In fig. 2 we show the matrix elements of the axial charge for the relativistic
and non relativistic cases of eqs. (11) and (14) respectively as a
function of q. We show the results for the #math42#1p1/2→2s1/2
and #math43#1p3/2→1d3/2 transitions on 16O. One
observes that the strength of the latter transition is about a factor 2
larger than for the first one. In both cases the relativistic
calculation of the matrix element yields larger values than the
non-relativistic one derived from the equivalent non relativistic potential.
We also observe that the matrix elements are weakly dependent on the momentum
transfer q up to values of #math44#q #tex2html_wrap_inline1382# 100 MeV/c. In fig. 3 we
show the ratio of the relativistic versus non relativistic matrix elements as
a function of q for the two transitions in 16O.
The values of the ratios at q = 0, relevant to β decay, are about 1.33
and 1.20 for the #math45#1p1/2→2s1/2 and #math46#1p3/2→1d3/2 transitions, respectively.
The strength of #math47#Σs(r ) from eq. (6) at r = 0 is in our case #math48#Σs #tex2html_wrap_inline1392# -384 MeV, close to the value typical for nuclear matter at saturation (
#math49#Σs #tex2html_wrap_inline1394# -400 MeV) which we have considered in our
estimates of section 2. In the perturbative approach of section 2 we would
have obtained a ratio of 1.41 whereas the non perturbative nuclear
matter estimate would even yield a ratio of 1.69 for this value of
#math50#Σs. We can see that the calculations performed
directly for the finite nuclei yields results which are significantly
smaller than those estimates from nuclear matter. The reason for this
difference is the fact that the calculation of matrix elements for
finite nuclei requires a radial integration which is dominated by the
integrand at the surface. This is due to the fact that the integrand
contains a product of wavefunctions for a particle- and a hole-state.
The nuclear density at the surface, however, is smaller than for r = 0
or the saturation density of nuclear matter. Consequently also the
relativistic effects due scalar potential Σs, leading to an
enhacement of the small component of the Dirac spinor, will be smaller
at these relevant densities than at the center of the nucleus or at the
saturation density of nuclear matter. Similar, although a bit smaller,
reductions with respect to the nuclear matter approach were also found
in the finite nuclei perturbative approach of [#8##1###], though the
results were found to be sensitive to the short range correlations
assumed. Here short range correlations are incorporated in the problem
in a selfconsistent way.
From these considerations we can also understand that the relativistic
renormalization of the axial charge operator in the case of the
#math51#1p3/2→1d3/2 transition
is smaller than the one in the #math52#1p1/2→2s1/2 case.
The smaller renormalization
in the case of the #math53#1p3/2→1d3/2 transition can be
interpreted in terms of the centrifugal barrier which pushes the d state
more to the surface of the nucleus where the potential #math54#Σs is
weaker. Furthermore we observe a slight increase of the renormalization
as a function of q. At a larger momentum transfer one tends to probe
more the higher densities in the interior of the nucleus.
These results are confirmed by our calculations for the nucleus 40Ca.
In fig. 4 we show the relativistic and non relativistic matrix elements for
the #math55#1d3/2→2p3/2 transition in 40Ca and in fig.
5 the
ratio of the relativistic to non relativistic matrix elements.
The ratio is of
the order of 1.23, rather independent on the momentum transfer. This
result for the renormalization of the axial charge is very similar
but slightly larger than the ratio obtained for the #math56#1p3/2→1d3/2 in 16O. Once again the centrifugal barrier is responsible for a
reduced renormalization compared to the expectations of nuclear matter
approach.