Hypernuclear decay of strangeness -2 hypernuclei Jordi Maneu In collaboration with Assumpta Parreño and Àngels Ramos Quantum Physics and Astrophysics Department
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