Parity Conservation inthe weak (beta decay) interaction

The parity operationThe parity operation involves the transformation

†

x Æ -x y Æ -y z Æ -z

r Æ r q Æ p -q( ) f Æ f + p( )

In rectangular coordinates --

In spherical polar coordinates --

In quantum mechanics

†

ˆ P y x,y,z( ) = ±1y -x,-y,-z( )For states of definite (unique & constant) parity -

If the parity operator commutes with hamiltonian -

†

ˆ P , ˆ H [ ] = 0 fi 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 mechanicsTo 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 --

†

ˆ P E = E ˆ P r r ⋅ r r ( ) =

r r ⋅ r r ( )Parity operation on a scaler quantity -

Parity operation on a polar vector quantity -

†

ˆ P r r = -

r r ˆ P r p = -

r p Parity operation on a axial vector quantity -

†

ˆ P r L = ˆ P

r r ¥ r p ( ) = -r r ¥ -

r p ( ) = -r L

Parity operation on a pseudoscaler quantity -

†

ˆ P 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 thatare pseudoscaler.

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

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

http://publish.aps.org/ tq puzzle

Parity experiments (Lee & Yang)

†

r p ⋅r I = 0

Parity reflectedOriginal

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

†

r p

†

r I

†

r p

†

r I

If parity is a good symmetry…• The the decay intensity should not

depend on .

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

†

r p ⋅r I = 0

†

r p ⋅r I ( )

†

r p ⋅r I ≠ 0

†

r p ⋅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/

†

Hb- = -

vc

†

n

†

Hn =1

†

b-

†

n

†

Hb- = -

vc

†

b-

†

Hn =1

†

n

†

Hn = -1

†

n

†

Hn = -1

NotobservedMeasure

Trecoil

†

GT : DI =1

A :1-13

vc

cosq

†

F : I = 0 Æ I = 0

V :1+vc

cosq

†

Hn =1

†

Hb- = -

vc

†

GT : DI =1

A :1-13

vc

cosq

†

F : I = 0 Æ I = 0

V :1+vc

cosq

Conclusions

GT is an axial-vector F is a vector

Violates parity Conserves parity

†

VN -N = Vs + Vw

ImplicationsInside the nucleus, the N-N interaction is

Conservesparity

Can violateparity

†

y =ys + Fyw ; F ª10-7The nuclear state functions are a superposition

\ Nuclear spectroscopy not affected by Vw

Generalized b-decay

†

H = g y p* gmyn( ) y e

*gmyn( ) + h.c. = gV

H = g y p* gmg 5yn( ) y e

*gmg5yn( ) + h.c. = gA

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

†

H ª g CVV + CA A( )

Or a linear combination of these two --

Generalized b-decay

†

H = g Ci y p* Oiyn( ) ye

*Oi 1+ g5( )yn[ ]i=A,V

 + h.c.

OV = gm ; OV = gmg5

The generalized hamiltonian for the weak interactionthat includes parity violation and a two-componentneutrino theory is --

†

g CA CVEmpirically, we need to find -- and

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

Generalized b-decay14O Æ 14N* (pure F)

†

ft =2p 3h7 log2

m5c4 M F2

g2CV2( )

g2CV2( ) = 1.4029 ± 0.0022 ¥10-49 erg cm3

n Æ p (mixed F and GT)

†

ft =2p 3h7 log2

g2m5c4 CV2( ) M F

2+ CA

2( ) MGT2È

Î Í ˘ ˚ ˙

Generalized b-decay

†

2CV2

CV2 + 3CA

2 = 0.3566 Æ CV2

CA2 =1.53

Assuming simple (reasonable) values for thesquare of the matrix elements, we can get (bytaking 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 ofthe form --

†

H µ g V - A( )

†

n Æ p + b- + n e

m- Æ e- + n e + nm

p - Æ m- + n m

Lo Æ p - + 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 ofthe form --

†

H µ g V - A( )

†

m- Æ e- + n e + nm 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

†

n Æ p + b- + n e fi n Æ p + b+ + ne C

All vectors unchanged

Time symmetry - T

†

n Æ p + b- + n e fi n ¨ p + b- + n e

ne + n Æ p + b-

T

All time-vectors changed (opposite)

(Inverse b-decay)

Symmetries in weak decayp+ at rest Note helicities

of neutrinos

†

r s p = 0

†

m-

†

nm

†

m+†

p +

†

fi

†

‹CP

†

p -

†

n m

†

m-

†

n m†

p -

Symmetries in weak decayp+ at rest Note helicities

of neutrinos

†

r s p = 0

†

p +

†

nm

†

fiCP

†

p -

†

m-

†

n m

†

m+

†

fi

†

n m†

p -†

m-

Conclusions1. Parity is not a good symmetry in the weak

interaction. (P)

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

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

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

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