[PPT] - Some theoretical issues in HAL QCD method Sinya Aoki Center for PowerPoint Presentation

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Some theoretical issues in HAL QCD method Sinya Aoki

Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University International Molecule-type Workshop Frontiers in Lattice QCD and related topics April 15- April 26, 2019, Yukawa Institute for Theoretical Physics, Kyoto University

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0. Introduction

Some Issues in the HAL QCD method

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HAL QCD method

A powerful method to investigate hadron interactions Strategy NBS wave function W~

k = 2

q ~ k2 + m2

N

→ X

lm

Clm sin(kx + δl(k)) kx Ylm(Ω~

x)

energy-independent non-local potential '

~ k(~

x)e−W~

kt = h0|N(~

r, t)N(~ r + ~ x, t)|NN, W~

ki

E~

k H0

'

~ k(~

x) = Z U(~ x, ~ y)'

~ k(~

y)d3y, E~

k =

~ k2 mN , H0 = r2 mN ,

W~

k ≤ Wth = 2mN + m⇡

Derivative expansion V (~ x, ~ r) = V0(x) + Vσ(x)(~ 1 · ~ 2) + VT (x)S12 + VLS(x)~ L · ~ S + O(~ r2) U(~ x, ~ y) = V (~ x, ~ r)(3)(~ x ~ y)

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Some issues

Q2. Validity of the derivative expansion ? small parameter ?
II. Definition of the HAL QCD potential with the derivative expansion
Q3. Is the HAL QCD potential Hermite ?
III. Hermite potential from non-Hermite potential
I. The HAL QCD potential from the moving system

Sinya Gongyo’s taik on 4/24.

Q5. Quark annihilation processes and resonances ?

Yutaro Akahoshi’s talk on 4/16.

Q1. The HAL QCD potential in the moving system ?
Q4. Partial wave mixings in the cubic box ?

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I. The HAL QCD potential

from the moving system

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Our Motivation

σ resonance from the I = 0 ππ scattering in the HAL QCD method “vacuum” has the same quantum numbers “vacuum” state appears in the NBS wave function in center-of-mass system the potential describes the vacuum as the “deeply bound state” of two pions ? Moving system no “vacuum” state in the NBS '

~ k(~

x)e−W~

kt = h0|N(~

r, t)N(~ r + ~ x, t)|NN, W~

ki

But → X

lm

Clm sin(kx + δl(k)) kx Ylm(Ω~

x)

true only in the CM system no definition of the potential directly from the boosted NBS

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Generalized definition of the potential in the CM system '

~ k(~

x, x4) = ⌧

N

✓~ x 2 , x4 2 ◆ N ✓ −~ x 2 , −x4 2 ◆

NN, W~

k

'

X

lm

Alm sin(kx + δl(k) + πl/2) kx Ylm(Ω~

x)

non-equal time NBS wave function in CM system k = |~ k| (Ek H0) '

~ k(~

x, x4) = Vx4(~ x, r)'

~ k(~

x, x4) the HAL QCD potential in the x4 scheme x4 = 0: equal time scheme Can we extract the generalized potential from the boosted NBS function ? For simplicity, we consider this problem for scalar field. also in Akahoshi’s talk

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CM and moving systems

Moving CM p, P p∗, P ∗ ~ p1 ~ p2

~ P = ~ p1 + ~ p2

~ p = (~ p1 − ~ p2)/2

~ p∗ − ~ p∗

~ P ∗ = 0

~ V

boost velocity

P ⇤ := (ΛP)0 = (P0 ~ V ~ P), P ⇤

k := (ΛP)k = (Pk VkP0) = 0,

where =

1

p

1~ V 2 ,

boost factor

as ~ V = ~ P/P0.

boost velocity depends on energy

(~ p⇤)2 := (~ p⇤

?)2 + (~

p⇤

k)2 = ~

p2 (~ P · ~ p)2 P 2 = P 2

0 ~

P 2 4 m2.

important relation

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relation of NBS wave functions between 2 systems

p,P (x, X) := h0|T{(x1)(x2)}|p1, p2i

NBS

where x := x1 x2, fy p2

i m2 = 0

at the total en

P0 =

q

~ p2

1 + m2 +

q

~ p2

2 + m2.

p,P (x, X) = h0|ei ˆ

P·XT{(x/2)(x/2)}ei ˆ P·X|p1, p2i

= eiP·X'p,P (x),

x := x1 − x2, X = (x1 + x2)/2

Scalar field

eiP·X'p,P (x) = eiP ∗·X∗'p∗,P ∗(x⇤).

Since P · X = P ⇤ · X⇤

CM moving

'p,P (x) = 'p∗,P ∗(x⇤),

p⇤ = (0, ~ p⇤), P ⇤ = (P ⇤

0 ,~

0) x⇤ =

⇣

(x0 ~ V ~ x), (~ xk ~ V x0), ~ x?

⌘

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Euclidean space-time (Wick rotation) x4 = ix0

X⇤

4

= (X4 i~ V ~ X), X⇤

k = (Xk + i~

V X4), X⇤

? = X?,

x⇤

4

= (x4 i~ V ~ x), x⇤

k = (xk + i~

V x4), x⇤

? = x?,

X⇤ = (X0 ~ V ~ X), ~ X⇤

?

= ~ X?, ~ X⇤

k = ( ~

Xk ~ V X0), x⇤ = (x0 ~ V ~ x), ~ x⇤

?

= ~ x?, ~ x⇤

k = (~

xk ~ V x0),

'p,P (x) = 'p∗,P ∗(x⇤),

p⇤ = (0, ~ p⇤), P ⇤ =

⇣

P ⇤

0 ,~

⌘

x⇤ =

⇣

(x4 i~ V ~ x), (~ xk + i~ V x4), ~ x?

⌘

.

(~ p⇤)2 = P 2

0 ~

P 2 4 m2, P ⇤

0 = P0

, ~ V = ~ P P0 , iP · X = P0X4 + i~ P ~ X = P ⇤

0 X⇤ 4.

moving CM complex in CM ! Minkowski Euclid

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HAL QCD potential from boosted NBS wave function using Boosted NBS generalized definition

r2

x⇤

= 2 ⇣ rxk + iV @x4

⌘2 + r2

x?.

V(x4iV xk)((xk + iV x4), x?, rx⇤)'p,P (x)

⇣ ⌘2

k

k ? r

= (~ p⇤)2 + 2 ⇣ rxk + iV @x4

⌘2 + r2

x?

m 'p,P (x)

x4 = 0 (equal-time boosted NBS) is required.

(Ep∗ H0)'p∗,P ∗(x⇤) = Vx∗

4(~

x, r)'p∗,P ∗(x⇤)

Moving Moving Moving CM CM CM

Vx∗

4( ~

x⇤, r)'p,P (x) = ( ~ p∗)2 + r2

x∗

m 'p,P (x)

'p,P (x) = 'p∗,P ∗(x⇤),

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ViV xk(xk, x?, rx⇤)'p,P (x) = (~ p⇤)2 + 2 ⇣ rxk + iV @x4

⌘2 + r2

x?

m 'p,P (x).

Each xk Minkowski time from Euclidean correlates ! potential in the x⇤

0 = −γxk scheme

(~ p⇤)2 = P 2

0 ~

P 2 4 m2, P ⇤

0 = P0

, ~ V = ~ P P0 ,

x4 = 0

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Time-dependent HAL QCD method

R ~

P (x, X)ei ~ P· ~ X

:= hT{(x1)(x2)}J†

(0, ~

P)i

~ ~

X

h { } i = ei ~

P· ~ X

X

P n

0 Wth

AneP n

0 X4'pn,P n(x) + · · · ,

A1(x, X) :=

⇣

@2

X4 ~

P 2⌘ R ~

P (x, X) =

X

n

AneP n

0 X4((P n

0 )2 ~

P 2)'pn,P n(x), A2(x, X) :=

⇣

@2

X4 ~

P 2⌘2 R ~

P (x, X) =

X

n

AneP n

0 X4((P n

0 )2 ~

P 2)2'pn,P n(x), B(x, X) := r2

x?A1(x, X),

C(x, X) =

⇣

@X4rxk + i|~ P|@2

x4

⌘2 R ~

P (x, X)

=

X

n

AneP n

0 X4(P n

0 )2 ⇣

rxk + iVn@x4

⌘2 'pn,P n(x),

(41) ViV xk(xk, x?, rx⇤)'p,P (x) = (~ p⇤)2 + 2 ⇣ rxk + iV @x4

⌘2 + r2

x?

m 'p,P (x).

R-correlator sum of NBS

(~ p⇤)2 = P 2

0 ~

P 2 4 m2, P ⇤

0 = P0

, ~ V = ~ P P0 ,

Vn = |~ P| P n

2

n =

(P n

0 )2

(P n

0 )2 − ~

P 2

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A2(x, X) 4 m2A1(x, X) + B(x, X) + C(x, X)

X ✓ ⇣

X

n

✓

= mVx⇤

4=0(xk = 0, x?, r)A1(x, X)

x4 = xk = 0 LO potentail

und state saturation. At the LO, we have

V LO

x⇤

4=0(0, x?)

= A2(x, X)/4 m2A1(x, X) + B(x, X) + C(x, X) mA1(x, X) .

= X

n

Ane−P n

0 X4

"( (P n

0 )2 ~

P 2 4 m2 + r2

x?

) + (P n

0 )2

(P n

0 )2 ~

P 2 (rxk + iVn@x4)2 # ⇣ (P n

0 )2 ~

P 2⌘ 'pn,P n(x)

= ( ~ p∗

n)2

= γ2

n

= m X

n

AneP n

0 X4ViγnVnxk(nxk, x?, rx⇤)((P n

0 )2 ~

P 2)'pn,P n(x) Set x4 = 0 Set xk = 0 = mVx∗

4=0(xk = 0, x?, rx∗)

X

n

AneP n

0 X4((P n

0 )2 ~

P 2)'pn,P n(x)

ViV xk(xk, x?, rx⇤)'p,P (x) = (~ p⇤)2 + 2 ⇣ rxk + iV @x4

⌘2 + r2

x?

m 'p,P (x).

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Future investigations

1. Check the formula for he simple system
2. resonance in the HAL QCD potential

I = 2 ππ moving system σ resonance in I = 0 ππ moving system

3. extension to fermions (Baryons)

lower components mix relativistic formulation for the “potential” ?

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II. Definition of the HAL QCD potential

with the derivative expansion

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(E~

k − H0)' ~ k(~

x) = Z U(~ x, ~ y)'

~ k(~

y)d3y, W~

k ≤ Wth

Of course, the scheme is not unique. One may use a different one. This equation does not fix the non-local potential due to the restriction of energies. Therefore this potential is ambiguous. We have to fix the definition of the potential (scheme) explicitly. For simplicity, let us consider the scalar particles and ignore the angular momentum dependent part of the potential.

We consider the expansion in terms of r2 (but not L2).
Terms with odd number of r are not included. This is our scheme.
The potential must be non-Hermitian. We can make it Hermitian as seen

later. We here propose a scheme to fix the potential completely using the derivative expansion.

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Definition of the potential choice of energy W n

k := 2mN + k

n(Wth − 2mN), k = 0, 1, · · · , n 2mN Wth W k W k

1

W k

2

· · · W k

n

approximated potential at the n-th order V (n)(x, r) :=

n

X

k=0

V (n)

k

(x)(r2)k (L2)k terms are ignored for simplicity. Coefficient functions V (n)

k

(x) can be determined from

n

X

k=0

V (n)

k

(x)(r2)k'pk(x) = (✏pk H0)'pk(x) W k

n = 2

q p2

k + m2 N

      'p0(x) r2'p0(x) · · · (r2)n'p0(x) 'p1(x) r2'p1(x) · · · (r2)n'p1(x) · · · · · · · · · · · · · · · · · · · · · · · · 'pn(x) r2'pn(x) · · · (r2)n'pn(x)              V (n) (x) V (n)

1

(x) · · V (n)

n

(x)        =       (✏p0 H0)'p0(x) (✏p1 H0)'p1(x) · · (✏pn H0)'pn(x)       a number of unknowns = a number of equations V (n)

k

(x) Def of potential V (x, r) := lim

n→∞ V (n)(x, r) = lim n→∞ n

X

k=0

V (n)

k

(x)(r2)k faithful to phase shifts

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In practice Take n = 1 with W 1

0 ' 2mN and W 1 1 ' Wth.

V LO(x) = −H0ϕ0(x) ϕ0(x) V LO(x) = (✏pth − H0)'pth(x) 'pth(x) δ0(p) p p0 = 0 pth H = H0 + V (1) (r) + V (1)

1

(r)r2 ˜ H = H0 + ˜ V (1) (r) + riV (1)

1

(r)ri δ0(p) p p0 = 0 pth good approximation in physical interval Next subject

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Demonstration Separable potential U(~ x, ~ y) = wv(~ x)v(~ y) v(~ x) = e−µx, x := |~ x| L=0 wave function ψ0

k(x) = eiδ(k)

kx  sin(kx + δ(k)) − sin δ(k)e−µx ✓ 1 + xµ2 + k2) 2µ ◆ = C eiδ(k) kx sin(kx + δR(k)) x ≤ R x > R R: IR cut-off phase shift δR(k) is exactly calculable. LO potential NLO potential V NLO (r) + V NLO

1

(r)r2 V LO (r) separable potential U(~ x, ~ y) from k2 = 0 or k2 = µ2

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LO at k2 = 0 NLO exact LO at k2 = µ2 NLO potential reproduces the exact phase shift rather well.

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LO at k2 = 0 LO ERE ar k2 = 0 LO ERE ar k2 = µ2 LO at k2 = µ2 exact NLO NLO ERE ERE =Effective Range Expansion NLO potential is equally good to or even better than NLO ERE.

k cot(δ0(k))

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III. Hermitian potential

from non-Hermitian potential

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The HAL QCD potential is non-Hermitian in general, since NBS wave functions are not orthogonal to each other. (If Hermite, eigenfunctions must be orthogonal.) Can we make non-Hermitan potential Hermite ?

H = H0 + U

U : non-Hermitian H0 = − 1 mN ∇2, U

Hψ = Eψ, E ∈ R

˜ Hφ = Eφ, ˜ H = R−1HR, ψ = Rφ

˜ H = H0 + V,

V : Hermitian

real eigenvalues

?

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1. LO in the derivative expansion (n=1)

U = V0 + V1∇i + 1 2V ij

2 ∇i∇j = V0 + ¯

V i

1∇i + 1

2∇jV ij

2 ∇j,

V ji

2

= V ij

2

˜ V i

1 := V i 1 (rjV ij 2 )/2

˜ H = H0 + ˜ V0 + 1 2riV ij

2 rj +

⇢ ˜ V i

1

2 mN R−1riR + V ij

2 R−1VjR

ri

˜ V0 := V0 − 1 mN R−1∇2R + V i

1R−1∇iR + 1

2V ij

2 R−1(∇i∇jR).

demanding

= 0 Hermite

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Rotationally symmetric case:

˜ V i

1 := ˆ

riV1(r), V ij

2

:= δijV2a + ˆ riˆ rjV2b, ˆ ri := ri r

dR(r) dr = mN 2 ¯ V1(r) 1 − mN 2 V2(r) R(r), V2 := V2,a + V2,b,

R(r) = exp

⎡ ⎢ ⎣mN

2

r

r∞

ds ¯ V1(s) 1 − mN 2 V2(s)

⎤ ⎥ ⎦

At this order, we can make the potential Hermitian without approximation.

˜ V0 = V0 − ˜ V1 r − ˜ V 0

1

2 + m 4 ˜ V 2

1

1 − m

2 V2

˜ V1 := V1 V2a V 0

2a + V 0 2b

2 ,

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2. NLO (n=2)

H1 = H0 + U1, H0 = H0 + U0

U1 = V3 + V4 = U1,4 + U1,3 + U1,2 + U1,1

˜ H1 := R−1H1R, R := R0(1 + R1)

˜ H1 ' ˜ H0 + R−1

0 U1R0 + [ ˜

H0, R1] + [R−1

0 U1R0, R1]

Hermitian

˜ U1 := R−1

0 U1R0 = U1,4 + ˜

U1,3 + ˜ U1,2 + ˜ U1,1 + ˜ U1,0

˜ H0 := R−1

0 H0R0 = ˜

V0 + 1 2riHij

0 rj

R−1 ' (1 R1)R−1

neglected as higher order (approximation) O(r4) O(r2) H H H U1,n : n-th derivative term Hermitian Hermitian (n=1) (1 − R1)R−1

0 (H0 + U1)R0(1 + R1)

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Take

R1 = R1,0 + R1,2, R1,2 := 1 2Rij

1,2rirj

= 0 = 0

X0 := [ ˜ H0, R10] = X0,1 + X0,0, X2 := [ ˜ H0, R12] = X2,3 + X2,2 + X2,1 + X2,0

R10 R12

The potential can be made Hermitian within the derivative expansion. Higher orders can be treated similarly. (detail skipped) Hermitian

[ ˜ H0, R1]

˜ H1 = ˜ H0 + U1,4 + ˜ U1,2 + ˜ U1,0 + X2,2 + X2.0 + X0,0 ˜ U1,3 + X2,3 ˜ U1,1 + X2,1 + X0,1

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Example of II & III

2+1 flavor QCD a = 0.09 fm (a−1 = 2.2 GeV) mπ = 0.51 GeV, mN = 1.32 GeV, mK = 0.62 GeV, mΞ = 1.46 GeV

Effective LO — Veff

small difference

From the difference, we can determine two terms.

n = 0 ΞΞ potential

smeared: more higher energy modes wall: low energy modes

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V0(r) + V1(r)r2 ˜ V0(r) + riV1(r)ri Hermitization Hermite non-Hermite V0(r) m2

πV1(r)

˜ V0(r)

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Scattering phase shift

δ0(k) k cot(δ0(k))

LO local potential after Hermitization is better than non-Hermitian one. Effects of NLO contributions gradually show up as energy increases.

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Our answers

Q2. Validity of the derivative expansion ? small parameter ?
Q3. Is the HAL QCD potential Hermite ?
Q5. Quark annihilation processes and resonances ?

Yutaro Akahoshi’s talk on 4/16.

Q1. The HAL QCD potential in the moving system ?
Q4. Partial wave mixings in the cubic box ?
I. The potential can be construed from the boosted NBS.

Sinya Gongyo’s talk on 4/24.

II. The derivative expansion is a part of the definition.
III. The HAL QCD potential is non-Hermite, but can be made Hermite.

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States and operators Lorentz tr. of EM op.

ˆ P 0

µ

:= U ˆ PµU 1 = (Λ1)µν ˆ Pν,

Lorentz tr. of states

ˆ Pµ|pi = pµ|pi U|pi = |Λpi, (Λp)µ = Λµνpν,

ˆ PµU|pi = UU 1 ˆ PµU|pi = UΛµν ˆ Pν|pi = ΛµνpνU|pi,

U|p1, p2i = |Λp1, Λp2i, U|0i = |0i, Scalar field op.

Uφ(x)U 1 = eiΛ−1 ˆ

P·xUφ(0)U 1eiΛ−1 ˆ P·x = ei ˆ P·Λxφ(0)ei ˆ P·Λx

= φ(Λx)

(x) = ei ˆ

P·x(0)ei ˆ P·x,

U(0)U 1 = (0),

Λ1A · B := gµ⌫(Λ1)µ↵A↵B⌫ = g↵A↵Λ⌫B⌫ = A · ΛB.

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Procedure

1. At non-zero boost ~

P, calculate 'p,P (x) at fixed energy or R ~

P (x, X).

2. Define projections as

(Pk)ij = PiPj ~ P 2 , P? = 1 Pk.

3. If P0 can be measured, calculate

~ V = ~ P P0 , = 1 p 1 ~ V 2 , (~ p⇤)2 = P 2

0 ~

P 2 4 m2.

4. Calculate ViV xk(xk, x?, r) from 'p,P or Vx⇤

4=0(0, x?, r) from R ~

P .

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IV. Partial wave decomposition

in the HAL QCD method

T. Miyamoto, et al. (HAL QCD), in preparation

SLIDE 36

Lattice QCD in the finite box Rotational symmetry Cubic symmetry angular momentum is conserved partial wave decomposition is possible different partial waves are mixed a finite number of irreducible representation

O(3, R) O(3, Z)

0 = A+

1 , 1 = T − 1 , 2 = E+ ⊕ T + 2 ,

3 = A−

2 ⊕ T − 1 ⊕ T − 2 , 4 = A+ 1 ⊕ E+ ⊕ T + 1 ⊕ T + 2 ,

l rep. basis polynomials independent elements A+

1

1 1 T −

1

ri i = 1, 2, 3 2 E+ r2

i − r2 j

(i, j) = (1, 2), (2, 3) 2 T +

2

rirj i ̸= j 3 A−

2

r1r2r3 3 T −

1

5r3

i − 3r2rj

i = 1, 2, 3 3 T −

2

ri(r2

j − r2 k)

(i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2) 4 A+

1

5(r4

1 + r4 2 + r4 3) − 3r4

4 E+ 7(r4

i − r4 j) − 6r2(r2 i − r2 j)

(i, j) = (1, 2), (2, 3) 4 T +

1

rir3

j − rjr3 i

i ̸= j 4 T +

2

7(rir3

j + rjr3 i ) − 6r2rirj

i ̸= j

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Motivation

'NBS(~ x)

S-wave projection continuous space

'L=0

NBS(r = |~

x|)

'L=0(r) = Z

s

dΩ Y ∗

00(✓, ) '(~

x, r = |~ x|) spherical surface integral discrete space 'A1(~ x) = 1 48 X

g∈O(3,Z)

'(g−1~ x) average over cubic group

0.0 0.5 1.0 1.5 2.0 r [fm] −500 500 1000 1500 2000 2500 3000 3500

3S1 ΛcN effective-central potential V eff C

(r) [MeV] 0.0 0.5 1.0 1.5 2.0 −30 −20 −10 10 20 30 40 mπ ≃ 700 MeV mπ ≃ 570 MeV mπ ≃ 410 MeV

A1 = 0 ⊕ 4 ⊕ 6 ⊕ · · · manifestation of higher (L =4,6, .. ) partial waves Can we remove these higher partial waves ? Ex.

SLIDE 38

Misner’s method

C. W. Misner, Class. Quantum Grav. 21 (2004) S243 - S247

ψ(r, θ, φ) = X

lm

glm(r)Ylm(θ, φ) glm(r) ? setup

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

SR,Δ

R Δ

Legendre polynomial

よって、r = R で球面調和関数の係数 glm(R) は次のように求められるこの球核の上では、スカラー場は次のように展開できるこのとき、係数 cnlm は SR,Δ上の体積積分で求めることができる

まず、連続空間での議論を行う

３次元空間上のスカラー場 ψ が次のように球面調和関数で展開されているとする：この時、r = R で球面調和関数の係数 glm(R) を取り出すことを考える

幅 2Δを持つ球核上の規格直行系をと定義する。この時、GnRΔ(r) (n = 0,…,∞) は次のような動径方向の完全系である

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この関係を持つ１例：

∵ A complete set in the shell SR,∆ = {~ x|R − ∆ ≤ |~ x| ≤ R + ∆} YR,∆

nlm (r, θ, φ) := GR,∆ n

(r)Ylm(θ, φ) Z R+∆

R−∆

r2dr GR,∆

n

(r)GR,∆

m

(r) = δn,m Ex.

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

SR,Δ

R Δ

GR,∆

n

(r) ≡ Pn ✓r − R ∆ ◆ 1 r r 2n + 1 2∆

Legendre polynomial

よって、r = R で球面調和関数の係数 glm(R) は次のように求められるこの球核の上では、スカラー場は次のように展開できるこのとき、係数 cnlm は SR,Δ上の体積積分で求めることができる

まず、連続空間での議論を行う

３次元空間上のスカラー場 ψ が次のように球面調和関数で展開されているとする：この時、r = R で球面調和関数の係数 glm(R) を取り出すことを考える

幅 2Δを持つ球核上の規格直行系をと定義する。この時、GnRΔ(r) (n = 0,…,∞) は次のような動径方向の完全系である

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この関係を持つ１例：

∵

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

SR,Δ

R Δ

Legendre polynomial

よって、r = R で球面調和関数の係数 glm(R) は次のように求められるこの球核の上では、スカラー場は次のように展開できる

ψ(r, θ, φ) = X

nlm

cnlm YR,∆

nlm (r, θ, φ)

このとき、係数 cnlm は SR,Δ上の体積積分で求めることができる

まず、連続空間での議論を行う

３次元空間上のスカラー場 ψ が次のように球面調和関数で展開されているとする：この時、r = R で球面調和関数の係数 glm(R) を取り出すことを考える

幅 2Δを持つ球核上の規格直行系をと定義する。この時、GnRΔ(r) (n = 0,…,∞) は次のような動径方向の完全系である

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この関係を持つ１例：

∵ in SR,∆

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

SR,Δ

R Δ

Legendre polynomial

よって、r = R で球面調和関数の係数 glm(R) は次のように求められるこの球核の上では、スカラー場は次のように展開できるこのとき、係数 cnlm は SR,Δ上の体積積分で求めることができる

cnlm = Z

SR,∆

d3r YR,∆

nlm (r, θ, φ) ψ(r, θ, φ)

まず、連続空間での議論を行う

３次元空間上のスカラー場 ψ が次のように球面調和関数で展開されているとする：この時、r = R で球面調和関数の係数 glm(R) を取り出すことを考える

幅 2Δを持つ球核上の規格直行系をと定義する。この時、GnRΔ(r) (n = 0,…,∞) は次のような動径方向の完全系である

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この関係を持つ１例：

∵

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

SR,Δ

R Δ

Legendre polynomial

よって、r = R で球面調和関数の係数 glm(R) は次のように求められるこの球核の上では、スカラー場は次のように展開できるこのとき、係数 cnlm は SR,Δ上の体積積分で求めることができる

まず、連続空間での議論を行う

３次元空間上のスカラー場 ψ が次のように球面調和関数で展開されているとする：この時、r = R で球面調和関数の係数 glm(R) を取り出すことを考える

幅 2Δを持つ球核上の規格直行系をと定義する。この時、GnRΔ(r) (n = 0,…,∞) は次のような動径方向の完全系である

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Z

SR,∆

d3r YR,∆

nlm (~

r)YR,∆

n0l0m0(~

r) = nn0ll0mm0

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この関係を持つ１例：

∵

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

SR,Δ

R Δ

Legendre polynomial

よって、r = R で球面調和関数の係数 glm(R) は次のように求められるこの球核の上では、スカラー場は次のように展開できるこのとき、係数 cnlm は SR,Δ上の体積積分で求めることができる

glm(R) = X

n

cnlmGR,∆

n

(R)

まず、連続空間での議論を行う

３次元空間上のスカラー場 ψ が次のように球面調和関数で展開されているとする：この時、r = R で球面調和関数の係数 glm(R) を取り出すことを考える

幅 2Δを持つ球核上の規格直行系をと定義する。この時、GnRΔ(r) (n = 0,…,∞) は次のような動径方向の完全系である

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sha1_base64="Be0rhAcq08NFdX8T1cWuH+68zl4=">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</latexit><latexit sha1_base64="Be0rhAcq08NFdX8T1cWuH+68zl4=">AFM3iclVLbtNAFL1tApTwaAsbJDYWVErhWoClUBISG1BKs+SB90Gsuxp83Q8YPxOKIMwfwAyxYgYQYs0XsGHDkXFDrFBiFWR2LDg2kmbkJBCx7Ln+vic+/1nWokeKwI2enrz+WPHD02cLxw4uSp04NDw2eW4jCRLiu7oQjlStWJmeABKyuBFuJHP8qmDL1a2b6flOpMxD4M7ajti676zGfAN7joKIXu4/8MoVeyBypx0VSTMaNfWgfCNsWjMfYsKJ9gUzK+o2quI/SqeiFIr3FhHKMrb3EcVpYz2yaBRzi8qMbnuFdmPJPKMP4fR3xkzL3bphUS9FbT09Ywm62HqePdST8ruJ7yOHSV+atNR2WxFXyqZhlWv3IfuZ/r/GqEB62ylM0+P1tIhjckiVTWmnCKNanx8r89ZO2ijslxa9XWKb2dbApdmXigbL1ot6TGq1yRj2mIZyg9YgfXQuvM1dKMm54jDio0ktxnltgL/GbQUreNtjm31IiC7I3/cw9tAImSDZsrqDUjMYgeaC4deAQUPQnAhAR8YBKAwFuBAjNcalIBAhNg6aMQkRjz7zsBAbUJshgyHES38LmJb2tNMD31DPO1C5mEXhLVFowSj6S12SXvCdvyFfyq6eXzjzSWrZxrza0LIHn5xb/PlPlY+7glpLdWDNCjbgWlYrx9qjDEm7cBv6+sOnu4vXF0b1RfKCfMP6n5Md8g47COo/3JfzbOEZFHApc7f3R0sXZ4oYTw/OTI10xzFAJyHCzCG/sqTMFtmIMyuLnJ3N2cm/Pyb/Of8p/zXxrU/r6m5iz8sfLfwO1gp8I</latexit><latexit sha1_base64="Be0rhAcq08NFdX8T1cWuH+68zl4=">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</latexit><latexit sha1_base64="Be0rhAcq08NFdX8T1cWuH+68zl4=">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</latexit>

この関係を持つ１例：

∵ exact in continuous space

SLIDE 39

discrete space

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパスこの時、有限個のデータ点しかないことによりYnlmの直行性は破れる

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

離散空間では、体積積分は球核上の全てのデータ点の重み付きの和に置き換えることができる点 (x,y,z) を中心とした単位立方体と球核 SR,Δ との重なりの大きさに対応した重み R Δ

⃗ x ω( ⃗ x )

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素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパスこの時、有限個のデータ点しかないことによりYnlmの直行性は破れる

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

離散空間では、体積積分は球核上の全てのデータ点の重み付きの和に置き換えることができる点 (x,y,z) を中心とした単位立方体と球核 SR,Δ との重なりの大きさに対応した重み

hf|gic ⌘ Z

~ x∈SR,∆

d3x f(~ x)g(~ x) hYR,∆

A

|YR,∆

B

ic = δAB

(A,B = nlm)

hf|gid ⌘ X

~ x∈SR,∆

!(~ x)f(~ x)g(~ x)

R Δ

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weight function

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパスこの時、有限個のデータ点しかないことによりYnlmの直行性は破れる

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

離散空間では、体積積分は球核上の全てのデータ点の重み付きの和に置き換えることができる点 (x,y,z) を中心とした単位立方体と球核 SR,Δ との重なりの大きさに対応した重み R Δ

hYR,∆

A

|YR,∆

B

id 6= δAB

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素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

Δ と nmax の決め方 R Δ

✓ 動径方向の直交基底 GnRΔ(r) はだけの離散誤差を持つ

Lattice QCD data の離散誤差と同じようにスケーリングさせるために、Δ は Δ～a (lattice spacing) とした方が良い

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nmax は、LQCD data の離散誤差より小さくなるように選ぶ（例：LQCD の離散誤差がの場合、少なくとも nmax >= 2 とする）

<latexit sha1_base64="2ZRGF+CoPRHNB1QhYiaKzhymbKI=">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</latexit><latexit sha1_base64="2ZRGF+CoPRHNB1QhYiaKzhymbKI=">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</latexit><latexit sha1_base64="2ZRGF+CoPRHNB1QhYiaKzhymbKI=">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</latexit><latexit sha1_base64="2ZRGF+CoPRHNB1QhYiaKzhymbKI=">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</latexit>

離散空間上でも直行性を保たせるために、次のように dual basis を定義する

dual basis を構成する際には、有限個のベースしか使えないことに注意

hYR,∆

A

|YR,∆

B

id = GAB = G∗

BA

よって、を得る

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sha1_base64="LxVWu3c3u+e5RLmlPaQUKFMsI=">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</latexit>

YR,∆

adj,A ≡

X

B

G−1

ABYR,∆ B

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✓ nmax, lmax は対象となるデータによって適切なものを選ぶ

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dual basis

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

Δ と nmax の決め方 R Δ

✓ 動径方向の直交基底 GnRΔ(r) はだけの離散誤差を持つ

Lattice QCD data の離散誤差と同じようにスケーリングさせるために、Δ は Δ～a (lattice spacing) とした方が良い

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nmax は、LQCD data の離散誤差より小さくなるように選ぶ（例：LQCD の離散誤差がの場合、少なくとも nmax >= 2 とする）

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離散空間上でも直行性を保たせるために、次のように dual basis を定義する

dual basis を構成する際には、有限個のベースしか使えないことに注意よって、を得る

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hYR,∆

dual,A|YR,∆ B

id = δAB

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✓ nmax, lmax は対象となるデータによって適切なものを選ぶ

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Since a number of points in the shell is finite, we have to truncate n ≤ nmax and l ≤ lmax to have G−1

AB.

This introduces an approximation !

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

Δ と nmax の決め方 R Δ

✓ 動径方向の直交基底 GnRΔ(r) はだけの離散誤差を持つ

Lattice QCD data の離散誤差と同じようにスケーリングさせるために、Δ は Δ～a (lattice spacing) とした方が良い

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nmax は、LQCD data の離散誤差より小さくなるように選ぶ（例：LQCD の離散誤差がの場合、少なくとも nmax >= 2 とする）

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離散空間上でも直行性を保たせるために、次のように dual basis を定義する

dual basis を構成する際には、有限個のベースしか使えないことに注意よって、を得る

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sha1_base64="ugQB+3wCUqUL7CQbHzyW+PXzxQ=">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</latexit><latexit 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✓ nmax, lmax は対象となるデータによって適切なものを選ぶ

X

B

=

nmax

X

n=0 lmax

X

l=0 l

X

m=−l

!

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SLIDE 40

A choice of ∆, nmax and lmax GR,∆

n

has O(∆nmax+2) discretization errors ∆ ∼ a, nmax ≥ 2 Our choice ∆ = a, nmax = 2, lmax = 6 NBS wave function Higher L contributions seem to be removed by the Misner’s method !

SLIDE 41

Laplacian term

素粒子・原子核・宇宙「京からポスト京に向けて」シンポジウム筑波大学東京キャンパス

“ひげ” 構造

I. Introduction
II. Misner’s method
III. Mockup data
IV. Lattice QCD data
V. Summary

✓従来の計算では、ラプラシアンは２回差分として定義していた ✓ミスナー法では、ラプラシアンは解析的に計算できる

実際の Lattice QCD データへの適用：ΛcN（1S0）系 NBS 波動関数のラプラシアン

~ r2glm(r) =

nmax

X

n=0

cR,∆

nlm

1 r @2 @r2 ⇥ rGR,∆

n

(r) ⇤

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✓ミスナー法の結果と比べると、ひげ構造が大きくなっていることがわかる

L >= 4 成分は波動関数のラプラシアンに大きなひげ構造を作る

Misner’s method Conventional HAL QCD ~ r2'A1(~ x) '

3

X

k=1

'A1(~ x + a~ k) + 'A1(~ x a~ k) 2'A1(~ x) a2 The finite difference approximation enhances higher partial wave contributions. analytic derivative

SLIDE 42

Potential The Misner’s method can remove large fluctuations caused by the contamination from higher partial waves to the S=0 component.

SLIDE 43

Fits Fit to the conventional HAL QCD data The Misner’s method

'

Statistical errors of the fit to the conventional HAL QCD data are not affected by contaminations from higher partial waves.

SLIDE 44

Scattering phase shift Almost identical between the conventional result and the Misner’s method. No improvement of statistical errors, but we have more confidence on validity of our results !