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 to

disparities in particle lifetimes, discovered in 1968 by SLAC.

2

Strange quark (S)

Mass (MeV) 95

Spin 12

Isospin 0

Charge (e) −13

Strangeness -1

Decay mode s ⟶ u + W∗−

Introduction

• Hyperon: Baryon with

non-zero strangeness

• Unstable with respect

to the weak interaction

• Hypernuclei: Hyperon-

nucleus bound system

• Strangeness physics: Study of YN and YY interactions in order

to obtain unified knowledge of BB interaction in SU(3)F

• Main handicap: Limited knowledge from YN and YY scattering

data 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 5960.97

Binding 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 known Nagara events, the KISO event shows

evidence of a deeply bound state of Ξ− − N14 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 the

baryons 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 1

2

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

Charge 0 -1 0 +1 -1 0

Mean life (10-10s) 2.632 1.479 7.4 × 10−10 0.8018 1.639 2.90

Introduction

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

7

Introduction

Free space

Medium

Consideration of the medium lowers the thresholds but does not

change its order

8

Introduction

Both Λ baryons will be found in the 1𝑠1

2 state, and due to the

wavefunction antisymmetry will couple to 𝑆0

1

𝐿 = 𝑙Λ⨂𝑙Λ = 0⨂0 = 0

𝑆 = 12⨂

12 = 0⨁1

ΛΛ 𝑆 ΛΛ

Ξ𝑁ΣΣ

𝑊 Λ𝑛Σ𝑁

𝑆 Λ𝑛Σ𝑁

Λ𝑛 𝑆 Λ𝑛

Σ𝑁

𝑊 𝑁1𝑁2

𝑆 𝑁1

′𝑁2′

𝑆0 𝑆0

1

𝑃03

1 𝑆0 𝑆0

1

𝑃03

1

𝑆1

𝑆13

𝐷13

𝑃13 , 𝑃1

1

3

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 model What does the weak two-body reaction look like? How do we

study it?

Strong correlations

Weak vertex Strong vertex

One-Meson-Exchange model

10

Strong correlations

The strong initial mixing may be treated with the G-matrix

equation, which includes the Pauli blocking operator

𝑉 𝜓 𝑐𝑜𝑟𝑟 = 𝐺 𝜙 𝑢𝑛𝑐𝑜𝑟𝑟⟶ 𝐺 = 𝑉 + 𝑉

𝑄

𝐸𝐺

The relevant matrices for the initial correlation are

𝐺ΛΛ→ΛΛ 𝐺ΛΛ→ΣΣ 𝐺ΛΛ→Ξ𝑁

𝐺ΣΣ→ΛΛ 𝐺ΣΣ→ΣΣ 𝐺ΣΣ→Ξ𝑁

𝐺Ξ𝑁→ΛΛ 𝐺Ξ𝑁→ΣΣ 𝐺Ξ𝑁→Ξ𝑁𝑇=0

𝐺ΣΣ→ΣΣ 𝐺ΣΣ→Ξ𝑁

𝐺Ξ𝑁→ΣΣ 𝐺Ξ𝑁→Ξ𝑁 𝑇=1

For the final states a T-matrix equation formalism is required

𝑉 𝜓 𝑐𝑜𝑟𝑟 = 𝑇 𝜙 𝑢𝑛𝑐𝑜𝑟𝑟⟶ 𝑇 = 𝑉 + 𝑉

1

𝐸𝑇

Similarly the T-matrices for the final states

𝑇Λ𝑁→Λ𝑁 𝑇Λ𝑁→Σ𝑁

𝑇Σ𝑁→Λ𝑁 𝑇Σ𝑁→Σ𝑁 𝑇=12

𝑇Σ𝑁→Σ𝑁 𝑇=32

11

Strong correlations

Pauli blocking operator

14

Strong correlations S0

1

15

Strong correlations Final state ΛN Final state ΣN

One-Meson-Exchange model

Strong correlations

Weak vertex Strong vertex

16

Strong correlations

ℒ𝑆 = Tr 𝐵 𝑖𝛾𝜇𝛻 𝐵 − 𝑀𝐵Tr 𝐵 𝐵 + 𝐷𝛾𝜇𝛾5Tr 𝐵 𝑢𝜇, 𝐵 + 𝐹𝛾𝜇𝛾5Tr 𝐵 𝑢𝜇 , 𝐵

For pseudoscalar mesons, formalism by Callan, Coleman, Wes i

Zumino (Phys. Rev. 177 1969)

Covariant derivative introduced to contemplate gauge invariance

SU(3) constants fitted to experiments

𝛻𝜇𝐵 = 𝜕𝜇𝐵 + Γ𝜇 , 𝐵

17

Weak transition: Strong vertices

𝐵 =

Σ0

2+

Λ

6Σ+ 𝑝

Σ− −Σ0

2+

Λ

6𝑛

Ξ− Ξ0 −2Λ

6

𝜙 =

𝜋0

2+

𝜂

6𝜋+ 𝐾+

𝜋− −𝜋0

2+

𝜂

6𝐾0

𝐾− 𝐾0 −2𝜂

6

𝑢𝜇 = 𝑖𝑢†𝜕𝜇𝑈𝑢† 𝑈 = 𝑢2 = 𝑒𝑖

2𝜙𝑓 Γ𝜇 =

1

2𝑢†𝜕𝜇𝑢 + 𝑢𝜕𝜇𝑢

†

Where:

The interaction Lagrangian for vector mesons may be obtained

from the generalization of Hidden Local Symmetry in SU(2) to the

SU(3) sector (Phys. Rev. Lett. 54 1985)

ℒ𝑉𝑆 = −𝑔 𝐵 𝛾𝜇 𝑉8

𝜇, 𝐵 +

1

4𝑀 𝐹 𝐵 𝜎𝜇𝜈 𝜕𝜇𝑉8

𝜈 − 𝜕𝜈𝑉8𝜇, 𝐵 + 𝐵 𝛾𝜇𝐵 𝑉8

𝜇

+ 𝐷 𝐵 𝜎𝜇𝜈 𝜕𝜇𝑉8𝜈 − 𝜕𝜈𝑉8

𝜇, 𝐵 + 𝐵 𝛾𝜇𝐵 𝑉0

𝜇+

𝐶0

4𝑀𝐵 𝜎𝜇𝜈𝑉0

𝜇𝜈𝐵

18

Weak transition: Strong vertices

𝐵 =

Σ0

2+

Λ

6Σ+ 𝑝

Σ− −Σ0

2+

Λ

6𝑛

Ξ− Ξ0 −2Λ

6

𝑉𝜇 =1

2

𝜌0 + 𝜔 2𝜌+ 2𝐾∗+

2𝜌− −𝜌0 + 𝜔 2𝐾∗0

2𝐾∗− 2𝐾∗0 2𝜙

Baryon matrix: Vector meson matrix:

19

Strong vertices: couplings 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

One-Meson-Exchange model

Strong correlations

Weak vertex Strong vertex

20

Strong correlations

Through a lowest-order chiral analysis two non-derivative

Lagrangians can be formulated for pseudoscalar mesons:

A parity and charge conjugation transformation implies that:

The expression for ℒ(𝑃)𝑊 presents a sign change: Only parity-

violating amplitudes can be obtained through a chiral

Lagrangian.

With ℎ =0 0 00 0 10 0 0

and 𝜉 ≈ 1 +𝑖

2𝑓𝜙

ℒ(𝑆)𝑊 = 𝐺𝐹𝑚𝜋

2 2𝑓𝜋 ℎ𝐷Tr 𝐵 𝜉†ℎ𝜉, 𝐵 + ℎ𝐹Tr 𝐵 𝜉†ℎ𝜉, 𝐵

ℒ(𝑃)𝑊 = 𝐺𝐹𝑚𝜋

2 2𝑓𝜋 ℎ𝐷𝛾5Tr 𝐵 𝜉†ℎ𝜉, 𝐵 + ℎ𝐹𝛾5Tr 𝐵 𝜉†ℎ𝜉, 𝐵

𝐵 → 𝑖𝛾2𝐵 𝑇 , 𝜉 → 𝜉† 𝑇

21

Weak transition: Weak vertices (PV)

The inclusion of vector mesons requires the use of the SU(6)W

group, which describes the product of the SU(3) flavour group

with the SU(2)W spin group (L. De la Torre, Ph. D. Thesis)

𝐵𝑎𝑏𝑐 ≡1

6 𝑆𝑎(1)𝑆𝑏(2)𝑆𝑐(3)𝑝𝑒𝑟𝑚𝑎,𝑏,𝑐

𝜙𝑏𝑎 = 휀𝑞𝑏𝑞 𝑎 with

휀 = 1 𝑎, 𝑏 even휀 = −1 otherwise

22

The final general expression accounting for weak baryon-

baryon-vector meson couplings is:

Weak vertices (PV)

23

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

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

𝐵′𝑀𝑖(𝑞) 𝐻𝑊 𝐵 ~ 𝛿 𝑝 𝑛 − 𝑝 𝐵′ − 𝑞 𝐵′ 𝐴𝑖

𝜇𝑛 𝑛 𝐻𝑊 𝐵

𝑝𝐵0 − 𝑝𝑛

0

𝑛

+ 𝛿 𝑝 𝐵 − 𝑝 𝑛′ − 𝑞 𝐵′ 𝐴𝑖

𝜇𝑛′ 𝑛′ 𝐻𝑊 𝐵

𝑝𝐵0 − 𝑞0 − 𝑝𝑛

0

𝑛

lim𝑞→0

𝐵𝑃𝑉

𝐵′𝑀𝑖 = −𝑖

𝐹𝜋𝐵′ 𝐹𝑖 , 𝐻6 𝐵 = 𝐴𝐵𝐵′

Weak transition shifted over to the baryonic line

Using the soft-meson reduction theorem:

Starting from the transition amplitude for a non-leptonic meson

emission and introducing a complete set of states

Generator associated to the emitted

meson 24

Weak vertices (PC): Pole Model

Possible meson-pole contributions should in theory be taken into

account

• Uncertainty in the relative phase between baryon and meson

contributions

• Becomes even more model-dependent

• Of low importance in comparison with baryon-pole terms 25

Weak vertices (PC): Pole Model

27

Weak vertices: PC couplings Weak parity-conserving baryon-baryon-pseudoscalar meson

couplings

D = 1.18 MeV; F = 0.7 MeV

28

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

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

𝑡ΛΛ→𝑌𝑁 ≈ 𝑑3𝑟Ψ𝑘∗𝜒𝑀𝑆

†𝑆 𝜒𝑀𝑇

†𝑇𝑉(𝑟 )Φ𝑁𝑟𝐿𝑟𝑟𝑒𝑙 𝑟

2𝑏𝜒𝑀𝑆0

𝑆0 𝜒𝑀𝑇0

𝑇0

How do we obtain the decay rate using the elements we

currently have?

Decay rate

𝑉 𝑞 = 𝐶𝑆𝜎 1𝑞 𝑝𝛼

𝑞 2 + 𝑚𝛼2 𝐶𝑊

𝑃𝑉 + 𝐶𝑊𝑃𝐶𝜎 2𝑞

𝑀𝑓𝑖 ∝ 𝑘1𝑚1𝑘2𝑚2; Ψ𝑅𝐴−2 𝑂ΛΛ→𝑌𝑁 𝑍ΛΛ

𝐴

Γ𝑛(𝑝) = 𝑑𝑝 1

(2𝜋)3

𝑑𝑝 2(2𝜋)3

2𝜋𝛿 𝑀𝐻 − 𝐸𝑅 − 𝐸1 − 𝐸2

1

2𝐽 + 1𝑀𝑓𝑖

2

𝑀𝑖{𝑅}

1 {2}

34

Fourier

Transform

i

q2 - mK

2=

i

q0

2 - q2 - mK

2

n.r .¾ ®¾ -

i

q2 + mK

2

Thus the potential may be written as:

V(q) = -CX-K+L

s1q1

q2 + mK

2fpGFmp

2 1

fhD + hF( ) + 3F +

D

3

æ

èçö

ø÷3 -1

mn - mL

ALp

2+ F - D( )

3 -1

mn - mL

AS

+ p

2

æ

èç

ö

ø÷

s 2q

2 f

é

ëêê

ù

ûúú

Derivation of the interaction potential through an example:

GpK-n

W = - fpGFmp

2 1

fhD + hF( ) - 3F +

D

3

æ

èçö

ø÷3 -1

mn - mL

ALp

2+ F - D( )

3 -1

mn - mL

AS+ p

2

æ

èç

ö

ø÷

s 2q

2 f

GX-K+L

S = CX-K+L

E + MLX-

2E

s1

E + MLX-

k + k '( )æ

èç

ö

ø÷ q0 - s1 +

s1k '( )s1 s1k( )E + M

LX-( )2

æ

è

çç

ö

ø

÷÷

q

é

ë

êê

ù

û

úú

GX-K+L

S = -CX-K+L

s1q

Decay rate

35

1. Introduction

2. One-Meson-Exchange Model

• Strong correlations

• Weak transition

Strong vertices

Weak vertices

3. Decay rate

4. Results

5. Summary

37

Results The expression for the decay rate is Γ = ΓΛ𝑁→𝑁𝑁 + ΓΛΛ→Λ𝑁 + ΓΛΛ→Σ𝑁

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, ΓΛ𝑁→𝑁𝑁 𝐻𝑒ΛΛ6 = 0.95ΓΛ

(0)≈ 2Γ 𝐻𝑒Λ

5

ΓΛΛ→𝑌𝑁

For the ΛN → NN channel we have:

38

Results (Preliminary) In the case of the ΛΛ → YN with the ΛΛ-ΛΛ diagonal coupled

channel:

The decay rate in this situation is ΓΛΛ→𝑌𝑁 𝐻𝑒ΛΛ6 = 2.96 × 10−2ΓΛ

(0),

or roughly 3% of ΓΛ𝑁→𝑁𝑁 𝐻𝑒ΛΛ6

In units of ΓΛ = 3.8 x 109

ΛΛ → Λn ΛΛ → ΣN

Σ0𝑛 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

39

Results (Preliminary) Lastly, for the ΛΛ → YN including diagonal and non-diagonal

channels (ΛΛ-ΛΛ, ΛΛ-ΞN):

The decay rate with the additional channel taken into account

is ΓΛΛ→𝑌𝑁 𝐻𝑒ΛΛ6 = 3.4 × 10−2ΓΛ

(0), or roughly 3.5% of ΓΛ𝑁→𝑁𝑁 𝐻𝑒ΛΛ

6

In units of ΓΛ = 3.8 x 109

ΛΛ → Λn ΛΛ → ΣN

Σ0𝑛 Σ−𝑝

𝛑 1.28 x 10-4 2.35 x 10-3 4.7 x 10-3

K 2.12 x 10-2 3.08 x 10-4 6.16 x 10-4

𝛈 8.66 x 10-4 2.78 x 10-7 5.57 x 10-7

𝛒 3.93 x 10-6 1.88 x 10-6 3.76 x 10-6

K* 3.84 x 10-3 7.67 x 10-5 1.53 x 10-4

𝛚 6.16 x 10-5 3.44 x 10-8 6.88 x 10-8

𝛑 + 𝐊 2.10 x 10-2 2.09 x 10-3 4.18 x 10-3

All 2.85 x 10-2 1.91 x 10-3 3.82 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

41

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 through

the G and T-matrix formalism.

• Use of effective Lagrangian for the derivation of coupling

constants in the strong and weak sectors (SU(6)W for vector,

pole model for PC couplings).

• Presented results showing that the ΛΛ-ΛΛ channel amounts for

the 3% of the total decay rate

• Presented results showing that the ΛΛ-ΛΛ, ΛΛ-ΞN channels

amount for the 3.5% of the total decay rate

• Work in progress for the rest (ΛΛ-41ΣΣ) of the coupled channels

42

Thank you for your attention