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8 Chapter 2. Reaction Observables and Kinematics
e'
⎜ f>
K A-1
K f
ieJ
ν
(K A-1,K f ,K A)
u (K',S') e
p
-ieγ µ
u (K,S)
iD (Q) F µν
e
K A
e
⎜ i>
Figure 2.2 Lowest order Feynman diagram for the exclusive A(⃗e, ⃗e ′ ⃗p)B scattering process.
J ν denotes the matrix element of the electromagnetic nuclear current operator between the
initial and final hadronic states.
invariant matrix element M fi reads
M fi = ie ( ɛɛ
′ ) 1/2
Q 2 m 2 j e (K ′ , S ′ ; K, S) µ J µ (K A−1 , K f ; K A ) fi , (2.3)
e
with the electromagnetic current for the electron
( )
m
j e (K ′ , S ′ 2 1/2
; K, S) µ = −e e
ɛɛ ′ ū e (K ′ , S ′ )γ µ u e (K, S) . (2.4)
In the impulse approximation the nuclear electromagnetic current in momentum
space J µ (K A−1 , K f ; K A ) fi = J µ (Q) fi can be written as
J µ (Q) fi =< K f S f |Ĵ µ |K i S i >= ū f Γ µ (K f , K i )u i , (2.5)
with Γ µ the electromagnetic vertex function for the nucleon and u i (u f ) the nucleon
(distorted) spinors.
In the actual measurements, the momentum of the recoiling nucleus is not measured,
while those of the electron and the ejected proton are. The recoil momentum
can be eliminated from Eq. (2.2) through integration over ⃗ k A−1 . This results in the
following fivefold differential cross section :
d 5 σ
dɛ ′ = m2 eM f M A−1 k ′ k f
dΩ e ′dΩ f (2π) 5 M A k
f rec
−1
∑
|M fi | 2 , (2.6)
if