300
20. Introduction to the Phenomenology of Weak Interactions
tensor ∈ αβγδ is just the generalization of ∈ ijk to four dimensions, and is defined
by
{
{ +1
for ∈ 0123 and all even permutations of 0, 1, 2, 3
∈ αβγδ =
−1
for∈
f
{
1023 and all odd permutations o 0, 1, 2, 3
0
o t h e r w i s e .
(20.60)
Its appearance here is a direct consequence of parity violation. Notice that
this definition has the consequence that
∈ 0123 = +1
(20.61)
but
∈
0123 = −1.
(20.62)
We will also need to contract two ∈ tensors. By looking at the possible combinations, it should be easy to convince yourself of the result
|
|
|
| δ
δ
∈ ijk ∈ ilm =
jl
jm
|
|
| δ kl δ km |
(20.63)
i.e.
∈ ijk ∈ ilm = δ jl δ km − δ kl δ jm .
(20.64)
For the four-dimensional ∈ tensor one can show (see problem 20.6)
|
| γ
γ |
∈ αβ ∈
μνγδ
μν
=
| δ
δ |
|
−2! |
α
β
δ
δ
α
δ
δ |
(20.65)
β
where the minus sign arises from (20.62) and the 2! from the fact that the two
indices are contracted.
We can now evaluate N μν . We obtain, after some rearrangement of indices,
the result for the ν μ → μ
− tensor:
N μν = 8[(k μ
′ k ν + k ν
′ k μ + (q
2 /2)g μν ) − i∈ μναβ k
α k
′β ].
(20.66)
For the electron tensor E
μν we have a similar result (divided by 2):
E
μν = 4[(p
′μ p
ν + p
′ν p
μ + (q
2 /2)g
μν ) − i∈
μνγδ p γ p
′
δ ].
(20.67)
Next, we have to perform the contraction N μν E
μν in (20.53). In the case
of elastic e
− μ
− scattering considered in section 8.7, the analogous contraction
between the tensors L μν and M
μν was simplified by using the conditions
q
μ L μν = q
ν L μν = 0 (see (8.189)), which followed from electromagnetic current
conservation at the electron vertex (see (8.188)): q
μ u ¯(k
′ )γ μ u(k) = 0. Here, the
analogous vertex is u ¯(μ, k
′ )γ μ (1 − γ 5 )u(ν μ , k). In this case, when we contract
this with q
μ = (k − k
′ )
μ we find a non-zero result:
(m νμ − m μ )¯ u(μ, k
′ )u(ν μ , k) + (m μ + m νμ )¯ u(μ, k
′ )γ 5 u(ν μ , k),
(20.68)
20. Introduction to the Phenomenology of Weak Interactions
tensor ∈ αβγδ is just the generalization of ∈ ijk to four dimensions, and is defined
by
{
{ +1
for ∈ 0123 and all even permutations of 0, 1, 2, 3
∈ αβγδ =
−1
for∈
f
{
1023 and all odd permutations o 0, 1, 2, 3
0
o t h e r w i s e .
(20.60)
Its appearance here is a direct consequence of parity violation. Notice that
this definition has the consequence that
∈ 0123 = +1
(20.61)
but
∈
0123 = −1.
(20.62)
We will also need to contract two ∈ tensors. By looking at the possible combinations, it should be easy to convince yourself of the result
|
|
|
| δ
δ
∈ ijk ∈ ilm =
jl
jm
|
|
| δ kl δ km |
(20.63)
i.e.
∈ ijk ∈ ilm = δ jl δ km − δ kl δ jm .
(20.64)
For the four-dimensional ∈ tensor one can show (see problem 20.6)
|
| γ
γ |
∈ αβ ∈
μνγδ
μν
=
| δ
δ |
|
−2! |
α
β
δ
δ
α
δ
δ |
(20.65)
β
where the minus sign arises from (20.62) and the 2! from the fact that the two
indices are contracted.
We can now evaluate N μν . We obtain, after some rearrangement of indices,
the result for the ν μ → μ
− tensor:
N μν = 8[(k μ
′ k ν + k ν
′ k μ + (q
2 /2)g μν ) − i∈ μναβ k
α k
′β ].
(20.66)
For the electron tensor E
μν we have a similar result (divided by 2):
E
μν = 4[(p
′μ p
ν + p
′ν p
μ + (q
2 /2)g
μν ) − i∈
μνγδ p γ p
′
δ ].
(20.67)
Next, we have to perform the contraction N μν E
μν in (20.53). In the case
of elastic e
− μ
− scattering considered in section 8.7, the analogous contraction
between the tensors L μν and M
μν was simplified by using the conditions
q
μ L μν = q
ν L μν = 0 (see (8.189)), which followed from electromagnetic current
conservation at the electron vertex (see (8.188)): q
μ u ¯(k
′ )γ μ u(k) = 0. Here, the
analogous vertex is u ¯(μ, k
′ )γ μ (1 − γ 5 )u(ν μ , k). In this case, when we contract
this with q
μ = (k − k
′ )
μ we find a non-zero result:
(m νμ − m μ )¯ u(μ, k
′ )u(ν μ , k) + (m μ + m νμ )¯ u(μ, k
′ )γ 5 u(ν μ , k),
(20.68)