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

28 Chapter 4. Off-Shell Electron-Proton Coupling
This is the same matrix element one would obtain in the Feynman gauge when
using the replacement given in Eq. (4.5). Another noncovariant gauge, the Weyl, or
temporal, gauge defines the matrix element as
M Weyl = − ı
(
⃗j
Q 2 · ⃗J − (⃗q · ⃗J)(⃗q
)
· ⃗j)
ω 2 . (4.16)
Similarly, this expression is obtained from Eq. (4.14) upon replacement of Eq. (4.7).
The substitution of Eq. (4.9) is simply obtained by setting the gauge parameter ξ
equal to zero in Eq. (4.13); this is the so-called Landau gauge. And again, the same
expression is found upon substituting the recipe of Eq. (4.9) into Eq. (4.14). For the
prescription of Eq. (4.10) there is no deeper-lying justification on the basis of the
most general covariant expression of Eq. (4.13).
The electromagnetic vertex function of Eq. (4.11) is based on the following. It is
shown in Ref. [44] that the electromagnetic interactions of any two-body system (i.e.
electron-proton system) described by a relativistic two-body equation will always
conserve current provided that the following conditions are met
• the electromagnetic currents for the interacting off-shell nucleons (and mesons)
satisfy the appropriate Ward-Takahashi identity
• the interacting incoming and outgoing two-body system satisfies the same twobody
relativistic equation
• the interaction current is built up from the relativistic kernel by coupling the
virtual photon to all possible places (or possible orders) in the kernel
It is obvious that in a many-body calculation not all of these conditions can be met,
so that the occurence of current non-conserving terms becomes almost unavoidable.
In analogy with the CC1 and CC2 current operators, we have chosen to restore
current conservation by means of the recipes of Eqs. (4.5) and (4.7).
In Sec. 7 we will focus on the uncertainties induced by the off-shell effects and
quantify the importance of these ambiguities for the description of the A(e, e ′ p)
process at various energy scales.
Another issue of current interest related to the off-shellness of nucleons, is the
question whether the electromagnetic form factors are modified by the presence of
a nuclear medium. In Ref. [40] variations due to the off-shellness of the proton
were found up to 10 % relative to the on-shell form factors for intermediate energy
kinematics. When going off-shell the magnitudes of the form factors are generally
increased, and, consequently so are the cross sections. On the other hand it was
found that the ratio of the form factors G M /G E was rather insensitive to off-shell
effects.