hypernucleardecayof strangeness-2...

Hypernuclear decay of strangeness -2 hypernuclei

Jordi Maneu

In collaboration with Assumpta Parreño and Àngels Ramos

Quantum Physics and Astrophysics Department

1. Introduction

2. One-Meson-Exchange Model

• Strong correlations

• Weak transition

§ Strong vertices

§ Weak vertices

3. Decay rate

4. Results

5. Summary

What are strange quarks?

+ ?

Type of elementary particle, existence theorized in 1964 due todisparities in particle lifetimes, discovered in 1968 by SLAC.

2

Strange quark (S)

Mass (MeV) 95

Spin 12#

Isospin 0

Charge (e) −1 3#

Strangeness -1

Decay mode s ⟶ u +W∗,

Introduction

• Hyperon: Baryon withnon-zero strangeness

• Unstable with respectto the weak interaction

• Parity and strangenessare not conserved

• Hypernuclei: Hyperon-nucleus bound system

• Strangeness physics: Study of YN and YY interactions in orderto obtain unified knowledge of BB interaction in SU(3)F

• Main handicap: Limited knowledge from YN and YY scatteringdata

3

Introduction

4

Why should we care about hypernuclei?• ∆S useful for separating PV and PC components• Glue-like role in nuclei.

Introduction

𝐇𝐞𝟒 𝐇𝐞𝚲𝟓 𝐇𝐞𝚲𝚲

𝟔

Mass (MeV) 3738.93 4852.08 5960.97Binding energy (MeV) 16.7485 19.2721 26.0639

• Nuclear radius shrinkage in the presence of hyperons (Int. J. Mod. Phys. E22 2013)

• Hyperons in the core of neutron stars?• Apart from well established Nagara event ( He77

8 ), the KISO eventshows evidence of a deeply bound state of Ξ, − N;< system

(J. Phys.: Conf. Ser. 668 2016) (Phys. Rev. C59 1999)

5

PANDA at FAIR• Anti-proton beam• Λ Λ -hypernuclei• γ-ray spectroscopy• W hypernuclei

KAOS @ MAMI • Electro-production• Λ-hypernuclei

Jlab• Electro-production• Λ-hypernuclei FINUDA at DAFNE

• e+e- collider• Stopped-K- reaction• Λ- and Σ-hypernuclei• γ-ray spectroscopy J-PARC

• High-intensity proton beam• Λ and Λ Λ – hypernuclei• γ-ray spectroscopy for Λ- hypernuclei• Σ and X hypernuclei

HypHI at GSI• Heavy ion beams• Λ-hypernuclei at extreme isospin• Magnetic moments

SPHERE at JINR• Heavy ion beams• Single Λ-hypernuclei

COSY @ Jülich• proton beam• ΛN interaction

KEK• Stopped K- reactions• Λ and Λ Λ - hypernuclei

STAR/PHENIX@RHIC• HI colider• anti Λ-hypernuclei• exotic states

ALICE @ LHC • HI colider• anti Λ-hypernuclei• exotic states

Introduction

Updated from J. Pochodzalla, Int. Journal Modern Physics E, Vol 16, no. 3 (2007) 925-936

The hyperons involved in the hypernuclear decay are thebaryons from the octet, excluding the proton and the neutron

6

𝚲 𝚺, 𝚺𝟎 𝚺? 𝚵, 𝚵𝟎

Mass (Mev) 1115.683 1197.449 1192.642 1189.37 1321.71 1314.86

Composition uds dds uds uus dss uss

Isospin 0 1 1 1 12#

12#

Strangeness -1 -1 -1 -1 -2 -2

Charge 0 -1 0 +1 -1 0

Mean life (10-10s) 2.632 1.479 7.4×10,;F 0.8018 1.639 2.90

Introduction

Strangeness -2 means possible initial states are ΛΛ, ΞN and ΣΣ

7

Introduction

2000

2100

2200

2300

2400

2500

ΛΛ ΞN ΣΣ

Thre

shol

d (M

eV)

Free space

Medium

Consideration of the medium lowers the thresholds but does notchange its order

8

Introduction

Both Λ baryons will be found in the1𝑠H I⁄ state, and due to thewavefunction antisymmetry willcouple to 𝑆F;

𝐿 = 𝑙7⨂𝑙7 = 0⨂0 = 0

𝑆 = ;P⨂

;P = 0⨁1

1s1/2

ΛΛT ΛΛ

Ξ𝑁ΣΣ

W Λ𝑛Σ𝑁

T Λ𝑛Σ𝑁Λ𝑛

T Λ𝑛Σ𝑁

W𝑁;𝑁P

T𝑁;Y𝑁PY

𝑆F 𝑆F;

𝑃F[; 𝑆F

𝑆F;

𝑃F[;

𝑆;

𝑆;[ 𝐷;[ 𝑃;[ , 𝑃;;

[

1. Introduction

2. One-Meson-Exchange Model

• Strong correlations

• Weak transition

§ Strong vertices

§ Weak vertices

3. Decay rate

4. Results

5. Summary

One-Meson-Exchange modelWhat does the weak two-body reaction look like? How do westudy it?

� �

B1

Y

Y’

B2

N

„

N

S W

Strong correlations

Weak vertexStrong vertex

One-Meson-Exchange model

10

Strong correlations

� �� �

The strong initial mixing may be treated with the G-matrixequation, which includes the Pauli blocking operator

𝑉|𝜓⟩bcdd = 𝐺|𝜙⟩ghbcdd ⟶ 𝐺 = 𝑉 + 𝑉𝑄𝐸 𝐺

The relevant matrices for the initial correlation are𝐺77→77 𝐺77→ll 𝐺77→mn𝐺ll→77 𝐺ll→ll 𝐺ll→mn𝐺mn→77 𝐺mn→ll 𝐺mn→mn opF

𝐺ll→ll 𝐺ll→mn𝐺mn→ll 𝐺mn→mn op;

11

Strong correlations

Pauli blocking operator

G=

V+

V

Gq

For the final states a T-matrix equation formalism is required𝑉|𝜓⟩bcdd = 𝑇|𝜙⟩ghbcdd ⟶ 𝑇 = 𝑉 + 𝑉

1𝐸 𝑇

Similarly the T-matrices for the final states𝑇7n→7n 𝑇7n→ln𝑇ln→7n 𝑇ln→ln op; P#

𝑇ln→ln op[ P#

12

Strong correlations

T=

V+

V

T

15

Strong correlationsSF;

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10r (fm)

Initial state ΛΛHarmonic oscillator

From GΛΛ - ΛΛFrom GΛΛ - ΞNFrom GΛΛ - ΣΣ

-0.5

0.0

0.5

1.0

1.5 1S0-1S0 j0ΛN - ΛNΣN - ΛN

1S0-1S0 j0ΣN - ΣNΛN - ΣN

-0.5

0.0

0.5

1.0

1.5

1 2 3 4 5r (fm)

3P0-3P0 j1ΛN - ΛNΣN - ΛN

1 2 3 4 5r (fm)

3P0-3P0 j1ΣN - ΣNΛN - ΣN

16

Strong correlationsFinal state ΛN Final state ΣN

Weak transition: Strong vertices

� �

B1

Y

Y’

B2

N

„

N

S W

Strong correlations

Weak vertexStrong vertex

17

Strong correlations

One-Meson-Exchange model

ℒT = Tr 𝐵v 𝑖𝛾y𝛻 𝐵 − 𝑀|Tr 𝐵v𝐵 + 𝐷𝛾y𝛾}Tr 𝐵v 𝑢y, 𝐵 + 𝐹𝛾y𝛾}Tr 𝐵v 𝑢y, 𝐵

For pseudoscalar mesons, formalism by Callan, Coleman, Wes iZumino (Phys. Rev. 177 1969)

Covariant derivative introduced to contemplate gauge invariance

SU(3) constants fitted to experiments

𝛻y𝐵 = 𝜕y𝐵 + Γy, 𝐵

18

Weak transition: Strong vertices

𝐵 =

ΣF

2�+Λ6�

Σ? 𝑝

Σ, −ΣF

2�+Λ6�

𝑛

Ξ, ΞF −2Λ6�

𝜙 =

𝜋F

2�+𝜂6�

𝜋? 𝐾?

𝜋, −𝜋F

2�+𝜂6�

𝐾F

𝐾, 𝐾F −2𝜂6�

𝑢y = 𝑖𝑢�𝜕y𝑈𝑢� 𝑈 = 𝑢P = 𝑒�P� �� Γy =

12 𝑢�𝜕y𝑢 + 𝑢𝜕y𝑢�

Where:

The interaction Lagrangian for vector mesons may be obtainedfrom the generalization of Hidden Local Symmetry in SU(2) to theSU(3) sector (Phys. Rev. Lett. 54 1985)

ℒ�T = −𝑔 � 𝐵v𝛾y 𝑉�y, 𝐵 +

14𝑀 �𝐹 𝐵v𝜎y� 𝜕y𝑉�� − 𝜕�𝑉�

y, 𝐵 + 𝐵v𝛾y𝐵 𝑉�y

+ 𝐷 𝐵v𝜎y� 𝜕y𝑉�� − 𝜕�𝑉�y, 𝐵 � + 𝐵v𝛾y𝐵 𝑉F

y +𝐶F4𝑀 𝐵v𝜎y�𝑉F

y�𝐵 �

19

Weak transition: Strong vertices

𝐵 =

ΣF

2�+Λ6�

Σ? 𝑝

Σ, −ΣF

2�+Λ6�

𝑛

Ξ, ΞF −2Λ6�

𝑉y =12

𝜌F + 𝜔 2� 𝜌? 2� 𝐾∗?

2� 𝜌, −𝜌F + 𝜔 2� 𝐾∗F

2� 𝐾∗, 2� 𝐾∗F 2� 𝜙

Baryon matrix: Vector meson matrix:

20

Strong vertices: couplings

Pseudoscalar Vector

Coupling Analytic value N. value Coupling Analytic value N. value Coupling Analytic value N. value

p⇡�n D + F 1.88 ⌅�K⇤+⇤T �g(D�3F )

8p3M

-0.004 p!8pTg(D+F )

8p3M

-0.232

⌅�K+⇤ � 1p2

⇣Dp3�p3F

⌘0.37 ⌅�K⇤+⇤V �g

p3

2 0.866 p!8pV �gp3

2 0.866⌅�K+⌃0 1p

2(D + F ) 1.33 p⇢�nT

g(D+F )

4p2M

-0.569 n!8nTg(D+F )

8p3M

-0.232p⇡0p 1p

2(D + F ) 1.33 p⇢�nV � gp

20.707 n!8nV �g

p3

2 0.866

p⌘p � 1p2

⇣Dp3�p3F

⌘0.38 ⌅�K⇤+⌃0

Tg(D+F )

8M -0.403 ⌅0K⇤0⇤T �g(D�3F )

8p3M

-0.004⌅�K0⌃� D + F 1.88 ⌅�K⇤+⌃0

V �g2 0.5 ⌅0K⇤0⇤V �g

2 0.5n⇡0n � 1p

2(D + F ) -1.33 ⌅�K⇤0⌃�

Tg(D+F )

4p2M

-0.569 ⌅0K⇤0⌃0T �g(D+F )

8M 0.403

n⌘n � 1p2

⇣Dp3�p3F

⌘-1.34 ⌅�K⇤0⌃�

V � gp2

0.707 ⌅0K⇤0⌃0V �g

6 0.167

⌅0K0⇤ 1p2

⇣Dp3+p3F

⌘-0.376 p⇢0pT

g(D+F )8M -0.403 n⇢+pT

g(D+F )

4p2M

-0.569⌅0K0⌃0 � 1p

2(D + F ) -1.33 p⇢0pV �g

2 0.5 n⇢+pV � gp2

0.707n⇢0nT �g(D+F )

8M 0.403 nK⇤�⌃�T

g(D�F )

4p2M

-0.279n⇢0nV

g2 -0.5 nK⇤�⌃�

Vgp2

-0.707

Strong baryon-baryon-meson couplings

D = 1.18 MeV; F = 0.7 MeV D = 2.4 MeV; F = 0.82 MeV

Pseudoscalar mesons Vector mesons

Different fits for pseudoscalar and vector mesons result in different constants.

Weak transition: Weak vertices

� �

B1

Y

Y’

B2

N

„

N

S W

Strong correlations

Weak vertexStrong vertex

21

Strong correlations

One-Meson-Exchange model

• PV amplitudes• PC amplitudes

Through a lowest-order chiral analysis two non-derivativeLagrangians can be formulated for pseudoscalar mesons:

With ℎ =0 0 00 0 10 0 0

and 𝜉 ≈ 1 + �P� �𝜙

ℒ��W = 𝐺�𝑚 P 2� 𝑓  ℎ¢Tr 𝐵v 𝜉�ℎ𝜉, 𝐵 + ℎ�Tr 𝐵v 𝜉�ℎ𝜉, 𝐵

ℒ�£W = 𝐺�𝑚 P 2� 𝑓  ℎ¢𝛾}Tr 𝐵v 𝜉�ℎ𝜉, 𝐵 + ℎ�𝛾}Tr 𝐵v 𝜉�ℎ𝜉, 𝐵

22

Weak transition: Weak vertices (PV)

No CP!

The inclusion of vector mesons requires the use of the SU(6)Wgroup, which describes the product of the SU(3) flavour groupwith the SU(2)W spin group (L. De la Torre, Ph. D. Thesis)

𝐵¤¥b ≡16 § 𝑆¤(1)𝑆¥(2)𝑆b(3)

�

ª«d¬¤,¥,b

𝜙¥¤ = 𝜀𝑞¥𝑞v¤with ³𝜀 = 1 𝑎, 𝑏even𝜀 = −1 otherwise

23

Weak vertices: PV couplingsWeak parity-violating baryon-baryon-meson couplings

Pseudoscalar Vector

Coupling Analytic value N. value Coupling Analytic value N. value

pK�np2(hD + hF ) 0.58 pK⇤�n 1

6p2(bT � bV ) +

59p2cV -3.05

⌅�⇡+⇤⇣

hDp3�p3hF

⌘-1.86 ⌅�⇢+⇤ 1

6p3

��1

2bT + 12bV + cV

�0.08

⌅�⇡+⌃0 �(hD + hF ) -0.41 ⌅�⇢+⌃0 136 (�3bT � 3bV + 10cV ) -1.26

⌅�⇡0⌃� hD + hF 0.41 pK⇤0pp26

�12bV � 1

3cV�

0.99⌅�⌘⌃� �

p3(hD + hF ) -0.71 ⌅�⇢0⌃� � 5

18cV 1.26pK0p

p2 (hD � hF ) -1.99 ⌅�!8⌃� 5

6p3cV -2.19

nK0n 2p2hF 2.57 nK⇤0n

p23

��1

4bV + 23cV

�-2.06

⌅0⇡0⇤ 1p6(hD � hF ) -0.576 ⌅0!8⌃0 1

4p6(�bT � bV ) +

56p6cV -1.55

⌅0⇡0⌃0 1p2(hD + hF ) 0.290 ⌅0⇢0⌃0 � 1

4p2bT + 1

12p2bV � 5

18p2cV -0.37

⌅0⌘⇤ � 1p2(hD � hF ) 0.997 ⌅0!8⇤

112

p2(�bT + bV � 2cV ) -0.097

⌅0⌘⌃0 3p6(hD + hF ) 0.502 ⌅0⇢0⇤ 1

12p6(�3bT � bV + 2cV ) -1.04

⌅0K⇤+p 16p2(�bT + bV ) -1.27

⌅0⇢+⌃� 16p2(�bT + bV ) -1.27

hD = -0.5 MeV; hF = 0.91 MeV bV = -8.36 x 10-7; bT = 8.36 x 10-7; cV = 7.08 x 10-7

Pseudoscalar mesons Vector mesons

lim»→F

𝐵��𝐵′𝑀� = −

𝑖𝐹 

𝐵′ 𝐹�, 𝐻8 𝐵 = 𝐴||¿

Essentially the weak transition is shifted over to the baryonic line

Using the soft-meson reduction theorem:

Generator associated to the emitted meson

24

Weak vertices (PC): Pole Model

28

Weak vertices: PC couplingsWeak parity-conserving baryon-baryon-pseudoscalar meson couplingsCoupling Analytic value Numeric value

pK�n �12

⇣p3F + Dp

3

⌘ p3�1

mn�m⇤

2fA⇤pp2

� 12(F �D)

p3�1

mn�m⌃0

2fA⌃+pp2

1.23

⌅�⇡+⇤ Dp3

p3�1

m⌅��m⌃�2fA⌃+p +

2fp2(D � F )

p3+1

m⇤�m⌅0

12p2

�A⇤p �

p3A⌃+p

��0.34

⌅�⇡+⌃0 Fp3�1

m⌅��m⌃�2fA⌃+p +

1p2(D � F )

p3�1

m⌃0�m⌅0

2f2p2

�A⇤p +

p3A⌃+p

��1.36

⌅�⇡0⌃� �12(F �D)

p3�1

m⌃��m⌅�2fA⌃+p � F

p3�1

m⌅��m⌃�2fA⌃+p 2.09

⌅�⌘⌃� �12

⇣Dp3+p3F

⌘ p3�1

m⌃��m⌅�2fA⌃+p +

Dp3

p3�1

m⌅��m⌃�2fA⌃+p �3.62

pK0p � 1p2(F �D) 1�

p3

mp�m⌃+2fA⌃+p �0.37

nK0n �12

⇣Dp3+p3F

⌘ p3�1

mn�m⇤

2fA⇤pp2

� 12 (D � F )

p3�1

mn�m⌃0

2fA⌃+pp2

0.86

⌅0⇡0⌃0 �12 (D � F ) 1

m⌃0�m⌅0

�fp2

�A⇤p +

p3A⌃+p

�� Dp

31

m⌅0�m⇤

fp2

�A⇤p �

p3A⌃+p

�0.77

⌅0⌘⌃0⇣�1

6 (D + 3F ) 1m⌃0�m⌅0

+ Dp3

1m⌅0�m⇤

⌘�fp2

�A⇤p +

p3A⌃+p

��0.57

⌅0⇡0⇤ �12 (D � F ) 1

m⇤�m⌅0

fp2

�A⇤p �

p3A⌃+p

�� Dp

31

m⌅0�m⌃0

�fp2

�A⇤p +

p3A⌃+p

�0.08

⌅0⌘⇤⇣�1

6 (D + 3F ) 1m⇤�m⌅0

� Dp3

1m⌅0�m⇤

⌘fp2

�A⇤p �

p3A⌃+p

�0.18

D = 1.18 MeV; F = 0.7 MeV

29

Weak vertices: PC couplingsWeak parity-conserving baryon-baryon-vector meson couplings

Coupling Analytic value Numeric value

pK⇤�nT � (D�3F )

8p3

p3�1

mn�m⇤

2fA⇤pp2

+ (D�F )8

p3�1

mn�m⌃0

2fA⌃+pp2

0.54

pK⇤�nV

p32

p3�1

mn�m⇤

2fA⇤pp2

+ 12

p3�1

mn�m⌃0

2fA⌃+pp2

�0.57

⌅�⇢+⇤TD

4p3

p3�1

m⌅��m⌃�2fA⌃+p � (D�F )

4p2

p3+1

m⇤�m⌅0

2f2p2

�A⇤p �

p3A⌃+p

�0.20

⌅�⇢+⇤V � 1p2

p3+1

m⇤�m⌅0

2f2p2

�A⇤p �

p3A⌃+p

�2.45

⌅�⇢+⌃0T �F

4

p3�1

m⌅��m⌃�2fA⌃+p +

(D�F )

4p2

p3�1

m⌃0�m⌅0

2f2p2

�A⇤p +

p3A⌃+p

�0.62

⌅�⇢+⌃0V �

p3�1

m⌅��m⌃�2fA⌃+p � 1p

2

p3�1

m⌃0�m⌅0

2f2p2

�A⇤p +

p3A⌃+p

�2.64

⌅�⇢0⌃�T

(D�F )8

p3�1

m⌃��m⌅�2fA⌃+p +

F4

p3�1

m⌅��m⌃�2fA⌃+p �0.90

⌅�⇢0⌃�V

12

p3�1

m⌃��m⌅�2fA⌃+p +

p3�1

m⌅��m⌃�2fA⌃+p 1.11

⌅�!⌃�T

(D�F )

8p3

p3�1

m⌃��m⌅�2fA⌃+p +

D4p3

p3�1

m⌅��m⌃�2fA⌃+p �0.51

⌅�!⌃�V � 1

2p3

p3�1

m⌃��m⌅�2fA⌃+p � 1p

3

p3�1

m⌅��m⌃�2fA⌃+p 0.64

pK⇤0pT(D�F )

4p2

1�p3

mp�m⌃+2fA⌃+p �0.31

pK⇤0pV1p2

1�p3

mp�m⌃+2fA⌃+p �0.78

nK⇤0nT � (D+3F )

8p3

p3�1

mn�m⇤

2fA⇤pp2

� (D�F )8

p3�1

mn�m⌃0

2fA⌃+pp2

0.23

nK⇤0nV

p32

p3�1

mn�m⇤

2fA⇤pp2

� 12

p3�1

mn�m⌃0

2fA⌃+pp2

�1.34

⌅0K⇤+pTF�D

41

m⌅0�m⌃0

�fp2

�A⌃+p +

p3A⇤p

�+ �(D+F )

4p2

1mp�m⌃+

A⌃+p +D+3F8p3

1m⌅0�m⇤

fp2

�A⇤p �

p3A⌃+p

�0.693

⌅0K⇤+pV �12

1m⌅0�m⌃0

�fp2

�A⌃+p +

p3A⇤p

�+ 1p

21

mp�m⌃+A⌃+p �

p32

1m⌅0�m⇤

fp2

�A⇤p �

p3A⌃+p

��0.866

⌅0⇢�⌃�T �F

41

m⌅0�m⌃0

�fp2

�A⌃+p +

p3A⇤p

�+ D�F

4p2

1m⌃��m⌅�

2fA⌃+p � D4p3

1m⌅0�m⇤

fp2

�A⇤p �

p3A⌃+p

�0.276

⌅0⇢�⌃�V �1

21

m⌅0�m⌃0

�fp2

�A⌃+p +

p3A⇤p

�+ 1p

21

m⌃��m⌅�2fA⌃+p 1.28

⌅0!⌃0T

⇣F�D

81

m⌃0�m⌅0+ D

41

m⌅0�m⌃0

⌘fp2

�A⌃+p +

p3A⇤p

�0.46

⌅0!⌃0V

⇣�1

21

m⌃0�m⌅0+ 1

21

m⌅0�m⌃0

⌘�fp2

�A⌃+p +

p3A⇤p

�-0.58

⌅0⇢0⌃0T

D�F8

1m⌃0�m⌅0

fp2

�A⌃+p +

p3A⇤p

�� D

4p3

1m⌅0�m⇤

fp2

�A⇤p �

p3A⌃+p

�-0.58

⌅0⇢0⌃0V

12

1m⌃0�m⌅0

fp2

�A⌃+p +

p3A⇤p

�-0.29

⌅0!⇤T

⇣D�F

81

m⇤�m⌅0� D

121

m⌅0�m⇤

⌘fp2

�A⇤p �

p3A⌃+p

�-0.53

⌅0!⇤V

⇣�1

21

m⇤�m⌅0+ 1

m⌅0�m⇤

⌘fp2

�A⇤p �

p3A⌃+p

�2.00

⌅0⇢0⇤TF�D

81

m⇤�m⌅0

fp2

�A⇤p �

p3A⌃+p

�+ �D

4p3

1m⌅0�m⌃0

�fp2

�A⌃+p +

p3A⇤p

�0.46

⌅0⇢0⇤V �12

1m⇤�m⌅0

fp2

�A⇤p �

p3A⌃+p

�0.67

D = 2.4 MeV;F = 0.82 MeV

1. Introduction

2. One-Meson-Exchange Model

• Strong correlations

• Weak transition

§ Strong vertices

§ Weak vertices

3. Decay rate

4. Results

5. Summary

𝑡77→Án ≈ Â𝑑[𝑟ΨÆ∗𝜒ÈÉ

�T 𝜒ÈÊ�Ë 𝑉(𝑟)ΦnÎÏÎ

d«Ð d⃗P� ¥

𝜒ÈÉÑ

TÑ 𝜒ÈÊÑ

ËÑ�

�

How do we obtain the decay rate using the elements wecurrently have?

Decay rate

𝑉 �⃗� = 𝐶T�⃗�;�⃗�𝑝Ò

�⃗�P + 𝑚ÒP 𝐶W�� + 𝐶W�£�⃗�P�⃗�

𝑀�� ∝ 𝑘;𝑚;𝑘P𝑚P;ΨÖ×,P 𝑂77→Án 𝑍77×

Γh(ª) = Â𝑑�⃗�;(2𝜋)[

�

�

Â𝑑�⃗�P(2𝜋)[ § 2𝜋𝛿 𝑀Û − 𝐸Ö − 𝐸; − 𝐸P

12𝐽 + 1 𝑀��

P�

ÈÝ{Ö}; {P}

�

�

35

FourierTransform

iq2 −mK

2 = iq02 − q2 −mK

2n.r.⎯ →⎯ − i

q2 +mK2

Thus the potential may be written as:

V (q) = −CΞ−K +Λ

σ1q 1q2 +mK

2 fπGFmπ2 1fhD + hF( )+ 3F + D

3⎛⎝⎜

⎞⎠⎟

3 −1mn −mΛ

AΛp

2+ F −D( ) 3 −1

mn −mΛ

AΣ+p

2⎛

⎝⎜⎞

⎠⎟

σ 2q

2 f⎡

⎣⎢⎢

⎤

⎦⎥⎥

Derivation of the interaction potential through an example:

ΓpK −nW = − fπGFmπ

2 1fhD + hF( )− 3F + D

3⎛⎝⎜

⎞⎠⎟

3 −1mn −mΛ

AΛp

2+ F −D( ) 3 −1

mn −mΛ

AΣ+p

2⎛

⎝⎜⎞

⎠⎟

σ 2q

2 f

ΓΞ−K+Λ

S =CΞ−K+Λ

E '+MΛ( ) E +MΞ−( )4EE '

!σ1!k

E +MΞ−

+!σ1!k '

E '+MΛ

%

&''

(

)**q0 −

!σ1 +

!σ1!k '( ) !σ1 !σ1

!k( )

E '+MΛ( ) E +MΞ−( )

%

&

''

(

)

**!q

+

,

--

.

/

00

ΓΞ−K +ΛS = C

Ξ−K +Λ

E +MΛΞ−

2E

σ1

E +MΛΞ−

k +k '( )⎛

⎝⎜⎞

⎠⎟q0 −

σ1 +

σ1

k '( ) σ1

σ1

k( )

E +MΛΞ−( )2

⎛

⎝⎜⎜

⎞

⎠⎟⎟q

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ΓΞ−K +ΛS = −C

Ξ−K +Λ

σ1q

Decay rate

36

1. Introduction

2. One-Meson-Exchange Model

• Strong correlations

• Weak transition

§ Strong vertices

§ Weak vertices

3. Decay rate

4. Results

5. Summary

38

ResultsThe expression for the decay rate is Γ = Γ7n→nn + Γ77→7n + Γ77→ln

In units of ΓΛ = 3.8 x 109

ΛN → NN

nn np

𝛑 9.72 x 10-2 1.07

K 8.10 x 10-2 0.463

𝛈 3.84 x 10-3 7.46 x 10-3

𝛒 7.25 x 10-3 2.46 x 10-2

K* 3.77 x 10-3 2.48 x 10-2

𝛚 1.08 x 10-3 9.40 x 10-4

𝛑 + 𝐊 2.36 x 10-1 4.7 x 10-1

All 2.97 x 10-1 6.49 x 10-1

The total decay rate, Γ7n→nn 𝐻𝑒778 = 0.95Γ7

(F) ≈ 2Γ 𝐻𝑒7}

Γ77→ÁnFor the ΛN → NN channel we have:

39

ResultsIn the case of the ΛΛ → YN with the ΛΛ-ΛΛ diagonal coupledchannel:

The decay rate in this situation is Γ77→Án 𝐻𝑒778 = 2.96×10,PΓ7

(F), orroughly 3% of Γ7n→nn 𝐻𝑒77

8

In units of ΓΛ = 3.8 x 109

ΛΛ → ΛnΛΛ → ΣN

ΣF𝑛 Σ,𝑝

𝛑 1.16 x 10-4 2.19 x 10-3 4.38 x 10-3

K 1.79 x 10-2 2.64 x 10-4 5.28 x 10-4

𝛈 7.18 x 10-4 2.05 x 10-7 4.10 x 10-7

𝛒 1.03 x 10-5 7.31 x 10-7 1.46 x 10-6

K* 3.42 x 10-3 1.41 x 10-6 2.82 x 10-6

𝛚 3.61 x 10-5 3.71 x 10-8 7.42 x 10-8

𝛑 + 𝐊 1.74 x 10-2 1.95 x 10-3 3.91 x 10-3

All 2.42 x 10-2 1.80 x 10-3 3.60 x 10-3

40

ResultsLastly, for the ΛΛ → YN including diagonal and non-diagonalchannels (ΛΛ-ΛΛ, ΛΛ-ΞN):

The decay rate with the additional channel taken into accountis Γ77→Án 𝐻𝑒77

8 = 2.34×10,PΓ7(F), a reduction from 3% to 2.3% of

Γ7n→nn 𝐻𝑒778

In units of ΓΛ = 3.8 x 109

ΛΛ → ΛnΛΛ → ΣN

ΣF𝑛 Σ,𝑝

𝛑 2.63 x 10-4 2.11 x 10-3 4.22 x 10-3

K 1.75 x 10-2 3.03 x 10-4 6.06 x 10-4

𝛑 + 𝐊 1.55 x 10-2 1.98 x 10-3 3.96 x 10-3

All 1.80 x 10-2 1.82 x 10-3 3.63 x 10-3

1. Introduction

2. One-Meson-Exchange Model

• Strong correlations

• Weak transition

§ Strong vertices

§ Weak vertices

3. Decay rate

4. Results

5. Summary

43

Summary• Calculation of the hypernuclear decay rate due to growing

interest in high strangeness systems

• Calculation of previously un-derived coupling constants

• Initial and final strong interaction taken into account throughthe G and T-matrix formalism.

• Use of effective Lagrangian for the derivation of couplingconstants in the strong and weak sectors (SU(6)W for vector,pole model for PC couplings).

• Presented results showing that the inclusion of strong BB mixing(ΛΛ-ΛΛ, ΛΛ-ΞN) reduces the ΛΛ-YN rate from 3% to 2.3%

• To be compared with forthcoming experimental results fromthe J-PARC facility, hopefully accurate enough to distinguishthis variation.

44

Thank you for your attention