素粒子宇宙起源研究機構

Calculation of light nuclei from lattice QCD

Takeshi Yamazaki
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe
Nagoya University
K.-I. Ishikawa, Y. Kuramashi, A. Ukawa
Refs: PRD81:111504(R)(2010); PRD84:054506(2011); PRD84:054506(2012)

QUCS 2012 @ the Reception Hall in Nara Prefectural New Public Hall,
December 13–16 2012

Supports from Scientific Research on Innovative Areas「素核宇宙融合による計算科学に基づいた重層的物質構造の解明」

1. Researcher of A01 group at Univ. of Tsukuba
2. Proposed project grants
2009-2010
Study of light nuceli using lattice quantum chromodynamics
2011-2012
Study of calculation method of light nuclei
using lattice quantum chromodynamics
3. · · ·Appreciate all supports1Outline
1. Introduction
2. Problems of multi-nucleon bound state
3. Simulation parameters
4. Nf = 2 + 1 results at mπ = 0.5 GeV
4He, 3He, NN channels

5. Summary and future work3

1. Introduction
Strong interaction!
protons and neutrons → nuclei
Bindquarks and gluons → protons and neutrons
Spectrum of nuclei
success of Shell model: Jensen and Mayer(1949)
degrees of freedom of protons and neutronsSpectrum of proton and neutron (nucleons)
success of non-perturbative calculation of QCD
such as lattice QCD
degrees of freedom of quarks and gluons

Motivation: Understand property of nuclei from (lattice) QCD
Shell model"#$%
quarksand gluons →%"protons and neutrons# → nuclei$(lattice) QCD

4

Lattice QCD result at physical pion mass ’10 PACS-CS
target = physical pion mass(mh/mΩ)lat / (mh/mΩ)exp−1

0.20

0.100.00
original
target-0.10π Kηss ρ K* φNΛΣΞΔΣ∗ Ξ∗ Ω0Consistent nucleon mass with experiment5

1. Introduction
Strong interaction!
protons and neutrons → nuclei
Bindquarks and gluons → protons and neutrons
Spectrum of nuclei
success of Shell model: Jensen and Mayer(1949)
degrees of freedom of protons and neutrons
Spectrum of proton and neutron (nucleons)
success of non-perturbative calculation of QCD
such as lattice QCD
degrees of freedom of quarks and gluons

Motivation: Understand property of nuclei from (lattice) QCD
Shell model"#$%
quarksand gluons →%"protons and neutrons# → nuclei$(lattice) QCD

61. Introduction
Strong interaction!
protons and neutrons → nuclei
Bindquarks and gluons → protons and neutrons
Spectrum of nuclei
success of Shell model: Jensen and Mayer(1949)
degrees of freedom of protons and neutrons
Spectrum of proton and neutron (nucleons)
success of non-perturbative calculation of QCD
such as lattice QCD
degrees of freedom of quarks and gluons

Motivation: Understand property of nuclei from (lattice) QCD directly"

Shell model#$%
quarksand gluons → protons
and neutrons → nuclei#$%"(lattice) QCD

6-a1. Introduction
Motivation :Understand property of nuclei from (lattice) QCD directly
through different approach from nuclear force, HALQCD

If we can study nuclei from QCD, we may be able to1. reproduce spectrum of nuclei
2. predict property of nuclei hard to calculate or observe
such as neutron rich nuclei

So far only few works for multi-baryon bound states
Before studying such difficult problems, we should check→ Can we calculate known binding energy in a-few-nucleon systems?
7Multi-baryon bound state from lattice QCDNot observed before ’09 (except H-dibaryon ’88 Iwasaki et al.)Recent studies of lattice QCD for bound state of multi-baryon systems
1. 4He and 3He channels’10 PACS-CS Nf = 0 mπ = 0.8 GeV PRD81:111504(R)(2010)’12 HALQCD Nf = 3 mπ = 0.47 GeV, mπ > 1 GeV 4He’12 NPLQCD Nf = 3 mπ = 0.81 GeV
2. H dibaryon in ΛΛ channel (S=−2, I=0)
’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 GeV’12 NPLQCD Nf = 3 mπ = 0.81 GeV
3. NN channels’11 PACS-CS Nf = 0 mπ = 0.8 GeV PRD84:054506(2011)’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV (Possibility)’12 NPLQCD Nf = 3 mπ = 0.81 GeVOther channels: ’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV; ’12 Nf = 3 mπ = 0.81 GeV
Related studies: BB forces HALQCD [Doi and Inoue]; ΩΩ scattering Buchoff et al.
8Expoloratory study of 3-, 4-nucleon systems
PACS-CS Collaboration, PRD81:111504(R)(2010)

Identification of bound state from volume dependence of ∆E
00-0.01

ΔEL[GeV]-0.02

4He-0.03

-0.04

-0.04
S1S2exp.

-0.06
-0.07

02e-054e-05
33He-0.02

-0.03

-0.05

ΔEL[GeV]-0.01

S1
S2exp.

-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 experimentContaining several systematic errors, e.g., mπ = 0.8 GeV8-a

Multi-baryon bound state from lattice QCD
Not observed before ’09 (except H-dibaryon ’88 Iwasaki et al.)Recent studies of lattice QCD for bound state of multi-baryon systems
1. 4He and 3He channels’10 PACS-CS Nf = 0 mπ = 0.8 GeV PRD81:111504(R)(2010)’12 HALQCD Nf = 3 mπ = 0.47 GeV, mπ > 1 GeV 4He’12 NPLQCD Nf = 3 mπ = 0.81 GeV
2. H dibaryon in ΛΛ channel (S=−2, I=0)
’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 GeV’12 NPLQCD Nf = 3 mπ = 0.81 GeV
3. NN channels’11 PACS-CS Nf = 0 mπ = 0.8 GeV PRD84:054506(2011)’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV (Possibility)’12 NPLQCD Nf = 3 mπ = 0.81 GeVOther channels: ’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV; ’12 Nf = 3 mπ = 0.81 GeV
Related studies: BB forces HALQCD [Doi and Inoue]; ΩΩ scattering Buchoff et al.
9Expoloratory study of NN nuclei
PACS-CS Collaboration, PRD84:054506(2011)

Identification of bound state from volume dependence of ∆E0 and ∆E1
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-05

4e-05
1/L

31

S0 ΔE[GeV]-0.012S1 ΔE[GeV]-0.014O1O2-0.002-0.014
6e-05

8e-05

-0.016

Observe bound state in both channels∆E3S = 9.1(1.1)(0.5) MeV
1Same order to experiment02e-054e-05
1/L

36e-05∆E1S = 5.5(1.1)(1.0) MeV
0Not observed in experimentmight be caused by heavy quark mass in calculation
9-a8e-05

Multi-baryon bound state from lattice QCD
Not observed before ’09 (except H-dibaryon ’88 Iwasaki et al.)Recent studies of lattice QCD for bound state of multi-baryon systems
1. 4He and 3He channels’10 PACS-CS Nf = 0 mπ = 0.8 GeV PRD81:111504(R)(2010)’12 HALQCD Nf = 3 mπ = 0.47 GeV, mπ > 1 GeV 4He’12 NPLQCD Nf = 3 mπ = 0.81 GeV
2. H dibaryon in ΛΛ channel (S=−2, I=0)
’11 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV’12 HALQCD Nf = 3 mπ = 0.47–1.17 GeV’11 Luo et al. Nf = 0 mπ = 0.5–1.3 GeV’12 NPLQCD Nf = 3 mπ = 0.81 GeV
3. NN channels’11 PACS-CS Nf = 0 mπ = 0.8 GeV PRD84:054506(2011)’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV (Possibility)’12 NPLQCD Nf = 3 mπ = 0.81 GeVOther channels: ’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV; ’12 Nf = 3 mπ = 0.81 GeV
Related studies: BB forces HALQCD [Doi and Inoue]; ΩΩ scattering Buchoff et al.
10

Multi-baryon bound state from lattice QCD
Not observed before ’09 (except H-dibaryon ’88 Iwasaki et al.)Recent studies of lattice QCD for bound state of multi-baryon systems
1. 4He and 3He channels’10 PACS-CS Nf = 0 mπ = 0.8 GeV PRD81:111504(R)(2010)’12 HALQCD Nf = 3 mπ = 0.47 GeV, mπ > 1 GeV 4He’12 NPLQCD Nf = 3 mπ = 0.81 GeV
2. H dibaryon in ΛΛ channel (S=−2, I=0)
’11 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV’12 HALQCD Nf = 3 mπ = 0.47–1.17 GeV’11 Luo et al. Nf = 0 mπ = 0.5–1.3 GeV’12 NPLQCD Nf = 3 mπ = 0.81 GeV
3. NN channels’11 PACS-CS Nf = 0 mπ = 0.8 GeV PRD84:054506(2011)’12 NPLQCD Nf = 2 + 1 mπ = 0.39 GeV (Possibility)’12 NPLQCD Nf = 3 mπ = 0.81 GeV
Purpose: Extend our works to Nf = 2 + 1 QCD with smaller mπ and a10-a2. Problems of multi-nucleon bound state
Traditional method for example 4He channel&††#0|O4He(t)O4He(0)|0$ =#0|O4He|n$#n|O4He|0$e−Ent −−−→ A0 e−E0t
nt%1

Difficulties for multi-nucleon calculation
1. Statistical error'() *3
Statistical error ∝ exp NN mN − mπ t2→ heavy quark mass mπ = 0.5 GeV + large # of measurements
2. Calculation cost
Wick contraction for 4He = p2n2 = (udu)2(dud)2: 518400→ reduction using p(n) ↔ p(n), p ↔ n, u(d) ↔ u(d) in p(n)Multi-meson: ’10 Detmold and SavageMulti-baryon: ’12 Doi and Endres; Detmold and Orginos

3. Identification of bound state on finite volumeFinite volume effect of attractive scattering state ∆E0 = E0 −NN mN < 0→ Volume dependence of ∆E = E0 − NN mN
11

2. Problems of multi-nucleon bound state
Traditional method for example 4He channel&††#0|O4He(t)O4He(0)|0$ =#0|O4He|n$#n|O4He|0$e−Ent −−−→ A0 e−E0tnt%1

Difficulties for multi-nucleon calculation
1. Statistical error'() *3
Statistical error ∝ exp NN mN − mπ t2→ heavy quark mass mπ = 0.5 GeV + large # of measurements
2. Calculation cost PACS-CS PRD81:111504(R)(2010)
Wick contraction for 4He = p2n2 = (udu)2(dud)2: 518400 → 1107
→ reduction using p(n) ↔ p(n), p ↔ n, u(d) ↔ u(d) in p(n)Multi-meson: ’10 Detmold and Savage

Multi-baryon: ’12 Doi and Endres; Detmold and Orginos

3. Identification of bound state on finite volume
Finite volume effect of attractive scattering state ∆E0 = E0 −NN mN < 0→ Volume dependence of ∆E ’86,’91 Lüscher, ’07 Beane et al.
Spectral weight: ’04 Mathur et al., Anti-PBC ’05 Ishii et al.
12Identification of bound state in finite volumeobserve small ∆EL = E − 2m < 0 at one L is not enough

2m
bound

02e-054e-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
13

Identification of bound state in finite volumeobserve 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
13-a3. Simulation parameters
Nf = 2 + 1 QCD
Iwasaki gauge action at β = 1.90
a−1 = 2.194 GeV with mΩ = 1.6725 GeV ’10 PACS-CS
non-perturbative O(a)-improved Wilson fermion actionmπ = 0.51 GeV and mN = 1.32 GeV
ms ∼ physical strange quark mass

Finite volume dependence of ∆E0 with four volumes(4He, 3He, 3S1 and 1S0 channels)

L
32
40
48
64

L [fm]
2.9
3.6
4.3
5.8

NTraj
4000
2000
2000
1900

Nconf
200
200
200
190

Nmeas
192
192
192
256

Simulations:
PACS-CS, T2K-Tsukuba, HA-PACS at Univ. of Tsukuba, HA8000 at Univ. of Tokyoand K at AICS
14

4. Results∆EL in 4He and 3He channels at L. = 5.8 fm
R4He(t)C4He(t)∆EL = logwith R4He(t) =R4He(t + 1)
(CN (t))40.1

0.1

40.08

30.08

He0.06

He0.06

0.04

0.04
0.02

0.02
00-0.02

-0.02

-0.04
-0.06

048t

12

1620-0.04

0• Statistical error under control in t < 12• Relatively smaller error in 3He channel• Negative ∆EL in both channels

48t

12

20

1615

4. Results
4 He and 3 He channels ∆E

L = E 0 − NN m N

Identification of bound state from volume dependence of ∆E
0ΔEL[GeV]

04

He-0.02

-0.02

-0.04

-0.04
exp.
Nf=2+1 mπ=0.5GeV-0.06-0.08

ΔEL[GeV]

01e-05

2e-05
31/L

4e-05

-0.08

Heexp.
Nf=2+1 mπ=0.5GeV-0.06

3e-05

301e-05

2e-05
31/L

3e-05

4e-05

• ∆EL < 0 and small volume dependence• Infinite volume extrapolation with ∆EL = −∆Ebind + C/L316

4. Results
4 He and 3 He channels ∆E

L = E 0 − NN m N

Identification of bound state from volume dependence of ∆E
0

ΔEL[GeV]

04He-0.02

-0.02

-0.04

-0.04
exp.
Nf=2+1 mπ=0.5GeV
Nf=0 mπ=0.8GeV-0.06-0.08

ΔEL[GeV]

01e-052e-05
31/L

4e-05-0.08∆E4He = 43(12)(8) MeVHeexp.
Nf=2+1 mπ=0.5GeV
Nf=0 mπ=0.8GeV-0.06

3e-05

301e-05

2e-05
31/L

3e-05

4e-05

∆E3He = 20.3(4.0)(2.0) MeV

1. Observe bound state in both channels
2. Similar to quenched result
open symbols: quenched at mπ = 0.8 GeV, PRD81:111504(R)(2010)16-a

4. Results
NN (3S1 and 1S0) channels ∆EL = E0 − 2mN
Identification of bound state from volume dependence of ∆E0.0000.000exp
Nf=2+1 mπ=0.5GeV-0.004-0.004-0.008-0.008-0.012-0.012-0.016-0.0161

ΔEL[GeV] S03

ΔEL[GeV] S1-0.020-0.02401e-05

2e-05
31/L

3e-05

4e-05

-0.024• Negative ∆EL• Infinite volume extrapolation of ∆EL
γ2∆EL = −mN!

Nf=2+1 mπ=0.5GeV-0.020

01e-05

2e-05
31/L

3e-05

4e-05

’04 Beane et al., ’06 Sasaki & TY/√*2Cγ & exp(−γL $n )γ2√1+, ∆Ebind =2γLmN$n$nbased on Lüscher’s finite volume formula17

4. Results
NN (3S1 and 1S0) channels ∆EL = E0 − 2mN
Identification of bound state from volume dependence of ∆E
0.0000.000exp
Nf=2+1 mπ=0.5GeV
Nf=0 mπ=0.8GeV-0.004-0.004-0.008-0.008-0.012-0.012-0.016-0.0161

ΔEL[GeV] S0

3

ΔEL[GeV] S1-0.020-0.02401e-05

2e-05
31/L

3e-05

Bound state in both channels∆E3S = 11.5(1.1)(0.6) MeV
1

Nf=2+1 mπ=0.5GeV
Nf=0 mπ=0.8GeV-0.020

4e-05-0.024

01e-05

2e-05
31/L

3e-05

4e-05

∆E1S = 7.4(1.3)(0.6) MeV
0Not observed in experiment

Similar to quenched result at mπ = 0.8 GeV PRD84:054506(2011)17-aComparison of ∆E with previous works
NN (3S1 and 1S0) channels with (possible) bound states
0.040

0.040

30.020S1 ΔE[GeV]exp
Fukugita et al.
NPLQCD mixed
Ishii et al.
PACS-CS inf. V
NPLQCD 2+1 inf. V
This work 2+1 inf. V
NPLQCD 3 Vmax

0.000

0.000-0.020-0.020-0.04000.511.522mπ[GeV ]210.020

2.5-0.040

S0 ΔE[GeV]

00.5

11.522mπ[GeV ]Fukugita et al.
NPLQCD mixedIshii et al.
PACS-CS inf. V
NPLQCD 2+1 inf. V
This work 2+1 inf. V
NPLQCD 3 Vmax

2

2.5

filled(open) symbols: dynamical(quenched) results

Roughly consistent
Our quenched and Nf = 3 → Large sea quark effect?
This work and Nf = 3 → Large quark mass effect?

Need further study of systematic errors
such as quark mass dependence
18Comparison of ∆E with previous works
NN (3S1 and 1S0) channels with (possible) bound states
0.040

0.040

30.020S1 ΔE[GeV]exp
Fukugita et al.
NPLQCD mixed
Ishii et al.
PACS-CS inf. V
NPLQCD 2+1 inf. V
This work 2+1 inf. V
NPLQCD 3 Vmax

0.000

0.000

-0.020-0.020-0.04000.5

11.522mπ[GeV ]210.020

2.5-0.040

S0 ΔE[GeV]

00.5

11.5

22mπ[GeV ]Fukugita et al.
NPLQCD mixed
Ishii et al.
PACS-CS inf. V
NPLQCD 2+1 inf. V
This work 2+1 inf. V
NPLQCD 3 Vmax

2

2.5filled(open) symbols: dynamical(quenched) results

Roughly consistent
Our quenched and Nf = 3 → Large sea quark effect?
This work and Nf = 3 → Large quark mass effect?

Need further study of systematic errors
such as quark mass dependence195. Summary and future work
Extend our exploratory, quenched studies to Nf = 2 + 1 lattice QCD• Heavy quark mass of mπ = 0.5 GeV• Volume dependence of ∆E0∆E ,= 0 of 0th state in infinite volume limit→ bound state in 4He, 3He, 3S1 and 1S0 at mπ = 0.5 GeV• Similar result to quenched case → ∆E larger than experiment• Bound state in 1S0 not observed in experiment
Possibility in Nf = 2 + 1 at mπ = 0.39 GeV (’12 NPLQCD)
Deeper bound state in Nf = 3 at mπ = 0.8 GeV (’12 NPLQCD)

Further investigations in Nf = 2 + 1, e.g. mπ dependence(0.7 GeV ←) mπ = 0.5 GeV → 0.3 GeV → physical mπ (HPCI)20