. Results
1. Single state analysis
Effective energy shift ∆EL%= ENN −&2mN at L = 6.1 fm
R(t)CNN(t), R(t) =∆EL = log
R(t + 1)
(CN (t))2O1
O2

O1
O2

0.000

0.000

-0.005-0.005-0.010-0.0101S03

S1-0.015048t1216• ∆EL < 0 in 8<∼t ≤ 11
• consistent plateaus in 8<∼t ≤ 11

20-0.015

048t1216820

1. Single state analysis
Volume dependence of ∆EL
0.000

0.000
O1O2exp-0.002-0.004-0.004-0.006-0.006-0.008-0.008-0.010-0.0103-0.012-0.01602e-054e-05
1/L

S0 ΔE[GeV]-0.0146e-05

31-0.012

S1 ΔE[GeV]-0.014O1O2-0.002

8e-05-0.01602e-054e-05
1/L

36e-05• ∆EL < 0 in three volumes∆EL = −γ2mN')√'2(Cγexp(−γL $n )γ2√1+, ∆Ebind =2γLmN$n$n’04 Beane et al., ’06 Sasaki & TY

• Bound state in both channels ← incosistent with experiment∆E3S = 9.1(1.1)(0.5) MeV∆E1S = 5.5(1.1)(1.0) MeV
18e-051might be caused by heavy quark mass in calculation
92. Two-state analysis
Effective energy
shift for +ground and 1st excited states at L = 6.1 fm*Diag G−1(t0)G(t) = λ(t) with Gij (t) = )0|Oi(t)Oj (0)|0*%&Rα(t)λα(t)∆E L,α = Eα − 2mN = log, Rα(t) =Rα(t + 1)
(CN (t))20.015

0.015

O1 (single ana)
ΔEL,00.010

0.010

ΔEL,1
free 2N

0.005

O1 (single ana)
ΔEL,0ΔEL,1
free 2N

0.005

0.000

0.000

-0.005-0.005

3S1-0.010-0.01501-0.010

48

t

12

16

20-0.015

S0048

t

122016• ∆E L,0 < 0 and consistent with ∆EL• small, but ∆E L,1 > 0 as expected10

2. Two-state analysis
Volume dependence of ∆E L,10.025

310.020

S1S00.015
0.010
0.005

ΔEL,10.000
02e-054e-05
1/L• ∆E L,1 > 0 and 1/L3 tendency• Scattering length a0 fm
L[fm]
3.1 !
"+0.2
3S
−1.5(0.2) −1.4
11S0!

−1.8(0.3) +0.4
−12.9

3

"

6e-05

8e-05

6.1 !
"+0.05−1.05(24) −0.65!

−1.62(24) +0.01
−0.75

"

Observe expected properties of 1st excited state11

5. Summary and future work
Exploratory study of two-nucleon bound statein quenched lattice QCD• Unphysically heavy quark mass
• Volume dependence of energy shift of ground state• Properties of 1st excited state energy
1. ∆E += 0 of 0th state in infinite volume limit
2. Expected properties of 1st excited state→ bound state in 3S1 and 1S0 at mπ = 0.8 GeVBound state in 1S0 : observed in Nf = 2 + 1 QCD at mπ = 0.39 GeV(NPLQCD Collaboration, arXiv:1109.2889[hep-lat])Not observed in experiment12

Future work
Bound state in 1S0 vanishes as quark mass decreases?
• Quark mass dependence of ∆E and a0• Reduce systematic errors

Heavy quark massmπ = 0.8 GeV → 0.7 and 0.5 GeV
Quenched effectNf = 2 + 1 QCD
Lattice spacing dependence
Far future projectCalculation of NN, 3He, 4He channels
13Future work
Bound state in 1S0 vanishes as quark mass decreases?
• Quark mass dependence of ∆E and a0• Reduce systematic errors

Heavy quark massmπ = 0.8 GeV → 0.7 and 0.5 GeV
Quenched effectNf = 2 + 1 QCD
Lattice spacing dependence
Far future projectCalculation of NN, 3He, 4He channels
14Very preliminary results at mπ = 0.7 GeV
00-0.02

-0.02

-0.04

-0.04

Preliminary

-0.06
-0.08

Nf=2+1 mπ=0.7GeV
Nf=0 mπ=0.8GeV
ΔEL[GeV]
exp.

-0.1
-0.12

01e-052e-05
3Preliminary
3-0.0643e-05ΔEL[GeV]

-0.08

He

Nf=2+1 mπ=0.7GeV
Nf=0 mπ=0.8GeV
exp.

-0.1

4e-05

-0.12

01e-051/L

2e-05
33e-05

4e-05

1/L

0.000

0.000
Nf=2+1 mπ=0.7GeV
Nf=0 mπ=0.8GeVexp-0.004-0.008-0.004-0.008Preliminary-0.012-0.016-0.016

S1 ΔEL[GeV]-0.02001e-05

2e-05
1/L

3

Preliminary-0.0123-0.024

He

1S0 ΔEL[GeV]
Nf=2+1 mπ=0.7GeV
Nf=0 mπ=0.8GeV-0.0203e-05-0.02404e-05

1e-05

2e-05
1/L

33e-054e-05
15