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where f rec is the hadronic recoil factor
f rec = E ∣ ∣∣∣∣
A−1
1 + E (
f
1 − ⃗q · ⃗k f
E A E A−1 kf
2
)∣ ∣∣∣∣ =
∣ 1 + ωk ∣
f − qE f cos θ ∣∣∣∣ f
, (2.7)
M A k f
with θ f the angle between ⃗ k f and ⃗q (see Fig. 2.1). The squared invariant matrix
element M fi can be written as
∑
if
|M fi | 2 = (4πα)2
(Q 2 ) 2 η e(K ′ , S ′ ; K, S) µν W µν (Q) fi , (2.8)
where the electron tensor η e (K ′ , S ′ ; K, S) µν is defined by
η e (K ′ , S ′ ; K, S) µν ≡
∑ [ūe (K ′ , S ′ )γ µ u e (K, S) ] ⋆ [ ū e (K ′ , S ′ )γ ν u e (K, S) ] , (2.9)
if
and the nuclear tensor W µν (Q) fi , which contains all of the nuclear structure and
dynamics information, is given by
W µν (Q) fi ≡ ∑ if
J µ⋆ (Q) fi J ν (Q) fi . (2.10)
As it is more difficult to measure the polarization of the scattered electron than
it is to prepare a polarized electron beam, we will only consider the latter case from
here on. Then, the differential cross section contains two terms :
(
d 5 σ
dɛ ′ dΩ e ′dΩ f
) h
fi
= Σ fi + h∆ fi , (2.11)
where h reduces to the electron helicity in the extreme relativistic limit (ERL, ɛ ≫
m e ) for a longitudinally polarized beam. The first term Σ fi is independent of the
electron’s polarization, and, would also occur if no polarizations were considered;
the second term ∆ fi occurs only if the initial beam is polarized.
In the most general case, the contraction of the electron tensor η µν with the
nuclear one W µν results in an expression of the form [25]
4m 2 eη e (K ′ , S ′ ; K, S) µν W µν ∑
(Q) fi = v 0 V K R K fi, (2.12)
where the label K takes on the values L, T , T T , T L, T ′ , T L ′ , T T , T L and T L ′ .
These labels refer to the longitudinal and transverse components of the virtual photon
polarization, and hence correspond to the nuclear electromagnetic components
with respect to the direction of ⃗q. The R T T
fi , RT L
fi
and R T L′
fi
terms do not vanish
K