261
8.8. Electron–proton elastic scattering and nucleon form factors
we shall call e
− μ
+ , for definiteness. Energy and momentum conservation at
the μ
+ vertex gives the condition
′
p + q = p
(8.216)
with the mass-shell conditions (M is the μ
+ mass)
2
′ 2
p = p = M
2 .
(8.217)
Hence for elastic scattering we have the relation
2
2p · q = −q .
(8.218)
It is conventional to relate these invariants to the corresponding laboratory
frame (p
μ = (M, 0)) expressions. Neglecting the electron mass so that
2
k ≡ |k| = ω
(8.219)
k
′
ω
′
k
′
≡ | | =
(8.220)
we have
2
q = −2kk
′ (1 − cos θ) = −4kk
′ sin
2 (θ/2)
(8.221)
and
p · q = M (k − k
′ ) = M ν
(8.222)
0
where ν is the energy transfer q in this frame. To avoid unnecessary minus
signs, it is convenient to define
Q
2 = −q
2 = 4kk
′ sin
2 (θ/2)
(8.223)
and the elastic scattering relation between p · q and q
2 reads
ν = Q
2 /2M
(8.224)
or
k
′
1
=
.
(8.225)
k
1 + (2k/M ) sin
2 (θ/2)
Remembering, therefore, that for elastic scattering k
′ and θ are not independent variables, we can perform a change of variables (see appendix K) in the
laboratory frame
dΩ = 2π d(cos θ) = (π/k
′ 2 ) dQ
2
(8.226)
and write the differential cross section for e
− μ
+ scattering as
dσ
πα
2
1
=
[cos
2 (θ/2) + 2τ sin
2 (θ/2)].
(8.227)
dQ 2
4k 2 sin
4 (θ/2) kk ′
2 As after equation (8.126), note again that in the present context ‘k’ and ‘k ′ ’ are not
4-vectors but the moduli of 3-vectors.
8.8. Electron–proton elastic scattering and nucleon form factors
we shall call e
− μ
+ , for definiteness. Energy and momentum conservation at
the μ
+ vertex gives the condition
′
p + q = p
(8.216)
with the mass-shell conditions (M is the μ
+ mass)
2
′ 2
p = p = M
2 .
(8.217)
Hence for elastic scattering we have the relation
2
2p · q = −q .
(8.218)
It is conventional to relate these invariants to the corresponding laboratory
frame (p
μ = (M, 0)) expressions. Neglecting the electron mass so that
2
k ≡ |k| = ω
(8.219)
k
′
ω
′
k
′
≡ | | =
(8.220)
we have
2
q = −2kk
′ (1 − cos θ) = −4kk
′ sin
2 (θ/2)
(8.221)
and
p · q = M (k − k
′ ) = M ν
(8.222)
0
where ν is the energy transfer q in this frame. To avoid unnecessary minus
signs, it is convenient to define
Q
2 = −q
2 = 4kk
′ sin
2 (θ/2)
(8.223)
and the elastic scattering relation between p · q and q
2 reads
ν = Q
2 /2M
(8.224)
or
k
′
1
=
.
(8.225)
k
1 + (2k/M ) sin
2 (θ/2)
Remembering, therefore, that for elastic scattering k
′ and θ are not independent variables, we can perform a change of variables (see appendix K) in the
laboratory frame
dΩ = 2π d(cos θ) = (π/k
′ 2 ) dQ
2
(8.226)
and write the differential cross section for e
− μ
+ scattering as
dσ
πα
2
1
=
[cos
2 (θ/2) + 2τ sin
2 (θ/2)].
(8.227)
dQ 2
4k 2 sin
4 (θ/2) kk ′
2 As after equation (8.126), note again that in the present context ‘k’ and ‘k ′ ’ are not
4-vectors but the moduli of 3-vectors.