Parity Conservation in the weak (beta decay) interaction

The parity operation

The parity operation involves the transformation

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x → −x y → −y z → −z

r → r θ → π −θ( ) φ → φ + π( )

In rectangular coordinates --

In spherical polar coordinates --

In quantum mechanics

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ˆ Π ψ x,y,z( ) = ±1ψ −x,−y,−z( )

For states of definite (unique & constant) parity -

If the parity operator commutes with hamiltonian -

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ˆ Π , ˆ H [ ] = 0 ⇒ The parity is a “constant of the motion”

Stationary states must be states of constant parity

e.g., ground state of 2H is s (l=0) + small d (l=2)

In quantum mechanics

To test parity conservation - - Devise an experiment that could be done: (a) In one configuration (b) In a parity “reflected” configuration- If both experiments give the “same” results, parity is conserved -- it is a good symmetry.

Parity operations --

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ˆ Π E = E ˆ Π r r ⋅

r r ( ) =

r r ⋅

r r ( )

Parity operation on a scaler quantity -

Parity operation on a polar vector quantity -

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ˆ Π r r = −

r r ˆ Π

r p = −

r p

Parity operation on a axial vector quantity -

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ˆ Π r L = ˆ Π

r r ×

r p ( ) = −

r r × −

r p ( ) = −

r L

Parity operation on a pseudoscaler quantity -

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ˆ Π r p ⋅

r L ( ) = −

r p ⋅

r L ( )

If parity is a good symmetry…• The decay should be the same whether the process

is parity-reflected or not.

• In the hamiltonian, V must not contain terms that are pseudoscaler.

• If a pseudoscaler dependence is observed - parity symmetry is violated in that process - parity is therefore not conserved.

T.D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956).

http://publish.aps.org/ puzzle

Parity experiments (Lee & Yang)

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rp ⋅

r I = 0

Parity reflectedOriginal

Look at the angular distribution of decay particle (e.g., red particle). If this is symmetric above/below the mid-plane, then --

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rp

€

rI

€

rp

€

rI

If parity is a good symmetry…

• The the decay intensity should not depend on .

• If there is a dependence on and parity is not conserved in beta decay.

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rp ⋅

r I = 0

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rp ⋅

r I ( )

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rp ⋅

r I ≠ 0

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rp ⋅

r I ( )

Discovery of parity non-conservation (Wu, et al.)

Consider the decay of 60Co

Conclusion: G-T, allowed

C. S. Wu, et al., Phys. Rev 105, 1413 (1957)

http://publish.aps.org/

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Hβ − = −

v

c

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ν

€

Hν =1

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β−

€

ν €

Hβ − = −

v

c

€

β−

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Hν =1

€

ν

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Hν = −1

€

ν

€

Hν = −1

Not observedMeasure

Trecoil

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GT : ΔI =1

A :1−1

3

v

ccosθ

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F : I = 0 → I = 0

V :1+v

ccosθ

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Hν =1

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Hβ − = −

v

c

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GT : ΔI =1

A :1−1

3

v

ccosθ

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F : I = 0 → I = 0

V :1+v

ccosθ

Conclusions

GT is an axial-vector F is a vector

Violates parity Conserves parity

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VN −N = Vs + Vw

ImplicationsInside the nucleus, the N-N interaction is

Conserves parity

Can violate parity

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ψ =ψs + Fψ w ; F ≈10−7

The nuclear state functions are a superposition

Nuclear spectroscopy not affected by Vw

Generalized β-decay

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H = g ψ p* γ μψ n( ) ψ e

*γ μψ ν( ) + h.c. = gV

H = g ψ p* γ μγ 5ψ n( ) ψ e

*γ μγ 5ψ ν( ) + h.c. = gA

The hamiltonian for the vector and axial-vector weak interaction is formulated in Dirac notation as --

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H ≈ g CVV + CA A( )

Or a linear combination of these two --

Generalized β-decay

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H = g Ci ψ p* Oiψ n( ) ψ e

*Oi 1+ γ5( )ψν[ ]i=A,V

∑ + h.c.

OV = γμ ; OV = γμ γ5

The generalized hamiltonian for the weak interaction that includes parity violation and a two-component neutrino theory is --

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g CA CVEmpirically, we need to find -- and

Study: n p (mixed F and GT), and: 14O 14N* (I=0 I=0; pure F)

Generalized β-decay14O 14N* (pure F)

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ft =2π 3h7 log2

m5c4 M F

2g2CV

2( )

g2CV2

( ) =1.4029 ± 0.0022 ×10−49 erg cm3

n p (mixed F and GT)

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ft =2π 3h7 log2

g2m5c4 CV2

( ) M F

2+ CA

2( ) MGT

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Generalized β-decay

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2CV2

CV2 + 3CA

2= 0.3566 →

CV2

CA2

=1.53

Assuming simple (reasonable) values for the square of the matrix elements, we can get (by taking the ratio of the two ft values --

Experiment shows that CV and CA have opposite signs.

Universal Fermi InteractionIn general, the fundamental weak interaction is of the form --

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H ∝ g V − A( )

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n → p + β − + ν e

μ − → e− + ν e + ν μ

π − → μ − + ν μ

Λo → π − + p

semi-leptonic weak decay

pure-leptonic weak decay

semi-leptonic weak decay

Pure hadronic weak decay

All follow the (V-A) weak decay. (c.f. Feynman’s CVC)

Universal Fermi InteractionIn general, the fundamental weak interaction is of the form --

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H ∝ g V − A( )

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μ−→ e− + ν e + ν μ pure-leptonic weak decay

The Triumf Weak Interaction Symmetry Test - TWIST

BUT -- is it really that way - absolutely?

How would you proceed to test it?

Other symmetriesCharge symmetry - C

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n → p + β − + ν e ⇒ n → p + β + + ν e C

All vectors unchanged

Time symmetry - T

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n → p + β − + ν e ⇒ n ← p + β − + ν e

ν e + n → p + β −

T

All time-vectors changed (opposite)

(Inverse β-decay)

Symmetries in weak decay at rest

Note helicities of neutrinos

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rs π = 0

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μ−

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νμ

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μ+€

+

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⇒

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⇐ CP

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−€

ν μ

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μ−

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ν μ€

−

Symmetries in weak decay at rest

Note helicities of neutrinos

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rs π = 0

€

+

€

νμ

€

⇒CP

€

−

€

μ−

€

ν μ

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μ+

€

⇒

€

ν μ€

−€

μ−

Conclusions1. Parity is not a good symmetry in the weak

interaction. (P)

2. Charge conjugation is not a good symmetry in the weak interaction. (C)

3. The product operation is a good symmetry in the weak interaction. (CP) - except in the kaon system!

4. Time symmetry is a good symmetry in the weak interaction. (T)

5. The triple product operation is also a good symmetry in the weak interaction. (CPT)