240
8. Elementary Processes in Scalar and Spinor Electrodynamics
met in sections 8.2.1 and 8.2.3, particularly the latter. The cross section for
′
the scattering of an electron in spin state s to one in spin state s is (cf (6.110))
1
′
dσ ss ' =
|M e − s + (s, s
′ )|
2 (2π)
4 δ
4 (k
′ + p − k − p)
4Eω|v|
′
1 d
3 k
′ d
3 p
×
(8.115)
(2π) 6 2ω ′ 2E ′
where we have defined
k
μ = (ω, k)
k
′ μ = (ω
′ , k
′ )
′ μ
p
μ = (E, p)
p = (E
′ , p
′ ).
(8.116)
For the unpolarized cross section we are required, as in (8.46), to evaluate
the quantity
∑
2
∑
1
′ )|
2
(
e
) 2 1
′ )γ
μ
′ )
|M e − s + (s, s
=
u ¯(k
′ , s
u(k, s)¯ u(k, s)γ
ν u(k
′ , s
2
q 2
2
'
'
s,s
s,s
× (p + p
′ ) μ (p + p
′ ) ν
(8.117)
( ) 2
2
e
′ )
≡
L
μν (k, k
′ )T μν (p, p
(8.118)
q 2
where the boson tensor T μν is just (p + p
′ ) μ (p + p
′ ) ν and the lepton tensor
L
μν has been evaluated in (8.79). Using q
2 = (k − k
′ )
2 = 2m
2
− 2k · k
′ , the
expression (8.79) can be rewritten as
k
ν + k
′ ν
μν ].
L
μν (k, k
′ ) = 2[k
′ μ
k
μ + (q
2 /2)g
(8.119)
We then find (problem 8.12)
L
μν T μν = 8[2(p · k)(p · k
′ ) + (q
2 /2)M
2 ]
(8.120)
′
′
k
′
since k
′
· p = k · p and k · p =
· p from 4-momentum conservation, and
p
2 = p
′ 2 = M
2 (we are using m for the e
− mass and M for the s
+ mass).
We can now give the differential cross section in the CM frame by taking
over the formula (6.129) with
∑
′ )|
2
|M|
2
→
1
|M e − s + (s, s
2
'
s,s
so as to obtain
(
)
d¯ σ
2α
2
=
[2(p · k)(p · k
′ ) + (q
2 /2)M
2 ]
(8.121)
dΩ
W 2 (q 2 ) 2
CM
where α = e
2 /4π and W
2 = (k + p)
2 .
8. Elementary Processes in Scalar and Spinor Electrodynamics
met in sections 8.2.1 and 8.2.3, particularly the latter. The cross section for
′
the scattering of an electron in spin state s to one in spin state s is (cf (6.110))
1
′
dσ ss ' =
|M e − s + (s, s
′ )|
2 (2π)
4 δ
4 (k
′ + p − k − p)
4Eω|v|
′
1 d
3 k
′ d
3 p
×
(8.115)
(2π) 6 2ω ′ 2E ′
where we have defined
k
μ = (ω, k)
k
′ μ = (ω
′ , k
′ )
′ μ
p
μ = (E, p)
p = (E
′ , p
′ ).
(8.116)
For the unpolarized cross section we are required, as in (8.46), to evaluate
the quantity
∑
2
∑
1
′ )|
2
(
e
) 2 1
′ )γ
μ
′ )
|M e − s + (s, s
=
u ¯(k
′ , s
u(k, s)¯ u(k, s)γ
ν u(k
′ , s
2
q 2
2
'
'
s,s
s,s
× (p + p
′ ) μ (p + p
′ ) ν
(8.117)
( ) 2
2
e
′ )
≡
L
μν (k, k
′ )T μν (p, p
(8.118)
q 2
where the boson tensor T μν is just (p + p
′ ) μ (p + p
′ ) ν and the lepton tensor
L
μν has been evaluated in (8.79). Using q
2 = (k − k
′ )
2 = 2m
2
− 2k · k
′ , the
expression (8.79) can be rewritten as
k
ν + k
′ ν
μν ].
L
μν (k, k
′ ) = 2[k
′ μ
k
μ + (q
2 /2)g
(8.119)
We then find (problem 8.12)
L
μν T μν = 8[2(p · k)(p · k
′ ) + (q
2 /2)M
2 ]
(8.120)
′
′
k
′
since k
′
· p = k · p and k · p =
· p from 4-momentum conservation, and
p
2 = p
′ 2 = M
2 (we are using m for the e
− mass and M for the s
+ mass).
We can now give the differential cross section in the CM frame by taking
over the formula (6.129) with
∑
′ )|
2
|M|
2
→
1
|M e − s + (s, s
2
'
s,s
so as to obtain
(
)
d¯ σ
2α
2
=
[2(p · k)(p · k
′ ) + (q
2 /2)M
2 ]
(8.121)
dΩ
W 2 (q 2 ) 2
CM
where α = e
2 /4π and W
2 = (k + p)
2 .