The Weak Production of Hypernuclei

D.D. van Niekerk (M.Sc. project)B.I.S. van der VentelG.C. Hillhouse

Department of PhysicsStellenbosch UniversitySouth Africa

Stellenbosch, South Africa

Outline

Motivation

Our Model

Formalism

The Hadronic Vertex

Kinematics

The Transition Matrix

Leptonic Tensor

Hadronic Tensor

Constructing Wµ (Example)

Conclusion

Motivation Recent large scale interest in astrophysics and the role of neutrinos in

stellar processes (i.e. supernovae)

Neutrino osscillations (changing of flavour) BooNE / MiniBooNE (Fermilab)

J-PARC Super-Kamiokande (50 GeV)

Nucleon decay postulated by supersymmetry

Hyperon and hypernuclei production form important part of neutrino-

induced reaction cross sections

0

p K

n K

Our Model Based on relativistic Dirac equation

never been studied (nuclear process)

first attempt in a fully relativistic framework

Quasifree process (interaction takes place between neutrino and

single bound nucleon)

Bound state wave functions are calculated using relativistic mean

field formalism

Aim:

Obtain quantitive results that will give indication of nuclear

model uncertainties

Provide theoretical basis for interpretation of experimental

results

Types of Reactions: Charged Current (CC)

(S = strangeness)

ΔS = 0

ΔS = 1

Neutral Current

ΔS = 0

ΔS = 1 not observed

0n K

n K n

0 0n K

Y12 12ν + C + K + C

Formalism

Neutral Current (NC) Charged Current (CC)

1K( )p

( )P

( )P

( )X q

( )k

( ')k

0X Z X W

Modelling the Hadronic Vertex

Quasifree Region

Use form factors ( )p

( )X q

1meson ( )p

( )P

( )P

( )X q

1meson ( )pboundhyperon

boundnucleon

2( )p

( )p

Vertex

Approximation

K is kinematic factor determined from normalisation of flux etc.

First order diagram:

Lμν contains projectile information

Wμν contains nuclear information

(4) 3 3 3 21 1( ' ' ') ' ' ' | |d K k P k p P d p d P d k M

2 μνμν| M | L W

Kinematics CC

In CC reactions we can detect the outgoing muon.

1

1'1

32 2 2' '4

' 1'

2( )

i roots 'of ( )

1( )

(cos ') 2 2

1| |

| '( ) |p

k p kk

ip

f E

dE E M

dE d d

Mf E

Kinematics NC

In NC reactions we cannot detect the outgoing neutrino.

' max

' min

1

1'1

( ) 12

' '41' ( ) 1

2 2 2' ( )

i roots 'of ( )

1(cos ')

2 2

1( ) | |

| '( ) |

.k

k

p

E

k k

E

p k ip

f E

ddE d E

d

E M Mf E

Transition Matrix Element

Leptonic Current Parity not conserved

Left-handed neutrinos

Propagator Vector Boson (W+ or Z0)

Coupling strengths follow from GSW Theory (ηl and ηh)

5(1 )l

( ') ( ) ( )l f h iiM u k l k iD q K J

2boson

gD

M

2 2| | FM G L W

Leptonic Tensor Lepton spinor

normalised as

helicity representation

Neutrino: m = 0 and h = -1

Feynman trace techniques and identities of the gamma

matrices can be used to simplify the expression for Lμν

2

h

k

kh

k

kE m

u k h kE kE m

† 1u u

5 5( ') (1 ) ( ) ( ') (1 ) ( ) *

' ' ' '

S A

L u k k u k k

N k k k k k k g i k k

L L

Hadronic Tensor

The hadronic tensor is expanded in a basis consisting of the

independent four-momenta, the metric tensor and the Levi-Civita tensor

(

μν μν μ ν μ ν μ ν μ ν μ ν1 2 3 4 5

μ ν μ ν μ ν μ ν6 7

μ ν μ ν μ ν μ ν8 9

μ ν μ ν μναβ μναβ μναβ10 11 α β 12 α β 13 α β

μν μνS A

W =W g +W q q +W P P +W p' p' +W q P + P q

+W q p' + p' q +W P p' + p' P

+W q P - P q +W q p' - p' q

+W P p' - p' P +W ε q P +W ε q p' +W ε P p'

=W +W

( )

( ) )

( ) ( )

( )

1meson ( )p

( )P

( )P

( )X q

This expansion is model independent

The Wi expansion coefficients are the structure functions

Extract Wi:g W W

q q W X W

1

2

11

2

W g W

W X q q W

done once

The contraction of hadronic and leptonic tensors is done considering symmetric and anti-symmetric contractions separately

General equation Model is needed for guidance

2 2

21 3

| | ( ) ( )

' ( ') ( )

F S A

F

M G L W L W

G W k k W

Construction of hµ

Born Term Model (s,t and u channels)

Propagators: spin ½

spin 0

Vertices: Strong coupling (baryon-baryon-meson) in s,t,u channels

Coupling constant

Weak coupling (meson-meson) in t channel Phenomenological meson form factorsMecklenberg W., Acta Physica Austriaca 48, 293 (1976)

Weak coupling (baryon-baryon) in s,u channel Form factors

Weak Current Operator

μ μ μν μWK 1 2 ν A 5

iJ = f γ + f σ q - g γ γ

2m

p m

p m

2 2

2 2

1

p - m

*,N Ym m

W h h

+1K ( )p

12 ( )C P

12* ( )C P

( )X q

0K

nW

0K

K

n

0

W

W

np

0

K

s

t

u

12 - + 12ν + C μ + K + C

d np

- 0 +ν + n μ + Λ + KElementary process:

s-channel

neutron-proton vertex

1μ μ μνCC V 1 2 ν +2

ip J + n = p j + ij n = u' γ + σ q τ u

2mf f( )

Form Factors

0n

K W

p

- 0 +ν + n μ + Λ + K

CVC relates weak vector form factors to isovector form factors

of EM current

EM isovector current

Axial form factor determined phenomenologically

(p) (nμ μ) (p) (n)1 1 2 2

μν 3EM ν

τiN' J N = u' γ + σ q u

2mF - F F -

2F

(p) (n)1 1 1

(p) (n)2 2 2

f = F - F

f = F - F

-22Dipole

A,CC A A2A

qg = g 1 - = 1.26 G

m

Total (for s-channel)

s d cosμ1 2 A

μ μν μn Λ C ,Cν C 5f f

ih = p θ γ + σ q - γ

2m

p×

g γ

+ - 5 n2 2 K p

+ mG γ

p -m

(p) (n)1 1 1

(p) (n)2 2 2

DipoleA,CC A

f = F - F

f = F - F

g = 1.26 G

u-channel

Vertex:

Weak current i.t.o. SU(3) octet currents

μ μ μ μ μ μ μ μ μCC 1 2 1 2 c 4 5 4 5 c

ΔS=0 ΔS=1

J = V ± iV - A ± iA θ + V ± iV - A ± iA θ ( ) ( ) cos ( ) ( ) sin

0ˆ ( ) ?CCJ

n

0 K

W

- 0 +ν + n μ + Λ + K

where

and λi = 3X3 generators of SU(3) γμ = 4X4 Dirac matrices

μ μ μ μ μ μ μ μ μCC 1 2 1 2 c 4 5 4 5 c

ΔS=0 ΔS=1

J = V ± iV - A ± iA θ + V ± iV - A ± iA θ ( ) ( ) cos ( ) ( ) sin

u x

q x = d x

s x

( )

( ) ( )

( )

μ μi i

μ μi i 5

iV = q x λ γ q x

2i

A = q x λ γ γ q x2

( ) ( )

( ) ( )

EM current

For Oj any octet current operator

For EM current

Comparison yields

μ μ μEM 3 8

1J =V + V

3

( ) ( ) ( ) ( )i j k ijk i k ijk i kB O B = if U B FU B + d U B DU B

ˆ

ˆ

D D1

μ μ μνEM ν

μ μ μνE

2

D DF

M νF1 2

1 2

2 in J n = - D = u' γ + σ q u

3 2m

D ip J p

2- F F3

F FF + F +

3= F + = u' γ + σ

3q u

3 2m

D (n)1 1

D (n)2 2

3F = - F

23

F = - F2

F (p) (n)1 1 1

F (p) (n)2 2 2

1F = F + F

21

F = F + F2

For weak current

Belongs to same octet as EM current

Axial form factor From s-channel

F μ F μν F μ1 2 ν 5

D μ D μν D μ1 2 ν 5

iF = f γ + f σ q - g γ γ

2mi

D = f γ + f σ q - g γ γ2m

D,F D,F1,2 1,2f = F

ˆ F D1 1 A,C

μ μ μCC A C5 5p J + n = u' γ γ u = u' γ γg + g g u( )

(0) (0)D F D F Dipole1 1 1 1 Ag + g = g + g G

u-channel weak baryon-baryon vertex:

propagator:

strong baryon-baryon-meson vertex:

ˆ ( )

1

20 μ - D μ D μν D μ

CC Λ 1 2 ν 1 5 Σ

2 iΛ J + Σ = f γ + f σ q - g γ γ

3 2m

+ - 5K nΣG γ

p2 2

+ m

p -m

- 0 +ν + n μ + Λ + K

n

0 K

W

Total (for u-channel)

u d cos D

1

2μ μ μν μ

n Λ C ν 5D D

1 2 1

2 ih = p θ γf f g+ σ q - γ γ

3 2m

p

+ - 5 n2 2 K nΣ

+ mG γ

p -m

D (n)1 1

D (n)2 2

D D1 1 A,CC

3f = - F

23

f = - F2

g = g (0)g

Summary(4) 3 3 3 2

1 1( ' ' ') ' ' ' | |d K k P k p P d p d P d k M

2 2

21 3

| | ( ) ( )

' ( ') ( )

F S A

F

M G L W L W

G W k k W

11

2

W g W

W X q q W

s t u

*

+h h h h

W h h

Conclusion We are constructing a relativistic model for the

description of weak hypernuclei production of relevance

to experiments at Fermilab (BooNE) and J-PARC

Hadronic tensor parametrised in model independent way

to facilitate different hadronic models through structure

functions

Code written in Fortran 95 and Mathematica. In process

of obtaining results: We are investigating the relation between the structure

functions Wi and the kaon scattering angle as well as

dependence of Wi on the momentum transfer

Calculate the cross sectionemail: ddvniekerk@sun.ac.za