can be shown via the Dirac equation to satisfy a continuity condition
indicating that it is conserved,
The conservation of this electromagnetic current originates from the conservation of electric charge. One can now generalize the proton wave function into a nucleon wavefunction with proton and neutron parts using an isospin projection operator
Since the isoscalar term cannot depend on rotations in isospin space, the isovector term must be conserved independently. This indicates that the third component of the isospin part of the nucleon wave function is a conserved quantity. The conserved vector current (CVC) hypothesis states that if the third component of isospin current is conserved, then the other two should be also. Thus,
where
(the
basis has been chosen here to make the following more illustrative).
Weak current neutron beta decay has the form
which can be rewritten using the nucleon wavefunction from above
The vector term (the first term on the RHS) is thus assumed conserved by CVC and therefore not renormalized by any hadronic renormalization.
The axial-vector term does not have the same protection but is still considered
to be partially conserved due to the partially conserved axial-vector
current (PCAC) hypothesis. Introduced in 1960 by Gell-Mann and Levy, PCAC
states that, since the divergence of the axial-vector part of the pion current
is proportional to the pion mass (i.e.
)
and the pion mass is light relative to other hadrons, then the divergence of
the axial-vector current is nearly zero and thus "partially" conserved [Ren 90].