格子 QCD によるハドロン間相互作用 - 核力を QCD から導く -

格子 QCD によるハドロン間相互作用 - 核力を QCD から導く -

青木 慎也

筑波大学 数理物質科学研究科

京都大学基礎物理学研究所セミナー

1. Motivation

What binds protons and neutrons inside a nuclei ?

p n

gravity: too weak

Coulomb: repulsive between pp no force between nn, np

1935 H. Yukawa

introduced virtual particles (mesons) to explain the nuclear force

Yukawa potential V (r) = g²

4π

e^−mπr r

1949 Nobel prize

New force (nuclear force) ?

Nuclear force is a basis for understanding ...

•

•

•

Λ

Nuclear Forces from Lattice QCD

Chiral Dynamics 09, Bern, July 7, 2009

S. Aoki, T. Doi, T. Inoue, K. Murano, K. Sasaki (Univ. Tsukuba) T. Hatsuda, Y. Ikeda, N. Ishii (Univ. Tokyo)

H. Nemura (Tohoku Univ.)

T. Hatsuda (Univ. Tokyo) HAL QCD Collaboration

(Hadrons to Atomic Nuclei Lattice QCD Collaboration)

NN, YN, YY, 3N forces from LQCD

Neutron matter quark Matter?

Atomic nuclei Neutron star Hadrons

Phenomenological NN potential

(~40 parameters to fit 5000 phase shift data)

II I III

One-pion exchange

I

II

III

Jastrow(1951)

Taketani et al.(1951) Yiukawa(1935)

¾

Yukawa (1935)

S

repulsive

core

¾

Jastrow (1951)

SS...

¾

Taketani et al.

(1951)

Key features of the Nuclear force

Modern high precision NN forces (90’s-)

¾

Yukawa (1935)

S

repulsive

core

¾

Jastrow (1951)

SS...

¾

Taketani et al.

(1951)

Key features of the Nuclear force

Modern high precision NN forces (90’s-)

¾

Yukawa (1935)

S

repulsive

core

¾

Jastrow (1951)

SS...

¾

Taketani et al.

(1951)

Key features of the Nuclear force

Modern high precision NN forces (90’s-)

Repulsive core is important

stability of nuclei maximum mass of neutron star

!"#$%&'

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9:23;<!" QCD;<!"

核力の性質をクォークから説明できるか？

Note: Pauli principle is not essential for the “RC”.

!"#$%&'()*$+++$(,$(-.)%/0*/$1

1. Matter(nuclei) cannot be stable without the “repulsive core (RC)”.

2. Neutron star & supernova explosion cannot exist without the “RC”.

3. QCD description should be essential for the “RC”.

4. SU(3) ? (NN ! YN ! YY) ! basis of hypernuclear physics @ J-PARC

23&,/()*,

1. What is the physical origin of the repulsion ?

2. The repulsive core is universal or channel dependent ?

Note: RC is not related to Pauli principle

+

Origin of RC: “The most fundamental problem in Nuclear physics.”

Plan of my talk

1. Motivation

2. Strategy in (lattice) QCD to extract “potential”

3. More structure: tensor potential

4. Inelastic scattering: octet baryon interactions

1. Baryon-Baryon interactions in an SU(3) symmetric world 2. Proposal for S=-2 inelastic scattering

3. H-dibaryon

5. Summary and Discussion

2. Strategy in (lattice) QCD to extract “potential”

南部陽一郎、『クォーク』第２版（講談社、ブルーバックス、

）

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QCD

から核力を如何に定義し、如何に計算するか？

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9:23;<!" QCD;<!" Y. Nambu,

“Force Potentials in Quantum Field Theory”, Prog. Theor. Phys. 5 (1950) 614.

C. Hayashi and Y. Munakata,

“On a Relativistic Integral Equation for Bound states”, Prog. Theor. Phys. 7 (1952) 481.

K. Nishijima,

“Formulation of Field Theories for composite particles”, Phys. Rev. 111 (1958) 995.

佐々木健志、土井琢身、青木慎也

筑波大

石井理修 、初田哲男

東大

池田陽一

理研

、井上貴史

日大

村野啓子

、根村秀克

東北大

HAL QCD Collaboration

(equal time) Nambu-Bethe-Salpeter wave function is a key

ϕ

(r) = � 0 | N (x + r, 0)N (x, 0) | 6q, E �

E = 2!

k² + m²_N QCD eigen-state with energy E and #quark =6

N(x) = ε_abcq^a(x)q^b(x)q^c(x): local operator

E < E_th

inelastic contribution ∝ O(e⁻√

E_th² −E²|r|)

C.-J.D.Lin et al., NPB69(2001) 467 CP-PACS Coll., PRD71 (2005) 094504 N. Ishizuka, PoS(LAT2009)119

ϕ_E(r) = C!

e^ik·r + "

d³p

(2π)³ e^ip·r E_k + E_p 8E_p²

T(p, −p ← k, −k) p² − k² − i#

+ I(r)#

Ishii-Aoki-Hatsuda, PRL 90(2007)0022001 Aoki-Hatsuda-Ishii, PTP123(2010)89

off-shell T-matrix

N. Ishizuka, PoS(LAT2009)119

(Relativistic) Spinor structure is contained in C.

(Equal time) contains sufficient information.

同時刻、 非重心系

空間的）非同時刻、 重心系

Lorentz transformation

p + q k + (−k)

Asymptotic behavior r = |r| → ∞

ϕ_E(r) −→ !

l

C_l sin(kr − lπ/2 + δ_l(k))

kr partial wave

spinor structure

δ_l(k) is the scattering phase shift

S = e

S-matrix below inelastic threshold

Our definition of “potential” Ishii-Aoki-Hatsuda, PRL 90(2007)0022001 Aoki-Hatsuda-Ishii, PTP123(2010)89

[!_k − H₀]ϕ_E(x) = !

d³y U(x, y)ϕ_E(y)

!_k = k² 2µ

H₀ = −∇² 2µ

U(x, y) may be non-local but can be energy-independent.

˜

ϕ_E(y)

dual basis

!ϕ˜_E|ϕ_E^!" = δ_EE^{! !}

E≤Eth

|ϕ_E!"ϕ˜_E| = 1_E≤Eth

identity in the restricted space

U(x, y) = !

E≤Eth

[ε_k − H₀]ϕ_E(x) ˜ϕ_E(y) this construction is NOT unique.

L

no interaction

interaction range

Finite but large volume

Lueshcer’s formula

allowed value: k_n²

δ_l(k_n)

Finite volume

“potential” is expected to be short-range.

Velocity expansion

U(x, y) = V (x, ∇)δ³(x − y)

V (x, ∇) = V₀(r) + V_σ(r)(σ₁ · σ₂) + V_T (r)S₁₂ + V_LS(r)L · S + O(∇²)

tensor operator

LO LO LO NLO NNLO

Okubo-Marshak (1958)

S₁₂ = 3

r²(σ₁ · x)(σ₂ · x)− (σ₁ · σ₂) spins

we calculate observables such as the phase shift and the binding energy, using this approximated potential.

3. Results from lattice QCD

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•

x x

Monte-Calro simulations

L a

Quenched QCD : neglects creation-anihilation of quark-anitiquak pair Full QCD : includes creation-anihilation of quark-anitiquak pair

NBS wave function from lattice QCD

!0|n_β(y, t)p_α(x, t)J ^pn(t₀)|0" = !0|n_β(y, t)p_α(x, t)!

n

|E_n"!E_n|J ^pn(t₀)|0"

= !

n

A_n!0|n_β(y, t)p_α(x, t)|E_n"e^{−En(t−t0)} −→ A₀ϕ^E_αβ⁰(x − y)e^{−E0(t−t0)}

t → ∞

A_n = !E_n|J ^pn(t₀)|0"

Wall source J ^pn(t₀) = p^wall(t₀)n^wall(t₀) ^{q(x, t0) → qwall(t0) = !}

x

q(x, t₀) with Coulomb gauge fixing L = 0 ^{P = +}

spin 1

2 ⊗ 1

2 = 1 ⊕ 0

2S+1

L

S

S

NN wave function

m_π ! 0.53 GeV

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 wave function !(r)

r [fm]

1S₀

3S₁

-2 -1 0 1 2 -2

-1 0 1 2 0.5

1.0 1.5

!(x,y,z=0;¹S₀)

x[fm] y[fm]

!(x,y,z=0;¹S₀)

t − t_s = 6

normalized here attraction

repulsion

Quenched QCD

a=0.137fm

Ishii-Aoki-Hatsuda, PRL90(2007)0022001

m_π ! 0.53 GeV

Ishii-Aoki-Hatsuda, PRL90(2007)0022001

E � 0

Qualitative features of NN potential are reproduced !

Central potential V_c(r) from !"(r) at E ~ 0

(m_#"=0.53 GeV)

1S₀ ,³S₁

Equal-time BS amplitude

Central potential

(quenched) potentials

LO (effective) central Potential

a=0.137 fm L=4.4fm

V (r;¹ S₀) = V₀^(I=1)(r) + V_σ^(I=1)(r) V (r;³ S₁) = V₀^(I=0)(r) − 3V_σ^(I=0)(r)

This paper has been selected as one of 21 papers in Nature Research Highlights 2007

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Scheme/Operator dependence of “potential”

tools running coupling potential

physical observable deep inelastic scattering NN scattering phase shift

phenomena almost free parton repulsive core

interpretation asymptotic freedom no theoretical explanation so far

scheme MS-bar coupling potential

from BS wave function

Other examples:

QM: (wave function,potential) → observables QFT: (asymptotic field,vertex) → observables EFT: (choice of field, vertex) → observables

Leading Order V_C(r) = (E − H₀)ϕ_E(x)

ϕ_E(x) Local potential approximation

The local potential obtained at given energy E may depend on E.

V (x, ∇) = V_C(r) + V_T (r)S₁₂ + V_LS(r)L · S + {V_D(r), ∇²} + · · ·

K. Murano, S. Aoki, T. Hatsuda, N. Ishii, H. Nemura

mπ ! 0.53 GeV

Numerical check in quenched QCD

a=0.137fm

Convergence of the derivative expansion

Non-locality can be determined order by order in velocity expansion ( cf. ChPT)

●     PBC (E〜0 MeV)         ^● APBC (E〜46 MeV)

E-dependence of the local potential turns out to be very small at low

energy in our choice of wave function.

m_π ! 0.53 GeV

a=0.137fm

Quenched QCD

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) (

) ( ) ) (

( ⁰

x x H

r E V

E E

C G

G

\

\

“projection” to L=0

“projection” to L=2

mixing between and through the tensor force³

S

3S₁

3D₁

| φ ! = | φ

! + | φ

!

|φ_S! = P|φ! = 1 24

!

R∈O

R|φ!

|φ_D! = Q|φ! = (1 − P)|φ!

(H

+ V

+ V

S

) | φ ! = E | φ !

P (H

+ V

+ V

S

) | φ ! = EP | φ ! Q(H

+ V

+ V

S

) | φ ! = EQ | φ !

Tensor potential

     quenched QCD        E 〜 0 MeV 

Aoki, Hatsuda, Ishii, PTP 123 (2010)89 arXiv:0909.5585

Wave functions

Quenched

0.0 0.5 1.0 1.5 2.0

r [fm]

m_/ = 529 MeV (a)

3S₁

3D₁

remove angular dependence Y₂₀(θ, φ) ∝ 3 cos² θ − 1

0 50 100

0 0.5 1 1.5 2

V_T(r) V_C(r) V_C,eff

-50 0 50 100 150 200 250 300 350 400

0 0.5 1 1.5 2

V(r) [MeV]

r [fm]

Tensor Force and Central Force (t-t₀=5)

Potentials

No repulsive core in tensor

V

! V

V

m_π ! 0.53 GeV

Aoki, Hatsuda, Ishii, PTP 123 (2010)89 arXiv:0909.5585

Quenched

0 50 100

0 0.5 1 1.5 2

V_T(r) V_C(r) V_C,eff

-50 0 50 100 150 200 250 300 350 400

0 0.5 1 1.5 2

V(r) [MeV]

r [fm]

Tensor Force and Central Force (t-t₀=5)

Potentials

No repulsive core in tensor

V

! V

V

!"#$% &'()*+,

! Nconf=1000

! time-slice: t-t₀=6

! m_-=0.53 GeV, m_.=0.88 GeV, m_N=1.34 GeV

from

R.Machleidt, Adv.Nucl.Phys.19

The wave function

! /%0123456789:;<=>?@AB%C

! deuteronDEF1GHIJK9LM9NO=>?C

! PQ9KLMNRST6!"#$%9ABSsingle

particle spectrum(UVWXYZ[$)\]^_56`ab c=d]^\'eHfg<\hije5kHC

! centrifugal barrier9lF6QmnSopqrs\t=u u6v;<\Uw=xykC

(0z(9{|=}k5j~dÄÅH)

m_π ! 0.53 GeV

Aoki, Hatsuda, Ishii, PTP 123 (2010)89 arXiv:0909.5585

Quenched

Quark mass dependence Quenched

Fit function

• Rapid quark mass dependence of tensor potential

• Evidence of one-pion exchange

Quenched QCD

Full QCD

Quenched QCD

Full QCD Calculation

m_π = 570 MeV, L = 2.9 fm

a=0.1fm

* Large repulsive core than quenched

* Large tensor force than quenched

m_π ! 0.53 GeV _{a=0.137 fm}

L=4.4fm

Phase shift from V(r) in full QCD

1S₀

1S₀

3S₁

3S₁

a=0.1 fm, L=2.9 fm

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1S₀ ³S₁

They have reasonable shapes. The strength is much weaker.

calculation at physical quark mass is important. (future work)

4. Inelastic scattering:

octet baryon interactions

Baryon-Baryon interactions in an SU(3) symmetric world 1. First setup to predict YN, YY interactions not accessible in exp.

2. Origin of the repulsive core (universal or not)

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BB interactions

in a SU(3) symmetric world

x

Six independent potentials in flavor-basis

1. First step to predict YN, YY interactions not accessible in exp.

2. Origin of the repulsive core (universal or not)

6 independent potential in flavor-basis

BB interactions

in a SU(3) symmetric world

x

Six independent potentials in flavor-basis

1. First step to predict YN, YY interactions not accessible in exp.

2. Origin of the repulsive core (universal or not)

BB interactions

in a SU(3) symmetric world

x

Six independent potentials in flavor-basis

1. First step to predict YN, YY interactions not accessible in exp.

2. Origin of the repulsive core (universal or not)

m_u = m_d = m_s

0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V(r) [MeV]

r [fm]

V⁽²⁷⁾

-50 0 50 100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m_/=1014 MeV

m_/=835 MeV

0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V(r) [MeV]

r [fm]

V^(10*)

-50 0 50 100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m_/=1014 MeV

m_/=835 MeV

0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V(r) [MeV]

r [fm]

V^(8s)

0 2500 5000 7500 10000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m_/=1014 MeV

m_/=835 MeV

0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V(r) [MeV]

r [fm]

V⁽¹⁰⁾

-50 0 50 100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m_/=1014 MeV

m_/=835 MeV

Potentials

27, 10*: same as before, NN channel 8s, 10: strong repulsive core

Inoue for HAL QCD Collaboration

-1500 -1000 -500 0 500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V(r) [MeV]

r [fm]

V^{(1) mm//=1014 MeV=835 MeV}

0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V(r) [MeV]

r [fm]

V^(8a)

-50 0 50 100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m_/=1014 MeV

m_/=835 MeV

1: no repulsive core, attractive core ! No quark mass dependence

8a: week repulsive core, deep attractive pocket

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Bound state in 1(singlet) channel ? H-dibaryon ?

However, it is difficult to determine E precisely, due to contaminations from excited states.

Singlet potential with a certain value of E Schroedinger eq. predicts a bound state at E < -30 MeV

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V (r) = a₁e^−a2r2 + a₃ �

1 − e^−a4r2�2 �

e^−a5r r

�2

Finite size effect is very large on this volume.

(consistent with previous results.)

larger volume calculations are in progress.

Proposal for S=-2 In-elastic scattering

m_N = 939 MeV, m_Λ = 1116 MeV, m_Σ = 1193 MeV, m_Ξ= 1318 MeV S=-2 System(I=0)

M_ΛΛ = 2232 MeV < M_NΞ = 2257 MeV < M_ΣΣ= 2386 MeV

The eigen-state of QCD in the finite box is a mixture of them:

E = 2!

m²_Λ + p²₁ = !

m²_Ξ + p²₂ + !

m²_N + p²₂ = 2!

m²_Σ + p²₃

In this situation, we can not directly extract the scattering phase shift in lattice QCD.

|S = −2, I = 0, E"^L = c₁(L)|ΛΛ, E" + c₂(L)|ΞN, E" + c₃(L)|ΣΣ, E"

HAL’s proposal

Let us consider 2-channel problem for simplicity.

NBS wave functions for 2 channels at 2 values of energy:

Ψ^ΞN_α (x) = !0|Ξ(x)N(0)|E_α"

Ψ^ΛΛ_α (x) = !0|Λ(x)Λ(0)|E_α" α = 1, 2

They satisfy

(∇² + p²_α)Ψ^ΛΛ_α (x) = 0 (∇² + q²_α)Ψ^ΞN_α (x) = 0

| x | → ∞

We define the “potential” from the coupled channel Schroedinger equation:

! ∇²

2µ_ΛΛ + p²_α 2µ_ΛΛ

"

Ψ^ΛΛ_α (x) = V ^ΛΛ←ΛΛ(x)Ψ^ΛΛ_α (x) + V ^ΛΛ←ΞN(x)Ψ^ΞN_α (x)

! ∇²

2µ_ΞN + q²_α 2µ_ΞN

"

Ψ^ΞN_α (x) = V ^ΞN←ΛΛ(x)Ψ^ΛΛ_α (x) + V ^ΞN←ΞN(x)Ψ^ΞN_α (x) µ: reduced mass

X, Y = ΛΛ or ΞN

! V ^X←X(x)

V ^X←Y (x)

"

=

! Ψ^X₁ (x) Ψ^Y₁ (x) Ψ^X₂ (x) Ψ^Y₂ (x)

"₋1 !

(E₁ − H₀^X)Ψ^X₁ (x) (E₂ − H₀^X)Ψ^X₂ (x)

"

X != Y

diagonal off-diagonal

diagonal off-diagonal

E_α = p²_α

2µ_ΛΛ , q²_α 2µ_ΞN

α = 1, 2

Using the potentials:

!

V

(x) V

(x) V

(x) V

(x)

"

we solve the coupled channel Schroedinger equation in the infinite volume with an appropriate boundary condition.

For example, we take the incomming ΛΛ state by hand.

In this way, we can avoid the mixture of several “in”-states.

Lattice is a tool to extract the interaction kernel (“T-matrix” or “potential”).

|S = −2, I = 0, E"^L = c₁(L)|ΛΛ, E" + c₂(L)|ΞN, E" + c₃(L)|ΣΣ, E"

Preliminary results from HAL QCD Collaboration

Sasaki for HAL QCD Collaboration

2+1 flavor full QCD

Diagonal part of potential matrix

a=0.1 fm, L=2.9 fm

m_π � 870 MeV

Non-diagonal part of potential matrix

V_A−B ! V_B−A

Hermiticity ! (non-trivial check)

1. S=-2 singlet state may become the bound state in flavor SU(3) limit.

2. In the real world (s is heavier than u,d), some resonance may appear above ΛΛ but below ΞN threshold.

3. Trial demonstration:

3.1. Use potential in SU(3) limit

3.2. Introduce only mass difference from 2+1 simulation 3-3. H-dibaryon

Inoue for HAL QCD Collaboration

! ! !

!"#$%&'"(&))&

!

&+,-./-012

!

%

^" !

%

^" !

% ^"

⁾ !

$ ^{# #} $ ^{# #}_$!

$^$!

%

&'()( &^!_%^$%&*!&' &^!_%^#%&!&' &_%^!!"%&*& *+(,-!.)(!/0(1 Potentials in particle basis in SU(3) limit

“2+1 flavor”

! ! !

!"#$%&'"(&))& ^* !

&+,-../0123

! "#$%&'()!45617-0892&*+,()-.#/01#

23456789!::&*:;45<)9=>?@A<BCD9

;<%EFGHIJKL@MNNOPQRFS6789

! T#$%UV345W45617-0892&*X<-.9

::&%:;45<X<9&;<&%EFGHIJKL#6789

S = −2, I = 0,¹ S₀ scattering

resonance no resonance

“2+1 flavor”

bound state no bound state

“2+1 flavor”

SU(3) limit

5. New method for hadron

interactions in lattice QCD

Inelastic scattering II: particle production

ϕ_E(r) = e^ik·r + !

d³p

(2π)³ e^ip·r E_k + E_p 8E_p²

T(p, −p ← k,−k) p² − k² − i#

+ I(r)

NBS wave function elastic scattering

inelastic contribution

N N ← N N

N N π ← N N ∝ e^iq·r

E ≥ E_th = 2m_N + m_π

|q| = O(E − E_th)

Consider additional NBS wave function

ϕ_E,π(r, y) = !0|N(r + x, 0)π(y + x, 0)N(x, 0)|6q, E"

Note that

|6q, E! = c₁|N N, E!ⁱⁿ + c₂|N N π, E!ⁱⁿ + · · ·

Coupled channel equations

(E − H₀)ϕ_E(x) = !

d³y U₁₁(x;y)ϕ_E(y) + !

d³yd³z U₁₂(x;y,z)ϕ_E,π(y,z) (E − H₀)ϕ_E,π(x,y) = !

d³z U₂₁(x,y;z)ϕ_E(z) + !

d³zd³w U₂₂(x,y;z, w)ϕ_E,π(z, w)

Velocity expansion at LO, two values of E

(E_i − H₀)ϕ_Ei(x) = V₁₁(x)ϕ_Ei(x) + V₁₂(x, x)ϕ_Ei,π(x, x) (E_i − H₀)ϕ_Ei,π(x, y) = V₂₁(x, y)ϕ_Ei(x) + V₂₂(x, y)ϕ_Ei,π(x, y)

i = 1, 2

V₁₁(x) : N N ← N N

V₂₁(x, y) : N N π ← N N V₂₂(x, y) : N N π ← N N π V₁₂(x, x) : N N ← N N π

Solve Schroedinger equation with these potentials and a specific B.C.

•

•

•

•

•

General prescription

ex. Q = 6q : N N, N N π, N N ππ, N N K⁺K⁻, N NN N,¯ · · ·

In practice, of course, final states more than 2 particles are very difficult to deal with.

6. Theoretical understanding of

repulsive core

•

hadron interactions in lattice QCD. Finite size effect is smaller and quark mass dependence is milder than the phase shift.

•

Rotational symmetry is recovered.

•

•

Summary

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