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88 Chapter 7. Off-Shell Ambiguities
negative energy projections. The R L and R T structure functions are mainly governed
by their positive energy projection, while for the R T T and R T L response functions
the cross term R C can be larger than the positive energy projection R P . As the
differences between the CC1 and CC2 current operators and the gauge restoration
schemes are mainly concentrated in their action upon the lower components of the
wave functions, it should not come as a surprise that the ambiguities will be much
larger in the responses that generate a significant part of their strength from these
negative energy projections.
The observed feature that the cross sections are only marginally sensitive to
gauge ambiguities at higher energies can be explained on the basis of the following
considerations. A measure for the violation of current conservation is given by [102]
q µ J µ = ωJ 0 − ⃗q · ⃗J ≡ χ , χ = δω[J] , (7.4)
where the quantity [J] denotes (part of) the nuclear current density (i.e. u f γ 0 u i ).
This can be understood by considering the explicit expression for the CC2 current
operator (Eq. (4.3)) for example :
q µ J µ CC2
= q µ u f Γ µ CC2 u i ,
= F 1 u f (ωγ 0 − ⃗q · ⃗γ)u i ,
= F 1 (ω − ω ′ )u f γ 0 u i , (7.5)
where we have used the definition (7.1) of ω ′ . Rewriting now the matrix element in
the Coulomb gauge of Eq. (4.15) in terms of χ, leaves us with
M Coulomb =
ı
Q 2 j µJ µ −
ı ( ) ωj0 χ
Q 2 ⃗q 2 . (7.6)
Along similar lines the matrix element in the Weyl gauge of Eq. (4.16) can be
rewritten as
M Weyl =
ı
Q 2 j µJ µ −
ı ( ) j0 χ
Q 2 . (7.7)
ω
With these two expressions we can estimate the relative differences between the
cross sections obtained in the J0 and J3 scheme, respectively :
( )
ı
M Weyl j
Q
=
2 µ J µ − ı j0 δω[J]
Q 2 ω
( ) . (7.8)
M Coulomb
ı
j
Q 2 µ J µ − ı ωj0 χ
Q 2 ⃗q 2
For the purpose of getting order of magnitude estimates, we approximate j 0 [J] ≃
j µ J µ and find
M Weyl
M Coulomb
≃ 1 − δω ω
1 − ωδω
⃗q 2
∼ σCC1 . (7.9)
σCC2