328
20. Introduction to the Phenomenology of Weak Interactions
(c) Insert the Callan–Gross relation
(ν)
(ν)
2xF
= F
1
2
to derive the result quoted in section 20.7.2:
(
)
d
2 σ
(ν)
G
2
1 + (1 − y)
2
xF
(ν) 1 − (1 − y)
2
F
(ν)
3
=
sF
+
.
2
(ν)
dxdy
2π
2
2
F 2
20.10 The differential cross section for ν μ q scattering by charged currents
has the same form (neglecting masses) as the ν μ e
−
→ μ
− ν e result of problem
20.7, namely
dσ
G
2
F
(νq) =
.
dt
π
(a) Show that the cross section for scattering by antiquarks ν μ ¯
q has the
form
dσ
G
2
F
(ν¯ =
.
q)
(1 − y)
2
dt
π
(b) Hence prove the results quoted in section 20.7.2:

d
2 σ
G
2
F
(νq) =
sxδ(x − Q
2 /2M ν)
dxdy
π
and

d
2 σ
G
2
F
(νq) ¯ =
sx(1 − y
2 )δ(x − Q
2 /2M ν)
dxdy
π

(where M is the nucleon mass).

(c) Use the parton model prediction

d
2 σ
(ν)
G
2
F
=
sx[q(x) + q ¯(x)(1 − y)
2 ]
dxdy
π

to show that

(ν)
F
= 2x[q(x) + q ¯(x)]
2
and
(ν)
xF (x)
q(x) − q ¯(x)
3
=
.
(ν)
q(x) + q ¯(x)
F (x)
2
20.11 Verify the transformation laws (20.160).