Lattice QCD studies of baryon interactions fromHAL QCD method and Luscher’s finite volume method

Takumi Iritani @ YITPfor HAL QCD Collaboration

LATTICE2015, July 13-18, Kobe, Japan

S. Aoki, S. Gongyo, TI, T. Miyamoto(YITP Kyoto Univ.)

T. Doi, T. Hatsuda, Y. Ikeda (RIKEN Nishina)

T. Inoue (Nihon Univ.)

N. Ishii, K. Murano (RCNP Osaka Univ.)

H. Nemura, K. Sasaki (Univ. Tsukuba)

F. Etminan (Univ. Birjand)

2 Lattice QCD Approaches for Hadron Interactions

Luscher’s finite volume method — Luscher ’86, ’91energy shift of two-particle system in “finite box” ⇒ phase shift

tan δ(k) =1

πL

∑n∈Z3

1

|n|2 − (kL/2π)2

with ∆E = 2√k2 +m2 − 2m

HAL QCD method — Ishii-Aoki-Hatsuda ’07NBS wave function ⇒ potential ⇒ phase shift

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0

NN

wa

ve

fu

nctio

n ψ

(r)

r [fm]

1S03S1

-2 -1 0 1 2 -2-1

01

20.5

1.0

1.5

ψ(x,y,z=0;1S0)

x[fm] y[fm]

ψ(x,y,z=0;1S0)

0

100

200

300

400

500

600

0.0 0.5 1.0 1.5 2.0

VC

eff(r

) [M

eV

]

r [fm]

-50

0

50

100

0.0 0.5 1.0 1.5 2.0

1S03S1

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-10

0

10

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50

60

0 50 100 150 200 250 300 350δ [

de

g]

Elab [MeV]

explattice

1 / 15

Consistency of Two Methods■ I = 2 ππ scattering — Kurth-Ishii-Doi-Aoki-Hatsuda ’13

⇒ good agreementb[°]

ECM[MeV]

V=(1.84 fm)3

V=(2.76 fm)3

V=(3.7 fm)3

V=(5.5 fm)3

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0

0 50 100 150 200 250 300 350 400

2 / 15

NN Interactions from Luscher’s Method

Both 1S0 and 3S1 channels are bound or unbound states ?

0 0.2 0.4 0.6 0.8 1

mπ

2[GeV

2]

-40

-20

0

20

40

Fukugita et al. 0f [7]

NPLQCD mixed [8]

PACS-CS 0f V∞

[5]

NPLQCD 2+1f V∞

[6]

NPLQCD 3f Vmax

[2]

Yamazaki et al. 2+1f V∞

[3]

This work 2+1f V∞

∆E(1S

0)[MeV]

0 0.2 0.4 0.6 0.8 1

mπ

2[GeV

2]

-40

-20

0

20

40

experiment

Fukugita et al. 0f [7]

NPLQCD mixed [8]

PACS-CS 0f V∞

[5]

NPLQCD 2+1f V∞

[6]

NPLQCD 3f Vmax

[2]

Yamazaki et al. 2+1f V∞

[3]

This work 2+1f V∞

∆E(3S

1)[MeV]

Figs. Yamazaki-Ishikawa-Kuramashi-Ukawa ’15 [arXiv:1502.04182]

3 / 15

NN Interactions from HAL QCD Method

□ HAL QCD Coll. — NN potential and NN(1S0) phase shift

⇒ scattering states

-500

0

500

1000

1500

2000

2500

3000

3500

0 0.5 1 1.5 2 2.5

V(r

) [M

eV

]

r [fm]

Central and tensor potentials (mπ=700 MeV)

-100

-50

0

50

100

0 0.5 1 1.5 2 2.5

VC(r; 1S0)VC(r; 3S1-3D1)VT(r; 3S1-3D1)

-20

-10

0

10

20

30

40

50

60

0 50 100 150 200 250 300 350

δ [deg]

Elab [MeV]

explattice

◦ qualitatively consistent with the experimental phase shift

4 / 15

Aim of This WorkFor deeper understanding of hadron interactions from lattice QCD, i.e.,

HAL QCD method and Luscher’s finite volume method

⇒ we investigate baryon interactionsfrom both methods using the same lattice setup

■ analyze baryon potential ■ measure energy shift

-500

0

500

1000

1500

2000

2500

3000

3500

0 0.5 1 1.5 2 2.5

V(r

) [M

eV

]

r [fm]

Central and tensor potentials (mπ=700 MeV)

-100

-50

0

50

100

0 0.5 1 1.5 2 2.5

VC(r; 1S0)VC(r; 3S1-3D1)VT(r; 3S1-3D1)

0 4 8 12 16 20

t

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1S

0

Fig. Yamazaki et al. ’155 / 15

1 Introduction: HAL QCD Method and Luscher’s Method

2 Formulations and Lattice Setup

3 Lattice QCD Results for ΞΞ InteractionΞΞ Energy Shift in Finite VolumeΞΞ Potential

4 Summary

Time-dependent HAL QCD Method

• normalized 4pt correlator

R(r, t− t0) ≡ e2mt⟨0|T{B(x+ r, t)B(x, t)}J (t0)|0

⟩=

∑n

AnϕWn(r)e−∆Wn(t−t0)

• R-correlator satisfies “time-dependent” Schrodinger-like equation[1

4m

∂2

∂t2− ∂

∂t−H0

]R(r, t) =

∫dr′U(r, r′)R(r′, t)

• using velocity expansion, “potential” is given by

V LO(r) =1

4m

(∂/∂t)2R(r, t)

R(r, t)− (∂/∂t)R(r, t)

R(r, t)− H0R(r, t)

R(r, t)

• This method does not require the ground state saturation.

6 / 15

Luscher’s Finite Volume Method

∆E = 2√

k2 +m2N − 2mN

tan δ(k) =1

πL

∑n∈Z3

1

|n|2 − (kL/2π)2

we measure “energy shift” ∆E = ENN − 2mN at finite volume

⇒ use effective mass plot

∆E(t) = lnR(t)

R(t+ 1)

with

R(t) =GNN (t)

(GN (t))2

• standard method in “single particle state”

• NN(1S0) energy shift

0 4 8 12 16 20

t

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1S

0

Fig. Yamazaki et al. ’157 / 15

Lattice Setup

• 2+1 flavor improved Wilson quark + Iwasaki gauge configuration†

lattice spacing a = 0.08995(40) fm, a−1 = 2.194(10) GeV• 2-type quark sources

(exponential) smeared source — the same as Yamazaki et al.

wall source — commonly used in HAL QCD method

□ Analyze ΞΞ-channel potential and energy shift

volume smeared src. wall src.

403 × 48 200 conf. × 192 meas. 200 conf. × 48 meas.483 × 48 800 conf. × 256 meas. 800 conf. × 48 meas.643 × 64 327 conf. × 48 meas. 327 conf. × 128 meas.

Table: Lattice configurations (we mainly use 483 × 48 volume)

mπ = 0.51 GeV, mN = 1.32 GeV, mK = 0.62 GeV, mΞ = 1.46 GeV† Yamazaki, Ishikawa, Kuramashi, Ukawa, arXiv:1207.4277, 1502.04182.

8 / 15

1 Introduction: HAL QCD Method and Luscher’s Method

2 Formulations and Lattice Setup

3 Lattice QCD Results for ΞΞ InteractionΞΞ Energy Shift in Finite VolumeΞΞ Potential

4 Summary

ΞΞ Energy Shift in 483 Lattice Volume

• ΞΞ(1S0) (the same rep. as NN(1S0)) ⇒ negative energy shift• ΞΞ(3S1) ⇒ positive energy shift

⇒ but ∆E(t) depends on quark sources ?

□ smeared source ■ wall source

-15

-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

exp.src. ΞΞ(1S0)

exp.src. ΞΞ(3S1)

12.05(68)-7.63(39)

-15

-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

wall src. ΞΞ(1S0)

wall src. ΞΞ(3S1)

0.57(47)

-2.44(14)

9 / 15

Effective Mass of ΞΞ states

∆E = MΞΞ − 2mΞ

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-5

0

5

5 10 15 20

∆E

ΞΞ [MeV]

t

wall.src.ΞΞ(1S0)

exp.src.ΞΞ(1S0)

10 / 15

Effective Mass of ΞΞ states• Be careful with effective mass plot

⇒ fictious “plateau” appears

◦ at least t ≳ 16◦ smeared src.

— need more statistics?

∆E = MΞΞ − 2mΞ

-10

-5

0

5

5 10 15 20

∆E

ΞΞ [MeV]

t

wall.src.ΞΞ(1S0)

exp.src.ΞΞ(1S0)

-2.25(1.28)

Ξ effective mass

1450

1455

1460

1465

1470

5 10 15 20

meff

Ξ [GeV]

t

wall.src. Ξ

exp.src. Ξ

ΞΞ(1S0) effective mass

2895

2900

2905

2910

2915

2920

2925

2930

2935

5 10 15 20

meff

ΞΞ [MeV]

t

wall.src.ΞΞ(1S0)

exp.src.ΞΞ(1S0)

10 / 15

1 Introduction: HAL QCD Method and Luscher’s Method

2 Formulations and Lattice Setup

3 Lattice QCD Results for ΞΞ InteractionΞΞ Energy Shift in Finite VolumeΞΞ Potential

4 Summary

NBS Wave Function and ΞΞ(1S0) Potential

V (r) =(∂/∂t)2R(r,t)

4mR(r,t) − (∂/∂t)R(r,t)R(r,t) − H0R(r,t)

R(r,t)

◦ smeared. src. — strong t-dep.

⇒ ∼ O(100) MeV cancellation• t-dep. HAL method works well!!◦ wall src. — weak t-dep.

□ smeared src. ■ wall src.

11 / 15

ΞΞ Potential from HAL QCD method

• ΞΞ(1S0) — attractive pocket + repulsive core, cf. NN(1S0)• ΞΞ(3S1-

3D1) — weak attractivepotentials are qualitatively consistent with each other

◦ difference — NLO term ? U(r, r′) = [V0(r) + V1(r)∇2]δ(r − r′)

□ smeared source ■ wall source

12 / 15

Convergence of ΞΞ potential□ smeared src. shows t-dep. ■ wall src. “stable”

By increasing t, “smeared src.” results seem to converge “wall src.”

13 / 15

Finite Volume Energy from Potential: Luscher from HAL

“wall source” at t = 12 • Now, we have the potential V ΞΞc (r)

⇒ solve “finite volume” eigen energies†

use 403, 483 and 643 wall src. results

• consistent with “wall src.” ∆E(t) fit−2.25(1.28) MeV @ 483 × 48

• vol. dep. implies scattering states

energy eigenvalues

vol. E(g.s.) [MeV] E(1st) [MeV]

403 −4.55(1.18) 75.63(1.31)483 −2.58(22) 52.87(33)643 −1.13(9) 28.71(9)

-10

-8

-6

-4

-2

0

0.0×100

5.0×10-6

1.0×10-5

1.5×10-5

2.0×10-5

ΞΞ

(1S

0)

E(g

.s.)

[M

eV]

1/L3

"potential method"1/L

3 fit

† ref. Charron for HAL QCD Coll. arXiv:1312.1032.14 / 15

1 Introduction: HAL QCD Method and Luscher’s Method

2 Formulations and Lattice Setup

3 Lattice QCD Results for ΞΞ InteractionΞΞ Energy Shift in Finite VolumeΞΞ Potential

4 Summary

Summary

We have investigated baryon interactions fromboth HAL QCD method and Luscher’s finite volume method.

we have shown

we should be careful with “plateau” in energy shift ∆E(t).

time-dependent HAL QCD method works well without ground statesaturation.

we obtain the energy shift ∆E in finite volume for ΞΞ(1S0)-channel,which is consistent with the potential by HAL QCD method.

we have to check

convergence of smared src. to wall src. results

contributions of inelastic/elastic excited states in NBS wave functionand ∆E(t) plot

NLO of velocity expansion in HAL QCD method for smeared source

...

15 / 15

Appendix

Volume Dependence of Energy Shift from Wall Source◦ approximate the energy shift by “plateau” value

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0

5

10

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20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

40x48 wall.src.:#conf.200#src.48

ΞΞ(1S0)

ΞΞ(3S1)

-4.31 +- 0.80

2.90 +- 0.99

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-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

48x48 wall.src.:#conf.800#src.48

ΞΞ(1S0)

ΞΞ(3S1)

-2.44 +- 0.15

0.57 +- 0.47

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-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

64x64 wall.src.:#conf.325#rot.4#src.32

ΞΞ(1S0)

ΞΞ(3S1)

-1.01 +- 0.06

0.39 +- 0.17

wall. src.vol. ∆EΞΞ(

1S0)

403 −4.31(80) MeV483 −2.45(15) MeV643 −1.01(6) MeV

Volume Dep. of Energy Shift from Smeared Src.

-15

-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

40x48 exp.sloppy:#conf.200#src.192

ΞΞ(1S0)

ΞΞ(3S1)

-7.31 +- 1.04

-15

-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

48x48 exp.src.:#conf.800#src.256

ΞΞ(1S0)

ΞΞ(3S1)

-7.63 +- 0.3912.05 +- 0.68

-15

-10

-5

0

5

10

15

20

25

5 10 15 20

∆E

ΞΞ [MeV]

t

64x64 exp.src.:#conf.325#src.48

ΞΞ(1S0)

ΞΞ(3S1)

-2.87 +- 0.92

exp. src. (PRELIMINARY)

vol. ∆EΞΞ(1S0)

403 −7.31(1.04) MeV483 −7.63(39) MeV643 −2.83(92) MeV

Phase Shift Analysis

wall source results at 48×48

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-5

0

5

10

15

20

25

30

0 50 100 150 200

phase shift δ [deg.]

Ekin. [MeV]

48x48 wall.sloppy: ΞΞ(1S0) #conf.800 #src.48

t = 11t = 13t = 15

use (2 Gauss + [Yukawa]2)-type fit

Vc(r) = a0 exp(−a1r2)+a2 exp(−a2r

2)+a4(1−exp(a5r2))2

(exp(−a6r)

r

)2

NN Energy Shift and Statisticsit is difficult to identify “plateau” in effective mass plot

800 conf. × 64 smeared src.

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-10

0

10

20

30

5 10 15 20

∆ENN [MeV]

t

3S1:#meas 64

-11.68 +- 0.53

2610

2620

2630

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2650

2660

2670

5 10 15 20

meff

NN [MeV]

t

NN(3S1):#meas 64

2634.86 +- 1.93

800 conf. × 256 smeared src.

-20

-10

0

10

20

30

5 10 15 20

∆ENN [MeV]

t

3S1:#meas 256

-15.17 +- 1.40

2610

2620

2630

2640

2650

2660

2670

5 10 15 20

meff

NN [MeV]

t

NN(3S1):#meas 256

2624.30 +- 1.40

NN Energy Shift from Wall Source

-5

0

5

10

5 10 15 20

∆ENN [MeV]

t

40x48 wall.src.:#conf.200#src.48

NN(1S0)

NN(3S1)

-6.11 +- 0.99

-6.57 +- 0.94

-5

0

5

10

5 10 15 20

∆ENN [MeV]

t

64x64 wall.src.:#conf.325#rot.4#src.32

NN(1S0)

NN(3S1)

-0.91 +- 0.12

-1.49 +- 0.09

-5

0

5

10

5 10 15 20

∆ENN [MeV]

t

48x48 wall.src.:#conf.800#src.48

NN(1S0)

NN(3S1)

-2.97 +- 0.26

-3.73 +- 0.27

-10

-8

-6

-4

-2

0

0.0×100

5.0×10-6

1.0×10-5

1.5×10-5

2.0×10-5

∆ E

[M

eV

]

1/L3

NN(1S0)

NN(3S1)

1/L3 fit