parity conservation in the weak (beta decay)...
TRANSCRIPT
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Parity Conservation inthe weak (beta decay) interaction
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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 --
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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)
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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.
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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 ( )
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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
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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
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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 ( )
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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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†
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
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†
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
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†
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
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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 --
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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)
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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 ˘ ˚ ˙
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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.
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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)
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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?
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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)
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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 -
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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-
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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)