Baryon interactions from Lüscher's finite volume method ... · Baryon interactions from Luscher¨...

Baryon interactions fromLuschers finite volume method and HAL QCD method

Takumi IRITANI (Stony Brook Univ.)for HAL QCD Collaboration

Apr. 18, 2016 @ INT-16-1 Nuclear Physics from Lattice QCD

Ref. TI for HAL QCD Coll., PoS(Lattice2015), arXiv:1511.05246[hep-lat].

S. Aoki, S. Gongyo, D. Kawai, T. Miyamoto(YITP)

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

T. Inoue (Nihon Univ.)

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

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

F. Etminan (Univ. Birjand)

In This Talk: We Solve Luscher vs HAL QCD Puzzle

1 QCD to Nuclear Physics: Puzzle between Luschers method and HAL

2 Baryon Interaction from Lattice QCDLattice Formulations and SetupLuschers Method: Energy Shift in Finite VolumeHAL QCD Method: PotentialEnergy Eigenvalue in Finite Volume

3 HAL and Luscher Origin of Fake Plateau

4 Summary and Outlook

2 Methods for Hadron Interaction from Lattice QCDfirst principle calculations based on QCD

1 Luschers finite volume method Luscher 86, 911. energy shift of two-particle system in box ! 2. phase shiftEL = 2

k2 +m22m = k cot (k) = 1

L

nZ3

1

|n|2 (kL/2)2

2 HAL QCD method Ishii-Aoki-Hatsuda 071. NBS wave function ! 2. potential ! 3. 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 f

un

ctio

n

(r)

r [fm]

1S03S1

-2 -1 0 1 2 -2-1

01

20.5

1.0

1.5

(x,y,z=0;1S0)

x[fm] y[fm]

(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

-20

-10

0

10

20

30

40

50

60

0 50 100 150 200 250 300 350 [

de

g]

Elab [MeV]

explattice

1 / 22

Consistency of Luscher and HAL I = 2 scattering Good

[]

ECM[MeV]

V=(1.84 fm)3

V=(2.76 fm)3

V=(3.7 fm)3

V=(5.5 fm)3

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 50 100 150 200 250 300 350 400

Figure: Kurth-Ishii-Doi-Aoki-Hatsuda 13 2 / 22

NN Interactions from Lattice QCD

Luscher HAL QCD phys. point1S0 bound unbound unbound3S1 bound unbound bound

contradictions between Luscher and HAL QCD! What is the origin of the difference ?

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1

1S0

E

[M

eV]

m2 [GeV

2]

Fukugita et al., Nf=0

NPLQCD, mixed

Aoki et al., Nf=0Yamazaki et al., Nf=0, V

NPLQCD, Nf=2+1, V

HALQCD, Nf=3, V

NPLQCD, Nf=3, VmaxYamazaki et al., Nf=2+1, V

HALQCD, Nf=2+1, V

Yamazaki et al., Nf=2+1, V

CalLat, Nf=3, V

NPLQCD, Nf=2+1, V

EXP.

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1

3S1

E

[M

eV]

m2 [GeV

2]

Fukugita et al., Nf=0

NPLQCD, mixed

Aoki et al., Nf=0Yamazaki et al., Nf=0, V

NPLQCD, Nf=2+1, V

HALQCD, Nf=3, V

NPLQCD, Nf=3, VmaxYamazaki et al., Nf=2+1, V

HALQCD, Nf=2+1, V

Yamazaki et al., Nf=2+1, V

CalLat, Nf=3, V

NPLQCD, Nf=2+1, V

EXP.

3 / 22

there are systematic discrepancies between

Luschers finite volume method and HAL QCD method

but those works used different quark mass, source (smeared and wall), ...

" check baryon interactions from both methods with the same setup# which is correct ? what is the origin of discrepancies ?

Luscher vs HAL# measure energy shift # analyze potential

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

V

C(r

) [M

eV]

r [fm]

wall sourcet = 15t = 13t = 11

4 / 22

Luschers Finite Volume Methodenergy shift in finite box L3

EL = EBB 2mB = 2k2 +m2B 2mB

phase shift (k)

k cot (k) =1

L

nZ3

1

|n|2 (kL/2)2

Eucl

idea

n tim

e

measurement: plateau in effective mass

Eeff(t) = logR(t)

R(t+ 1) EL

R(t) =GBB(t)

{GB(t)}2 exp [ (EBB 2mB) t]

with GBB(t)(GB(t)): BB(B) correlators

effective mass plot= standard method in single particle state

NN(1S0) energy shift

Fig. Yamazaki et al. 12

5 / 22

Time-dependent HAL QCD Method

# Nambu-Bethe-Salpeter correlation function

R(r, t) 0|T{B(x+ r, t)B(x, t)}J (0)|0

{GB(t)}2

=

n

Ann(r)e(En2mB)t +O(e(Eth2mB)t)

Eucl

idea

n tim

e

source

sink

! in Luschers method: r R(r, t) = R(t) = exp [(EBB 2mB)t]" each n(r)eEnt 0|T{B(x+ r, t)B(x, t)}|2B,n satisfies

[k2nmB

H0]n(r) =

drU(r, r)n(r)

with non-local interaction kernel U(r, r)# R-corr. satisfies t-dep. Schrodinger-like eq. with elastic saturation

[1

4mB

2

t2 t

H0]R(r, t) =

drU(r, r)R(r, t)

6 / 22

Time-dependent HAL QCD Method

# Nambu-Bethe-Salpeter correlation function

R(r, t) 0|T{B(x+ r, t)B(x, t)}J (0)|0

{GB(t)}2

=

n

Ann(r)e(En2mB)t +O(e(Eth2mB)t)

Eucl

idea

n tim

e

source

sink

! in Luschers method: r R(r, t) = R(t) = exp [(EBB 2mB)t]# R-corr. satisfies t-dep. Schrodinger-like eq. with elastic saturation

[1

4mB

2

t2 t

H0]R(r, t) =

drU(r, r)R(r, t)

! potential using velocity expansion U(r, r) V (r)(r r)V (r) =

1

4mB

(/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 / 22

Difficulties in Multi-Baryons! Luschers method requires ground state saturationGNN (t) = c0 exp(E(NN)0 t)+ c1 exp(E

(NN)1 t)+ c0 exp(E

(NN)0 t)

S/N becomes worse as [mass number A] [light quark] [t ]

S/N exp[A

(mN

32m

) t

]

smaller gaps: E p 2/m O(1/L2)

Elastic

Inelastic

! HAL QCD methodonly elastic states saturation E-indep potential

[1

4mB

2

t2 t

H0]R(r, t) =

drU(r, r)R(r, t)

7 / 22

Lattice Setup# 2 + 1 improved Wilson + Iwasaki gauge

lattice spacing a = 0.08995(40) fm, a1 = 2.194(10) GeVm = 0.51 GeV, mN = 1.32 GeV, mK = 0.62 GeV, m = 1.46 GeV

# 2-type quark sourceswall sourcestandard of HAL QCD

smeared sourcethe same as Yamazaki et al.

NN int. by Luscher& Helium binding energy

WALL SOURCE SMEARED SOURCE

SINK SINK

" analyze (1S0)-channel the same rep. as NN(1S0) and better S/Nwith high stat. ex. 484: (#smeared src.)/(Yamazaki et al.) 5

volume # conf. # smeared src. # wall src.403 48 200 512 48483 48 200 4 256 4 48643 64 327 256 4 32

Yamazaki-Ishikawa-Kuramashi-Ukawa, arXiv:1207.4277. 8 / 22

(1S0) Effective Energy Shift Ploteffective mass plot

Eeff(t) = logR(t)/R(t+ 1) E 2mwith R(t) G(t)/{G(t)}2

wall sourceEL 3 MeV

smeared sourceEL 8 MeV

plateau dependson quark source

???

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

9 / 22

(1S0) Effective Energy Shift Ploteffective mass plot

Eeff(t) = logR(t)/R(t+ 1) E 2mwith R(t) G(t)/{G(t)}2

wall sourceEL 3 MeVsmeared sourceEL 8 MeVplateau dependson quark source

???-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

9 / 22

Effective Masses of and wall source

2800

2850

2900

2950

3000

3050

0 5 10 15 20

2

mef

f

o

r E

eff

[M

eV]

t

wall src. wall src. (

1S0)

smeared source

2800

2850

2900

2950

3000

3050

0 5 10 15 20

2

mef

f

o

r E

eff

[M

eV]

t

smeared src. smeared src. (

1S0)

Be careful with effective mass plotw/o plateaux in Eeff(t) & m

eff (t),

Eeff(t) = Eeff (t) 2meff(t)

shows a fake plateau bycancellation! we need much larger t,

and much more statistics

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

10 / 22

Effective Masses of and wall source

2890

2900

2910

2920

2930

2940

0 5 10 15 20

2

mef

f

o

r E

eff

[M

eV]

t

wall src. wall src. (

1S0)

smeared source

2890

2900

2910

2920

2930

2940

0 5 10 15 20

2

mef

f

o

r E

eff

[M

eV]

t

smeared src. smeared src. (

1S0)

Be careful with effective mass plotw/o plateaux in Eeff(t) & m

eff (t),

Eeff(t) = Eeff (t) 2meff(t)

shows a fake plateau bycancellation! we need much larger t,

and much more statistics

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

10 / 22

Fake Plateaux = Wrong Conclusion

-14

-12

-10

-8

-6

-4

-2

0

0100

510-6

110-5

210-5

210-5

E

L

[M

eV]

1/L3 [a

-3]

wall src.smeared src.

-0.04(14) MeV-4.60(75) MeV

! direct methodwall src. vs smeared src.

different plateaux

different conclusionsunbound or bound

L3 = 403

-10

-5

0

5

0 5 10 15 20

-7.20(59)-5.00(48)

E

eff

(t)

[M

eV]

at L

= 4

0

t

smeared src. (1S0)

wall src. (1S0)

L3 = 483

-10

-5

0

5

0 5 10 15 20

-2.44(14)

-7.71(52)

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

L3 = 643

-10

-5

0

5

0 5 10 15 20

-4.84(61)

-1.12(7)

E

eff

(t)

[M

eV]

at L

= 6

4

t

smeared src. (1S0)

wall src. (1S0)

11 / 22

HAL: NBS Wave Function and (1S0) Potential Vc(r)

0

2x10-5

4x10-5

6x10-5

8x10-5

0 0.5 1 1.5 2 2.5 3 3.5

NBS

cor

rela

tion

func

tion

r [fm]

smeared src.: t = 14 t = 13 t = 12

wall src.: t = 13 t = 15

wall src. weak t-dep. smeared. src. strong t-dep.

O(100) MeV of cancellation time-dep. HAL method works well

Vc(r) = H0R

R(/t)R

R+(/t)2R

4mR

# wall src.

-50

0

50

0 0.5 1 1.5 2 2.5 3 3.5

wall src. t = 12

VC

[MeV

]

r [fm]

total.lap.d/dt

d2/dt

2

" smeared src.

-150

-100

-50

0

50

100

150

0 0.5 1 1.5 2 2.5 3 3.5

smeared src. t = 12

VC

[MeV

]

r [fm]

total.lap.d/dt

d2/dt

2

12 / 22

HAL: NBS Wave Function and (1S0) Potential Vc(r)

0

2x10-5

4x10-5

6x10-5

8x10-5

0 0.5 1 1.5 2 2.5 3 3.5

NBS

cor

rela

tion

func

tion

r [fm]

smeared src.: t = 14 t = 13 t = 12

wall src.: t = 13 t = 15

wall src. weak t-dep. smeared. src. strong t-dep.

O(100) MeV of cancellation time-dep. HAL method works well

Vc(r) = H0R

R(/t)R

R+(/t)2R

4mR

# wall src.

-50

0

50

0 0.5 1 1.5 2 2.5 3 3.5

wall src. t = 12

VC

[MeV

]

r [fm]

total.lap.d/dt

d2/dt

2

" smeared src.

-50

0

50

0 0.5 1 1.5 2 2.5 3 3.5

VC

[MeV

]

r [fm]

total.lap.d/dt

d2/dt

2

12 / 22

(1S0) Potential: Wall Source vs Smeared Source! wall src. stable ! smeared src. t-depend

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

V

C(r

) [M

eV]

r [fm]

wall sourcet = 15t = 13t = 11

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

V

C(r

) [M

eV]

r [fm]

smeared sourcet = 15t = 13t = 11

! smeared src. converges to wall src. residual from NLO pot.

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0) V

C(r

)[M

eV]

r [fm]

smeared src. t = 12

wall src. t = 12

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0) V

C(r

)[M

eV]

r [fm]

smeared src. t = 15

wall src. t = 15

13 / 22

Residual Diff. of Pot.: Next Leading Order CorrectionDerivative expansion: U(r, r) = {V0(r) + V1(r)2}(r r) (for 1S0)

[1

4m

2

t2 t

H0]R(r, t) =

d3rU(r, r)R(r, t)

14m

(2/t2)R

R(/t)R

RH0R

R= V0(r)+V1(r)

2R(r, t)R(r, t)

V naivetotal (r, t)

we have Rsmear(r, t) and Rwall(r, t) lets solve V0(r) and V1(r)! wall source Good convergence of LO pot.

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

po

ten

tial

[M

eV]

r [fm]

V0(r)

V1(r) 2R

wall/R

wall

Vtot.wall

(r,11)

! smeared source with NLO pot. correction

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

po

ten

tial

[M

eV]

r [fm]

V0(r)

V1(r) 2R

smear/R

smear

Vtot.smear

(r,11)

14 / 22

HAL QCD meets Luscher: Energy Shift from Potential! HAL QCD gives reliable interaction w/o g.s. saturation

quark source, time, and volume independent! what is the true energy shift in finite volume1 INPUT: potential V (r)

2 SOLVE: eigenvalue problem

[H0 + V ] = E

in finite volume L3-40

-20

0

20

40

0 1 2 3 4 5

(1

S0)

V

C(r

) [M

eV]

r [fm]

643 64 @ t = 12

403 48 @ t = 12

483 48 @ t = 12

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

E0 = -2.58(0.22)vol. g.s. [MeV] 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)

eigenvalues using Vc(r, t = 12)

ref. B. Charron for HAL Coll. arXiv:1312.1032. 15 / 22

Conclusion: (1S0) is Unbound at current mass

k cot (k) =1

L

nZ3

1

|n|2 (kL/2)2 , E = 2m2 + k2 2m

volume dep. of E0

-8

-6

-4

-2

0

2

0100

510-6

110-5

210-5

210-5

E

0 [

MeV

]

1/L3 [a

-3]

HAL QCD method/L3-fit

phase shift

-10

-5

0

5

10

15

20

25

30

35

0 50 100 150 200p

has

e sh

ift [

deg

.]

ECM [MeV]

403 48 at t = 12

483 48 at t = 12

643 64 at t = 12

Luscher formula

# wall src. pot. solve eigenvalue & Luscher formula phase shift# wall src. pot. fit & solve Schrodinger eq. phase shift

using 2 Gaussians + (Yukawa)2 ansatz for Vc(r) fit16 / 22

Short Summary: Consistency of 2 Methodsground state saturation of multi-baryon is extremely difficulteven without g.s. saturation there may appear plateau it should be checked by other source or variational method

fake plateaux wrong conclusionHAL QCD method gives source indep. results w/o g.s. saturation.

%Luscher vs HAL Luscher(smeared) vs Lushcer(wall)Pot. with wall src.

Pot. with smear src.

Direct with wall src.

Direct with smear src.NLO pot. corr. Fake plateauxConflict

explain Eeff(t)

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

po

ten

tial

[M

eV]

r [fm]

V0(r)

V1(r) 2R

smear/R

smear

Vtot.smear

(r,11)

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

17 / 22

Wavefunction, Potential, Eigenvalues and Eienfunctions

-0.01

0

0.01

0.02

0 5 10 15 20 25 30 35 40 45

n( r

,t=12

)

r

4th A13rd A12nd A1

1stg.s.

0 5 10 15 20 25 30 35 40 45

R( r

,t)

r

smear @ t = 12t = 13t = 14

wall @ t = 12t = 13t = 14

-40

-20

0

20

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S 0

) V

C( r

) [M

eV]

r [fm]

643 64 @ t = 12403 48 @ t = 12483 48 @ t = 12

NBS wave function Potential

Eigenvalues & Eigenfunctions

HAL method

Solvefeed back

decompositionprojection

(elastic scattering)

ground state

& excited states

18 / 22

Exited States in Wavefunction! R-corr. decomposition by energy eigenmodes & from HAL pot.

Rwall/smear(r, t) =

n

awall/smearn n(r, t) exp (Ent)

R(p = 0, t) =

r

R(r, t) =

n

bwall/smearn eEnt

" ex. 1st excited statewall sourceb1/b0 0.01smeared sourceb1/b0 0.1

with energy gapE1 E0 50 MeVfor L3 = 483

contamination of excited states bn/b0

1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1E+00

0 50 100 150 200 250

|bn/b

0|

(1

S0)

at t

= 1

4

En [MeV]

483 wall:+

483 smear:+

19 / 22

Excited States Contamination and Fake Plateau

Eeff(t) = log

n bn exp (Ent)

n bn exp (En(t+ 1))# direct measurement well reproduced by low-lying 3 eigenmodes

! fake plateau of smeared src. around t = 14 18

-15

-10

-5

0

5

10

0 5 10 15 20

E

eff

(t)

[M

eV]

L =

48

t [a]

bwalln = 0,..,2 at t = 14

bsmearn = 0,..,2 at t = 14

wall src. (1S0)

smeared src. (1S0)

eigenvalues En, coefficients bsmear/walln for n = 0, 1, 2, at t = 14. 20 / 22

Excited States Contamination and Fake Plateau

Eeff(t) = log

n bn exp (Ent)

n bn exp (En(t+ 1))# direct measurement well reproduced by low-lying 3 eigenmodes

! fake plateau of smeared src. around t = 14 18" g.s. saturation of smeared source 100 lattice units 10 fm !!!

-15

-10

-5

0

5

10

0 5 10 15 20

E

eff

(t)

[M

eV]

L =

48

t [a]

bwalln = 0,..,2 at t = 14

bsmearn = 0,..,2 at t = 14

wall src. (1S0)

smeared src. (1S0)

-15

-10

-5

0

5

10

0 20 40 60 80 100

E

eff

(t)

[M

eV]

L =

48

t [a]

bwalln = 0,..,2 at t = 14

bsmearn = 0,..,2 at t = 14

wall src. (1S0)

smeared src. (1S0)

eigenvalues En, coefficients bsmear/walln for n = 0, 1, 2, at t = 14.20 / 22

Luschers Method by Sink Projectionintroducing some function f(r)

R(f)(t)

r

f(r)

x

0|B(r + x, t)B(x)J (0)|0

=

r

f(r)R(r, t)

and effective energy shift E(f)eff (t) = log R(f)(t)/R(f)(t+ 1)

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

21 / 22

Luschers Method by Sink Projectionintroducing some function f(r)

R(f)(t)

r

f(r)

x

0|B(r + x, t)B(x)J (0)|0

=

r

f(r)R(r, t)

and effective energy shift E(f)eff (t) = log R(f)(t)/R(f)(t+ 1)

! correct plateau by using f(r) = gs(r) & from HAL pot.

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

-10

-5

0

5

10

12 13 14 15 16

E

eff

(t)

[M

eV]

L =

48

t [a]

wall src. (1S0)

smeared src. (1S0)

g.s. proj. wallg.s. proj. smear

E0

21 / 22

Summary: Luscher vs and HAL QCD are Consitent

Direct method ground state saturation is extremely difficult do not trust plateau, it may be a wrong signal!HAL QCD works well without g.s. saturation.

NBS corr. and potential clarify excited states contaminationand origin of fake plateau.

using NBS corr., potential and wavefunctions improvement of Lushcers measurement feed back method

Pot. with wall src.

Pot. with smear src.

Direct with wall src.

Direct with smear src.NLO pot. corr. Fake plateauxConflict

explain Eeff(t)

22 / 22

5 Appendix

NN(1S0), NN(3S1), Triton and HeliumDo not trust plateaux.

-10

-5

0

5

0 5 10 15 20

E

eff

NN

(t)

[M

eV]

at L

= 4

8

t

smeared src. NN(1S0)

wall src. NN(1S0) -25

-20

-15

-10

-5

0

5

5 10 15 20

E

eff

NN

(t)

[M

eV]

at L

= 4

8

t

smeared src. NN(3S1)

wall src. NN(3S1)

-80

-70

-60

-50

-40

-30

-20

-10

0

0 5 10 15 20

E

eff

3H

e (t

) [M

eV]

at L

= 4

8

t

smeared src. 3Hewall src. 3He -80

-70

-60

-50

-40

-30

-20

-10

0

0 5 10 15 20

E

eff

4H

e (t

) [M

eV]

at L

= 4

8

t

smeared src. 4Hewall src. 4He

speed up for 3He/4He calc. unified contraction algorithm Doi-Endres 13

So-Called Plateau

-30

-20

-10

0

10

0 5 10 15 20

E

eff

(t)

[M

eV]

L =

48

t [a]

smeared src. (1S0)

wall src. (1S0)

Figure: (a) NPL (2015) dineutron L = 483 (b) (1S0) L3 = 483 with the samerange of E

Fit range and value in Yamazaki et al12 and our re-analysis of NN(1S0)

-20

-15

-10

-5

0

0 5 10 15 20

E

eff

NN

(t)

[M

eV]

L =

48

t [a]

non.rela. smeared src. NN(1S0)

non.rela. wall src. NN(1S0)

Yamazaki et al

Effective mass of Single Channel

0.55

0.6

0.65

0.7

0.75

0 5 10 15 20

mef

f

(t)

[a

-1] L

= 4

8

t [a]

smeared src. wall src.

Figure: (a) NPL (2015) nucleon L = 483 (b) our data mass L = 483 with thesame range of E

Effective Masses Plot

Eeff(t) = Eeff(t)2meff (t)

shows plateaueven without plateauxin meff (t) or E

eff(t)

-10

-5

0

5

0 5 10 15 20

E

eff

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

1450

1455

1460

1465

1470

0 5 10 15 20

mef

f

(t)

[M

eV]

at L

= 4

8

t

smeared. src. wall src.

(1S0)

2890

2900

2910

2920

2930

2940

0 5 10 15 20

Eef

f

(t)

[M

eV]

at L

= 4

8

t

smeared src. (1S0)

wall src. (1S0)

Weight of Eigenstates in Wavefunction

Rwall/smear(x, t) =

n

awall/smearn n(x, t)eEnt

with eigenfunctions n and eigenvalues Ensmeared higher states are significant |aexp.0 | |aexp.1 | |aexp.2 |wall g.s. dominant |awall0 | |awall1 | |awall2 |

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 50 100 150 200 250

anexp

En [MeV]

smear @ t = 12t = 13t = 14

10-7

10-6

10-5

10-4

10-3

10-2

0 50 100 150 200 250

anwall

En [MeV]

wall @ t = 12t = 13t = 14

Figure: (a) smeared source. (b) wall source.

Wavefunction and Eigenfunctions below Threshold

0 5 10 15 20 25 30 35 40 45

R(r

,t)

r

smear @ t = 12t = 13t = 14

wall @ t = 12t = 13t = 14

-0.01

0

0.01

0.02

0 5 10 15 20 25 30 35 40 45

n(r

,t=

12)

r

4th A13rd A12nd A1

1stg.s.

n-th A1 En [MeV]0 -2.581 52.492 112.083 169.784 224.73

Table: 483 wall source @ t = 12

A1 states (SO(3,Z) rep. of L = 0)below inelastic threshold (483 vol.)

! + + @ 0.29 GeV! + + @ 0.29 GeV

cf. m = 0.51 GeV

Excited States Contamination and Fake-Plateauex. R(t) = b0 exp(E0t) + b1 exp(E1t)

# at b1/b0 = 0.1 plateau-like behavior for t = [12 : 20]

-10

-8

-6

-4

-2

0

2

4

0 5 10 15 20

E

eff(t)

-

E0[M

eV]

t [a]

E1 - E0 = 55.1 MeV

b1/b0 = - 0.1b1/b0 = 0.01

t-dependence of Potential

-40

-20

0

20

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0) V

c(r)

[MeV

] L

= 4

8

r [fm]

wall src. t = 16wall src. t = 15wall src. t = 14wall src. t = 13wall src. t = 11wall src. t = 10

-40

-20

0

20

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0) V

c(r)

[MeV

] L

= 4

8

r [fm]

exp src. t = 15exp src. t = 14exp src. t = 13exp src. t = 11exp src. t = 10

-40

-20

0

20

40

10 11 12 13 14 15 16

(1S

0) V

c(r,t

) [M

eV] L

= 4

8

t [a]

wall src. 0.65 r 0.70smeared src. 0.65 r 0.70

-40

-20

0

20

40

10 11 12 13 14 15 16

(1S

0) V

c(r,t

) [M

eV] L

= 4

8

t [a]

wall src. 1.70 r 1.75smeared src. 1.70 r 1.75

Projection of Eigenstate: Sink Projection

R(f)(t)

r

f(r)

x

0|B(r + x, t)B(x)J (0)|0

=

r

f(r)R(r, t)

and (f(r): arbitrary func.)

E(f)eff (t) = logR(f)(t)

R(f)(t+ 1)

! ground statef(r) = gs(r)

-10

-5

0

5

10

12 13 14 15 16

E

eff

(t)

[M

eV]

L =

48

t [a]

wall src. (1S0)

smeared src. (1S0)

g.s. proj. wallg.s. proj. smear

E0

! 1st excited statef(r) = 1(r)

0

20

40

60

80

100

12 13 14 15 16

E

eff

(t)

[M

eV]

L =

48

t [a]

1st proj. wall1st proj. smear

E1

Next Leading Order of Derivative ExpansionDerivative expansion: U(r, r) = {V0(r) + V1(r)2}(r r) (for 1S0)

[1

4m

2

t2 t

H0]R(r, t) =

d3rU(r, r)R(r, t)

14m

(2/t2)R

R(/t)R

RH0R

R= V0(r)+V1(r)

2R(r, t)R(r, t)

Vtotal(r, t)

! Now, we have Rsmear and Rwall{V0(r) + V1(r)2Rsmear/Rsmear = V smeartotal (r, tsmear)V0(r) + V1(r)2Rwall/Rwall = V walltotal(r, twall),

! LO V0(r) and NLO V1(r) potentials are given byV1(r) =

V smeartotal (r, tsmear) V walltotal(r, twall)2Rsmear/Rsmear 2Rwall/Rwall

V0(r) = Vsmeartotal (r, tsmear) V1(r)

2Rsmear

Rsmear.

2R/R2R/R in denominator

Results: NLO Potential# results of HAL QCD method

! source independentVtotal(r, t) = V0(r) + V1(r)

2R(r, t)R(r, t)

-1000

-800

-600

-400

-200

0

200

400

0 0.5 1 1.5 2 2.5 3 3.5

V1(r

) [M

eV]

r [fm]

V1(r) at t = 11

! wall source Good convergence of LO pot.

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

po

ten

tial

[M

eV]

r [fm]

V0(r)

V1(r) 2R

wall/R

wall

Vtot.wall

(r,11)

! smeared source with NLO pot. correction

-40

-30

-20

-10

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5

(1S

0)

po

ten

tial

[M

eV]

r [fm]

V0(r)

V1(r) 2R

smear/R

smear

Vtot.smear

(r,11)

Results:NLO Potential (1) V0(r) and V1(r)V0(r) = V

exp.total V1(r)

2Rexp.

Rexp., V1(r) =

V exp.total Vwalltotal

2Rexp./Rexp. 2Rwall/Rwall

Figure: (a) diff. of Vtotal (b) diff. of 2R/R (c) V0(r) and V wall (d) V1(r)

Results:NLO Potential (2) V0 and V1 in smeared and wall

Figure: (a) V0(r) (b) V1(r) (c) wall src. summary (d) smeared src. summary

Results:NLO Potential (3) SVDsolving SVD with several ts

1 2Rexp.(r, t1)/Rexp.(r, t1)1 2Rwall(r, t1)/Rwall(r, t1)1 2Rexp.(r, t2)/Rexp.(r, t2)1 2Rwall(r, t2)/Rwall(r, t2)...

...

(V0(r)V1(r)

)=

V exp.total(r, t1)V walltotal(r, t1)V exp.total(r, t2)V walltotal(r, t2)

...

A

(V0(r)V1(r)

)= B UV t

(V0(r)V1(r)

)= B,

Figure: (a) V0(r) (b) V1(r) using SVD

Results:NLO Potential (4) Energy EigenvaluesV walltot , V0(r) and V0(r) + V1(r)2energy eigenvalues are consistent with each other wall source potential LO

volume at 483 48 t g.s. [MeV] 1st [MeV]V walltotal(r) 11 -2.30(20) 53.08(29)

12 -2.58(22) 52.87(33)13 -2.70(30) 52.80(41)

V exp.total(r) 13 -5.33(59) 49.86(58)14 -4.76(55) 50.46(54)

V0(r) 11,12,13 -2.47(23) 52.91(32)12,13,14 -2.62(28) 52.86(37)

V0(r) + V1(r)2 11,12,13 -2.56(1.56) 53.07(2.73)12,13,14 -2.74(2.52) 51.58(3.77)

Table: The energy eigenvalues from potential method.

QCD to Nuclear Physics: Puzzle between Lscher's method and HALBaryon Interaction from Lattice QCDLattice Formulations and SetupLscher's Method: Energy Shift in Finite VolumeHAL QCD Method: PotentialEnergy Eigenvalue in Finite Volume

HAL and Lscher Origin of Fake PlateauSummary and OutlookAppendixAppendix

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