9.2
Problems
293
equation (9.3):
∑ ∑
1 1
j
μ
′
e
2 W
μν (q, p) =

em
4πM 2 s X
× ′
| ˆ j
ν (0)|p; p, s>(2π)
4 δ
4 (p + q − p
′ )
em
where the sum X is over all possible hadronic final states. If we consider the
special case of elastic scattering, the sum over X is only over the final proton’s
degrees of freedom:
∑ ∑
1 1
μν
′ ′
′
e
2 W
=
μ (0)|p; p , s > ′
| ˆ j
ν (0)|p; p, s>
el
em
em
4πM 2 s s '
′
1 d
3 p
× (2π)
4 δ
4 (p + q − p
′ )
.
(2π) 3 2E ′
Now use equation (8.208) with F 1 = 1 and κ = 0 (i.e. the electromagnetic
current matrix element for a ‘point’ proton) to show that the resulting cross
section is identical to that for elastic eμ scattering.
(a) Perform the contraction L μν W
μν for inclusive inelastic electron–
proton scattering (remember q
μ L μν = q
ν L μν = 0). Hence verify
that the inclusive differential cross section in terms of ‘laboratory’
variables, and neglecting the electron mass, has the form
d
2 σ
α
2
=
[W 2 cos
2 (θ/2) + W 1 2 sin
2 (θ/2)].
dΩdk ′
4k 2 sin
4 (θ/2)
(b) By calculating the Jacobian
|
|
|
|
∂u/∂x ∂u/∂y
J = |
|
|
|
∂v/∂x ∂v/∂y
for a change of variables (x, y) → (u, v)
du dv = |J|dx dy
find expressions for d
2 σ/dQ
2 dν and d
2 σ/dx dy, where Q
2 and ν
have their usual significance, and x is the scaling variable Q
2 /2M ν
and y = ν/k.
9.3 Consider the description of inelastic electron–proton scattering in terms
of virtual photon cross sections:
(a) In the ‘laboratory’ frame with
0
p
μ = (M, 0, 0, 0)
and
q
μ = (q , 0, 0, q
3 )