Question 1:
The pion electromagnetic F2π structure function
is given in the parton model by the charge-squared
weighted sum of parton distributions in the pion,
F2π(x)
= x Σq
eq2
[ q π (x)
+ q—π (x) ].
(a)
Neglecting strange quarks in the pion, write down the
F2π structure functions for the
π+, π- and π0.
(b)
Using isospin symmetry and the fact that the valence q
and q— distributions
in the pion are equivalent
(i.e., uvπ+
= d—vπ+),
write the pion structure function in terms of the pion valence
and sea distributions,
and show that
F2π+
= F2π-
= F2π0.
Question 2:
(a)
Derive the leptonic tensors for charged current ν and
ν— scattering,
including masses for the charged leptons.
(Start from the expressions in terms of Dirac spinors, and
evaluate the traces explicitly.)
(b)
Using the Lorentz vectors
pμ and
qμ,
and the tensors
gμν and
ε μναβ,
show that the most general
hadronic tensor
Wμν
for the weak interaction contains 6 independent terms.
(c)
Neglecting lepton masses, prove that when contracted with the
leptonic tensor in part (a),
only 3 terms in
Lμν
Wμν
survive.
(Note that
ε μναβ
ε μνλρ
= 2 (gαρ
gβλ
- gαλ
gβρ),
where
gαρ pρ
= pα.)
(d)
Show that the differential cross section can be written in terms
of the variables x and y = ν/E as:
d2σ/dE'dΩ
= (E' / (2π M E y))
d2σ/dxdy.