A Classic Thesis Style - Johannes Gutenberg-Universität Mainz

128 theoretical background
for the lepton tensor, and
W 00 =
� �2 |�q|
W
ω
zz
W 0i = |�q|
ω Wzi
(A.37)
(A.38)
for the hadronic tensor.
Making use of these relations, we can get rid of all time components.
The remaining terms in the contraction of both tensors can be
conveniently grouped in the following form:
LµνW µν =L xx W xx + L yy W yy − Q2
ω 2 (Lzx W zx + L xz W xz ) + Q4
ω 4 Lzz W zz +
L xy W xy + L yx W yx − Q2
ω 2 (Lyz W yz + L zy W zy )
(A.39)
where repeated use of the defining equation of momentum transfer
Q 2 = |�q| 2 − ω 2 has been made.
a.7 explicit form of the leptonic tensor
According to the kinematics shown in Fig. 81 and assuming massless
electrons we have:
sin α = E′ e
sin θe
(A.40)
|�q|
cos α = Ee − E ′ e cos θe
|�q|
(A.41)
Introducing these values in the defining equation for the leptonic
tensor A.28 we get:
Lxx = 4pexp ′ ex + Q2
(A.42)
Since by definition the change in the electron momentum is along the
z direction, we can write:
pex = p′ ex = Ee sin α = EeE ′ e sin θe
|�q|
Inserting these values in A.42 we get:
Lxx = Q 2
�
4 E2eE ′2
e sin 2 θe
Q2 |�q| 2
�
+ 1
(A.43)
(A.44)