Decuplet-Decuplet interaction and recent development of partial - - PowerPoint PPT Presentation

SLIDE 1

Decuplet-Decuplet interaction and recent development 


• f partial wave decomposition on lattice

Shinya Gongyo (RIKEN)

• Apr. 24, 2019@FLQCD

HAL QCD Collaboration

K.Sasaki(YITP), S. Aoki (YITP), Y. Akahoshi (YITP),


• T. Doi (RIKEN), F. Etiminan (Birjand U.), 

• T. Hatsuda (RIKEN), Y. Ikeda (YITP), T. Inoue (Nihon U.), 

• T. Iritani (RIKEN), N. Ishii (RCNP), T. Miyamoto (YITP), H. Nemura (RCNP)

SG, K.Sasaki + (HAL QCD Coll.), PRL 120 (2018) 212001

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

SLIDE 2

Outline

First part: Dec-Dec interaction from lattice QCD

• Introduction:


Dibaryon candidates and model studies

• Results at heavy quark masses for ΔΔ(7S3)
• Results at (almost) physical quark masses for ΩΩ(1S0)


Second part: Partial wave decomposition on lattice

• fixed-r method
• Misner’s method
• numerical test and application to ΛcN system

2

SLIDE 3

Introduction

3

Baryon (B=1) Dibaryon (B=2) Proton, Neutron, Lambda, Omega,… Deuteron

• bserved in 1930s

Dibaryon = two baryon bound state or resonance + d*(2380) resonance

SLIDE 4

4

udd uud uus dds sds sus uds

n p Σ+ Σ0, Λ Σ- Ξ- Ξ0

ddd dud uud uuu sss dds uus sds sus

Δ- Δ0 Δ+ Δ++ Σ*+ Σ*-

uds Σ*0

Ξ*- Ξ*0 Ω- Octet(S=1/2) Decuplet(S=3/2)

In decuplet baryons, only Ω is stable under strong decay. In the case of heavier pion mass, Delta baryons become stable.

Δ p+ + π+ Δ p+ + π+

heavy pion

Our lattice simulation 
 ΔΔ => heavy pion ΩΩ => phys. pt.

SLIDE 5

5

10 ⊗ 10 = 28 ⊕ 27 ⊕ 35 ⊕ ¯ 10

10 ⊗ 8 = 35 ⊕ 8 ⊕ 10 ⊕ 27

8 ⊗ 8 = 27 ⊕ 8s ⊕ 1 ⊕ ¯ 10 ⊕ 10 ⊕ 8a

1) octet-octet system 2) decuplet-octet system 3) decuplet-decuplet system H-dibaryon(J=0)

Jaffe (1977)

ΩΩ system (J=0) ΔΔ system (J=3)

Zhang et al(1997)

Deuteron(J=1)

Introduction: SU(3) classification for Dibaryon candidates (B=2)

NΩ system and NΔ system (J=2)

Dyson, Xuong (1964) Kamae, Fujita(1977) Oka, Yazaki(1980)

Goldman et al (1987)

Dyson, Xuong (1964)

SLIDE 6

6

10 ⊗ 10 = 28 ⊕ 27 ⊕ 35 ⊕ ¯ 10

10 ⊗ 8 = 35 ⊕ 8 ⊕ 10 ⊕ 27

8 ⊗ 8 = 27 ⊕ 8s ⊕ 1 ⊕ ¯ 10 ⊕ 10 ⊕ 8a

1) octet-octet system 2) decuplet-octet system 3) decuplet-decuplet system H-dibaryon(J=0)

Jaffe (1977)

ΩΩ system (J=0) ΔΔ system (J=3)

Zhang et al(1997)

Deuteron(J=1)

Introduction: SU(3) classification for Dibaryon candidates (B=2)

NΩ system and NΔ system (J=2)

Dyson, Xuong (1964) Kamae, Fujita(1977) Oka, Yazaki(1980)

Goldman et al (1987)

Dyson, Xuong (1964)

d*(2380) resonance by Kamae et al, 1975 WASA@COSY, 2009

SLIDE 7

p + n(d) → d + π0 + π0(+pspectator)

7

d*(2380) resonance

WASA@COSY, PRL 106, 242302 (2011)

ΔΔ contributions d* resonance m~2.38 GeV Γ~70 MeV d* (2380) observed by WASA@COSY col. m~ 2.38 GeV, Γ ~ 70 MeV, Jπ = 3+, I=0

SLIDE 8

The potential is extracted from this equation

Baryon-Baryon interaction from lattice QCD

• HAL method-

Ψn (~ r) e−Ent = X

~ x

h0| B1(t,~ r + ~ x)B2(t, ~ x) |Eni

Nambu-Bethe-Salpeter (NBS) w.f.

8

c.f. anothor method: Luscher’s direct method

• ~

p2

n + r2

Ψn (~ r) = 2µ Z d~ r0U(~ r, ~ r0)Ψn(~ r0) Schroedinger type equation is satisfied Local operators B1&B2 for decuplet baryons Dµα = ✏abc

• qaT Cµqb

qc

α

Aoki, Hatsuda, Ishii, PTP123, 89 (2010)

V( ⃗ r)

non-local pot.

B1, B2 →

SLIDE 9

• I. ΔΔ system with J=3

9

SLIDE 10

Nf = 2+1 full QCD with L = 1.93fm, SU(3) limit (CP-PACS Conf)

10

[MeV] mps 1015 moct 2030 mdec 2220 p+ + π+

3045MeV

Δ

2220MeV

ΔΔ p+p+ π+ π+ ΔΔ p+p+π+π+

CP-PACS

• phys. pt.

d*: resonance d*: bound state

Δ p+ + π+

• phys. pt.

Δ: resonance Δ: bound state

CP-PACS

SLIDE 11

10 plet in decuplet-decuplet system

ΔΔ in Jp(I) =3+(0)

• Nf = 2+1 full QCD with L = 1.93fm, mπ=1015MeV, SU(3) limit
• In short range, there is no repulsive core
• Deep bound state is found

SG and K. Sasaki

m∆ ' 2225MeV

11

a≒1fm, r≒0.5fm

preliminary preliminary

• We assume that


decay to NN(3D3) is neglected

d* is supported from lattice QCD

SLIDE 12

• II. ΩΩ system

12

SLIDE 13

Numerical Setup at (almost) physical mass

2+1 flavor gauge configurations

• Iwasaki gauge action & O(a) improved Wilson quark action
• a= 0.0846 [fm], a-1 = 2333 [MeV]
• 963x96 lattice, L = 8.1[fm]
• 400 confs x 48 source positions x 4 rotations

Wall source is employed. only S-wave state is produced.

[MeV] phys. π 146 8% K 525 6% N 964 3% Ω 1712 2%

13

K computer

SLIDE 14

ΩΩ in J =0

3)Nf=2+1 full QCD with L = 8.1fm, mπ= 146MeV

• Short range repulsive core and attractive pocket are found
• Phase shift shows the presence of a bound state
• The state is very close to the unitary region (r/a<1)
• (1S0)

(lattice) NN (spin-triplet) (experiment) NN (spin-singlet) (experiment)

a(ΩΩ) = 4.6(6)(+1.2

−0.5)

fm, r(ΩΩ)

eff

= 1.27(3)(+0.06

−0.03) fm.

14

“most strange dibaryon”

SG, K.Sasaki + (HAL QCD Coll.), PRL 2018

SLIDE 15

ΩΩ in J =0

Binding energy and the Coulomb effect

• “most strange dibaryon”

H = r2 mΩ + V LQCD

ΩΩ

(r) + α r

H = r2 mΩ + V LQCD

ΩΩ

(r)

(B(QCD)

ΩΩ

, B(QCD+Coulomb)

ΩΩ

) = (1.6(6)MeV, 0.7(5)MeV)

15

SG and K. Sasaki et.al.(HAL), PRL(2018)

Q=-1

SLIDE 16

Conservative estimate at exact phys. pt.

• mπ=146 MeV -> 135 MeV, mΩ= 1712MeV -> 1672 MeV

attractive pockets 
 becomes deeper

η, 2K, 2π (σ)

(B(QCD)

ΩΩ

, B(QCD+Coulomb)

ΩΩ

) = (1.6(6)MeV, 0.7(5)MeV) → (1.3(5)MeV, 0.5(5)MeV)

These changes are within errors

• nly change the mass of kinetic term

H = r2 mΩ + V LQCD

ΩΩ

(r)

kinetic energy is increasing


• > B.E. is reduced

V.S.

B.E. ↑ B.E. ↓ conservative estimate:

16

SLIDE 17

Summary in first part

• heavy pion masses:


ΔΔ interaction in 7S3


• shows only attractive region

• bound state in J=3 channel (=d* resonance)
• physical pion masses:


ΩΩ interaction in 1S0


• short range repulsive and attractive pocket

• a very shallow bound state [di-Omega]

+ di-Omega (bound) found in future HIC ?
 (LHC RUN3/FAIR/J-PARC)

Dibaryon (B=2)

Deuteron(1930s) + d*(2380) resonance

di-Omega

<= supported <= predicted

SLIDE 18

Recent development of partial wave decomposition 


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

SLIDE 19

Origin of comb-like behavior

19

comb

• If higher partial wave components were negligible, 


the wave function and its potential should have been isotropic. 
 
 The comb-like behavior = higher partial wave contributions comb-like behavior

• ccurs at some points 


with |xi|=r which cannot be connected via cubic rotation.
 ex) (1,2,2), (3,0,0)

SLIDE 20

➡Continuum space O(3,R): Spherical surface integration

R O

The discrete points with the distance R

➡ Discrete space O(3,Z): A1+ projection


Cubic rotation average + Parity average

ψNBS( ⃗ x )

S-wave projectionψL=0

NBS(

⃗ x )

ψ A+

1 (

⃗ x ) ≡ PA+

1 ψ(

⃗ x ) = 1 48 ∑

g∈Oh

ψ(g−1 ⃗ x ) ψL=0(R) = ∫S dΩ Y*

00(θ, ϕ)ψ(

⃗ x ; r = R) A1+ representation includes also l ≥ 4 Using the different values, we can extract each component from A1+ projected NBS wave function. At 2 points s.t. 
 which cannot be connected via cubic rotation

x1, x2 |x1| = |x2| = R

A+

1 (~

x1) 6= A+

1 (~

x2)

SLIDE 21

Naive treatment: 
 Decomposition at fixed r

21

After A1+ projection ψ A+

1 (

⃗ x ) ≡ PA+

1 ψ(

⃗ x ) = YA+

1

00 (θ, ϕ)g00(r) + ∑ m=0,±4

YA+

1

4m(θ, ϕ)g4m(r) + ⋯,

R O

Suppose components are neglected.
 At , the eq. is written as

l ≥ 6

x1, x2

YA+

1

00 (x, y, z) = Y00(x, y, z) =

1 4π , YA+

1

40 (x, y, z) =

7 8 π x4 + y4 + z4 − 3(x2y2 + y2z2 + z2x2) r4 , YA+

1

4,+4(x, y, z) = YA+

1

4,−4(x, y, z) =

5 14 YA+

1

40 (x, y, z)

g4(r) ≡ g40(r) + 5 14 (g44(r) + g4−4(r))

( ψ A+

1 (

⃗ x 1) ψ A+

1 (

⃗ x 2)) = ( YA+

1

00 YA+

1

40 (

⃗ x 1) YA+

1

00 YA+

1

40 (

⃗ x 2)) ( g00(R) g4(R))

g00(R), g4(R) are obtained

SLIDE 22

Naive treatment: 
 Decomposition at fixed r

22

R O

In general case: spherical functions up to

   A+

1 (~

x1) . . . A+

1 (~

xN)    =     Y

A+

1

00

Y

A+

1

40 (~

x1) Y

A+

1

60 (~

x1) · · · . . . . . . . . . . . . Y

A+

1

00

Y

A+

1

40 (~

xN) Y

A+

1

60 (~

xN) · · ·          g00(R) g4(R) g6(R) . . .     

• Using SVD, the components are extracted from N points
• At least # points (N) ≧ # spherical functions (n)
• #points (N) at fixed r is not large.

Consider N points s.t.

|x1| = |x2| = ⋯ = |xN| = R

gl

Yn0

SLIDE 23

Misner’s method in continuum space

23

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

To overcome this problem, we utilize points inside a spherical shell.

SR,Δ

R Δ

Let us first consider continuum space.

GR,∆

n

(r) ≡ Pn ✓r − R ∆ ◆ 1 r r 2n + 1 2∆

Legendre polynomial

Z R+∆

R−∆

dr r2 GR,∆

n

(r)GR,∆

m

(r) = δnm

SR,∆ = {~ x | R − ∆ ≤ |~ x| ≤ R + ∆}

YR,∆

nlm (r, θ, φ) ≡ GR,∆ n

(r)Ylm(θ, φ)

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A complete orthonormal set of functions 


• n the shell

Z

SR,∆

d3x YR,∆

nlm (θ, φ)YR,∆ n0l0m0(r, θ, φ) = δnn0δll0δmm0

SLIDE 24

Misner’s method in continuum space

24

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

SR,Δ

R Δ

SR,∆ = {~ x | R − ∆ ≤ |~ x| ≤ R + ∆}

Inside the shell, the wave function is expanded by is determined by the integration over the shell:

YR,∆

nlm (r, θ, φ) ≡ GR,∆ n

(r)Ylm(θ, φ)

The components of the partial wave 
 inside the shell are obtained by

ψ(r, θ, φ) =

∞

X

n=0 ∞

X

l=0 l

X

m=−l

cR,∆

nlm YR,∆ nlm (r, θ, φ)

=

∞

X

l=0 l

X

m=−l

glm(r)Ylm(θ, φ)

cR,∆

nlm

cR,∆

nlm =

Z

SR,∆

d3x YR,∆

nlm (r, θ, φ) ψ(r, θ, φ)

glm(r) =

∞

X

n=0

cR,∆

nlm GR,∆ n

(r)

SLIDE 25

Misner’s method in discrete space

25

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

The volume integration is replaced by

• verlap region between the shell

and a lattice cube

R Δ

⃗ x ω( ⃗ x )

Z

SR,∆

d3x = ⇒ X

~ x

!R,∆(~ x)

An approximate choice of the weight function

!R,∆(~ x) = 8 < : a3 a2 ∆ + 1

2a − |R − r|

• for |r − R| < ∆ − 1

2a

for |r − R| > ∆ + 1

2a

• therwise

The inner product on lattice

hf|giSR,∆ ⌘ X

~ x

!R,∆(~ x) f(~ x) g(~ x)

SLIDE 26

Because of finite points, orthonormality is broken:

hYR,∆

nlm |YR,∆ n0l0m0iSR,∆ 6= δn,n0δl,l0δm,m0

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

Misner’s method in discrete space

Dual basis

A = n, l, m, B = n0, l0, m0

˜ YR,∆

A

(~ x) ≡ X

B

YR,∆

B

(~ x) G1

BA

hYR,∆

A

|YR,∆

B

iSR,∆ = GAB

h ˜ YR,∆

A

|YR,∆

B

iSR,∆ = X

C

G1

AChYR,∆ C

|YR,∆

B

iSR,∆ = X

C

G1

ACGCB = δAB

X

B

≡

nmax

X

n=0 lmax

X

l=0 l

X

m=l

This satisfies orthonormality for l ≦ lmax, n ≦ nmax: To get , the restriction of summation (lmax, nmax) is introduced.

G−1

BA

SLIDE 27

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

Misner’s method in discrete space

Suppose that the components higher than lmax, nmax are negligibly small:

(~ x) '

nmax

X

n=0 lmax

X

l=0 l

X

m=−l

cR,∆

nlm YR,∆ nlm (r, ✓, )

are obtained from

cR,∆

nlm = h ˜

YR,∆

nlm |ψiSR,∆.

cR,∆

nlm

Components of partial wave expansion in the shell are

glm(r) '

nmax

X

n=0

cR,∆

nlmGR,∆ n

(r), R ∆ < r < R + ∆

~ r2glm(r) =

nmax

X

n=0

cR,∆

nlm

1 r @2 @r2 ⇥ rGR,∆

n

(r) ⇤

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Using this form, Laplacian can be calculated analytically

GR,∆

n

(r) ≡ Pn ✓r − R ∆ ◆ 1 r r 2n + 1 2∆

SLIDE 28

Misner’s method vs fixed-r method

Zero-shell limit (fixed-r limit) for Misner method

YR,∆

nlm (r, θ, φ) ≡ GR,∆ n

(r)Ylm(θ, φ)

GAA0 ⌘ hYR,∆

A

|YR,∆

A0

iSR,∆ ! Glm,l0m0 ⌘ hYlm|Yl0m0i

A = n, l, m

hf|giSR,∆ = X

~ x

!R,∆(~ x) f(~ x) g(~ x) ! hf|gi|~

x|=R =

X

|~ x|=R

f(~ x) g(~ x)

˜ YR,∆

A

(~ x) → ˜ Ylm(✓, ) ≡ X

l0,m0

Yl0m0(✓, ) G1

l0m0,lm

glm = h ˜ Ylm|ψi|~

x|=R

Dual basis

   A+

1 (~

x1) . . . A+

1 (~

xN)    =     Y

A+

1

00

Y

A+

1

40 (~

x1) Y

A+

1

60 (~

x1) · · · . . . . . . . . . . . . Y

A+

1

00

Y

A+

1

40 (~

xN) Y

A+

1

60 (~

xN) · · ·          g00(R) g4(R) g6(R) . . .     

Fixed-r method

SLIDE 29

Misner’s method vs fixed-r method

Zero-shell limit (fixed-r limit) for Misner method

YR,∆

nlm (r, θ, φ) ≡ GR,∆ n

(r)Ylm(θ, φ)

GAA0 ⌘ hYR,∆

A

|YR,∆

A0

iSR,∆ ! Glm,l0m0 ⌘ hYlm|Yl0m0i

A = n, l, m

hf|giSR,∆ = X

~ x

!R,∆(~ x) f(~ x) g(~ x) ! hf|gi|~

x|=R =

X

|~ x|=R

f(~ x) g(~ x)

˜ YR,∆

A

(~ x) → ˜ Ylm(✓, ) ≡ X

l0,m0

Yl0m0(✓, ) G1

l0m0,lm

glm = h ˜ Ylm|ψi|~

x|=R

Dual basis

   A+

1 (~

x1) . . . A+

1 (~

xN)    =     Y

A+

1

00

Y

A+

1

40 (~

x1) Y

A+

1

60 (~

x1) · · · . . . . . . . . . . . . Y

A+

1

00

Y

A+

1

40 (~

xN) Y

A+

1

60 (~

xN) · · ·          g00(R) g4(R) g6(R) . . .     

Fixed-r method

Misner’s method= extension of fixed-r method to include points inside shell

SLIDE 30

test calculation 1: check the decomposition

30

ψ4(r) ≡ sin (r/3) r

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ψ6(r) ≡ sin (r/2) r

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ψ0(r) ≡ 2 − e− r2

60

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(~ r) ≡ 0(r)Y0,0(~ r) + ↵ 4(r)Y4,0(~ r) + 6(r)Y6,0(~ r)

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(α = 0.2, β = 0.1)

g00(r) g20(r) g40(r) g60(r)

Ex) We apply Misner’s method with Δ=a, 
 nmax=2, lmax=6 to this wave function.

black: test data red: reconstruction data

All components were reproduced by Misner’s method

SLIDE 31

test calculation 2: solve Hamiltonian

31

V = a exp [− x2 + y2 + z2 b2 ] + c exp [− x2 + y2 + z2 d2 ]

a = 0.3, b = 2, c = − 0.3, d = 7

Model potential

• 1. [H0 + V(r)] ψ A+

1

n

= Eψ A+

1

n

3.V(r) = [E − H0] ψn,l ψn,l

2.ψ A+

1

n

→ ψn,l(l = 0,4,...)

1st 2nd 8th l=0 l=4 l=6 l=0 l=4 l=6 l=6 l=4 l=0 wave functions

SLIDE 32

test calculation 2: solve Hamiltonian

32

V = a exp [− x2 + y2 + z2 b2 ] + c exp [− x2 + y2 + z2 d2 ]

a = 0.3, b = 2, c = − 0.3, d = 7

Model potential

• 1. [H0 + V(r)] ψ A+

1

n

= Eψ A+

1

n

3.V(r) = [E − H0] ψn,l ψn,l

2.ψ A+

1

n

→ ψn,l(l = 0,4,...)

8th l=0 l=4 l=0 l=4 l=0 L=0 L=4 L=6 this small difference may come 
 from less #pts for smaller r

Even l≧4 components reproduce model potential

reconstruction

SLIDE 33

Application to NBS wave functions

33

ΛcN（1S0）

• l≧4 contributions for A1+ projected R-correlator can be found
• The comb-like behavior is removed by Misner’s method

Comb-like behavior

Δ=a, 
 nmax=2, lmax=4

SLIDE 34

Application to Laplacian term

34

~ r2glm(r) =

nmax

X

n=0

cR,∆

nlm

1 r @2 @r2 ⇥ rGR,∆

n

(r) ⇤

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• Conventional method: 


Laplacian=> a finite second-order difference

=> comb-like fluctuation due to l≧4 is enhanced

• Misner method:


Laplacian=> analytically calculable after l=0 extraction
 => The fluctuation is removed ΛcN（1S0）

SLIDE 35

Application to HAL potential

35

• Conventional method: 


Enhancement of the fluctuation of laplacian due to l≧4 contributions
 => Potential has large fluctuation (comb-like behavior)

• Misner method:


The fluctuation is removed because of l=0 extraction. ΛcN（1S0）

SLIDE 36

Fit results

36

Fit to pot. from A1 proj. ≒ Fit to pot. from Misner method

The fluctuation is not affected the fit results largely. ΛcN（1S0）

SLIDE 37

Fit results

37

The phase shifts are identical with each other.

ΛcN（1S0）

SLIDE 38

Future works

38

Use Misner method to extract 
 higher partial waves

Many systems couple to higher partial waves

We have succeeded in L=0 extraction Summary