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Chapter 4
Off-Shell Electron-Proton
Coupling
The scattering process of an electron on a free (or, on-shell) nucleon can be computed
in a model-independent fashion. The electromagnetic coupling on a bound
(or, off-shell) nucleon, on the other hand, is not free of ambiguities. For one, it is
believed that the electromagnetic vertices for off-shell nucleons have a more complicated
structure than for free nucleons. This elusive feature is known as the Gordon
ambiguity and there has been a string of papers that have addressed this issue
[39, 40, 41, 42] resulting in a number of recipes for the form of the off-shell electronproton
coupling.
Another equally important issue, that is closely related to the Gordon ambiguity,
is the gauge invariance of the electromagnetic current in many-body systems. This
topic is closely related to the condition that the electromagnetic current should be
conserved. In trying to describe the nuclear reaction by means of the free electromagnetic
current, one is forced to make certain assumptions. These assumptions
lead the off-shell nucleons to affect the nuclear current in such a way that current
conservation is lost. As discussed in many works [41, 42, 43, 44, 45], some arbitrariness,
often referred to as the “off-shell ambiguity”, surrounds the choice for
the functional form of the electromagnetic vertex function Γ µ for a bound nucleon.
There exists no uniquely defined procedure to remedy this unphysical feature and
one is forced to introduce ad hoc prescriptions to restore current conservation, as
will be discussed below.
We express the matrix elements of the nucleon current in the usual form
< K f S f |J µ |K i S i >= ū f Γ µ (K f , K i )u i , (4.1)
where Γ µ is the electromagnetic vertex function for the nucleon and u i (u f ) are the
nucleon spinors. For a free nucleon, Γ µ can be expressed in several fully equivalent
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