Download Thesis in Pdf Format - Theoretical Nuclear Physics and ...

27
One can also construct a vertex function that garantuees current conservation for
any initial and final nucleon state. This can be achieved for example by adding an
extra term to the vertex [47]
Γ µ DON = F 1(Q 2 )γ µ + ı κ
2M F 2(Q 2 )σ µν q ν + F 1 (Q 2 )̸ qqµ
Q 2 , (4.10)
which is also equivalent to the Eqs. (4.2) - (4.4) in the free nucleon case. An operator
derived from the generalized Ward-Takahashi identity reads [44]
Γ µ W T
= γ µ − ı κ
2M F 2(Q 2 )σ µν q ν + [F 1 (Q 2 ) − 1]̸ qqµ + Q 2 γ µ
Q 2 . (4.11)
We now proceed with putting the above-mentioned recipes in a more fundamental
context. We write the transition matrix element corresponding with the electron
scattering process in the general form
M = j µ Π µν J ν , (4.12)
where Π µν is the photon propagator and j µ the electron current. The explicit form
of the propagator is gauge dependent, and, consequently, so is the matrix element.
In the covariant class of gauges one has that
M Lorentz =
ı (
Q 2 j µ J µ + (1 − ξ) (q µJ µ )(q µ j µ )
)
Q 2
, (4.13)
where ξ is a free gauge parameter.
Usually one works in the so-called Feynman gauge, where ξ is set equal to 1. In
this case, the matrix element reduces to
M Feynman =
ı
Q 2 j µJ µ . (4.14)
The equation (4.14) holds always true in a covariant Lorentz gauge since the electron
current j is conserved. A choice of one or another gauge should have no effect on the
results. In calculations dealing with finite nuclei however, the occurence of current
non-conserving terms cannot be excluded, so that the different gauge possibilities
may eventually affect the predictions. We will now show that the prescriptions of
Eqs. (4.5), (4.7) and (4.9) are connected to different gauge choices.
In the frequently adopted Coulomb gauge, which is a noncovariant gauge, the
matrix element of Eq. (4.12) is written as
M Coulomb = ı
⃗q 2 j 0J 0 −
ı
(
⃗j
Q 2 · ⃗J − (⃗q · ⃗J)(⃗q
)
· ⃗j)
⃗q 2 . (4.15)