Conclusions

We have analyzed in detail the renormalization of the axial charge in nuclei by evaluating the matrix elements of the axial charge operator with relativistic wave functions, solutions of the Dirac equation with the relativistic potential, and with non relativistic wave function, solutions of the Schrödinger equation with an equivalent non relativistic potential. We have found renormalization effects due to the use of the relativistic wave functions, enhancing the axial charge in the direction found in earlier perturbative approaches for nuclear matter. However, the quantitative results differ from the estimates derived for nuclear matter significantly. Using the G-matrix derived from a realistic meson exchange model of the NN interaction [#rupr##1###] a perturbative estimate of the heavy meson exchange current contribution to the axial charge at nuclear matter saturation density [#7a##1###] would yield a renormalization factor of 1.4 and a non perturbative treatment would lead to enhancement as large as 1.7. For finite nuclei the enhancement factors considerably smaller, of the order of 1.2 - 1.3. We argue that this reduction of the renormalization effect is due to the smaller densities at the surface of finite nuclei, which are relevant for the evaluation of actual matrix elements. From these considerations we can also understand the dependence of the renormalization factor on the momentum transfer and on the transition actually considered.

The amount of axial charge renormalization depends on the model for the NN interaction. We have employed a relativistic meson exchange model (Potential version A of the Bonn potential [#rupr##1###]), which has been derived to reproduce NN scattering data. It is fair to quote at this point that using this potential in the present case there is the assumption that the relativistic potential constructed to reproduce NN scattering of on shell nucleons can be extrapolated to deal with negative energy states and on shell and off shell conditions. This is certainly a strong assumption from which all the microscopically constructed relativistic potentials suffer, and indeed different parametrizations of the NN amplitude on shell lead to different relativistic potentials [#11##1###]. Some efforts have been done to constrain the relativistic potential to be consistent with the #tex2html_wrap_inline1414#
N elementary amplitudes [#12##1###] and this leads to potentials like the one obtained here but about one half their strength. Even then this potential is constructed at the level of the impulse approximation or low density limit, tρ, and many body effects should modify it. It is clear that many efforts are still necessary to be able to claim that an unambiguous microscopical relativistic potential has been determined. On the other hand one can take a more phenomenological approach and say that a certain relativistic potential has a wide degree of phenomenological success, providing fair nuclear binding energies, spin-orbit splitting, nucleon nucleus cross sections and polarization observables, etc. [#10##1###,#klein##1###]. The potential we have used is one of such and provides empirical support for the axial charge renormalization found, but this does not exclude the possibility of other potentials with the same degree of phenomenological success and still providing different axial charge renormalization. The ultimate answer to this question is tied to the progress in our understanding of the meaning and accurate strength of the relativistic potential. Meanwhile, by using a fair and plausible model we have done detailed calculations and showed that the results are sufficiently different from the perturbative results to encourage the use of the present approach in future works dealing with the problem.

Two of us, A. Gil and E. Oset wish to acknowledge the hospitality of the University of Tübingen and H. Müther the one of the University of Valencia. E. Oset acknowledges support from the Humboldt Foundation. The work has been partially supported by the EU, program, Human Capital and Mobility contract no. CHRX-CT 93-0323, the CICYT contract no. AEN 93-1205 and the Graduiertenkolleg ``Struktur und Wechselwirkung von Hadronen und Kernen'' of the Deutsche Forschungsgemeinschaft (DFG Mu 705/3)

<#370#>Appendix: Matrix elements of the axial charge operator.<#370#>

<#371#>A) Relativistic case:<#371#> We write here the matrix element for the #math57#γ⁰γ_s operator between relativistic wave functions

#math58#

We distinguish 4 cases

#math59#

and the resulting matrix element is

#math60#

#math61#

where A_i is given for each of the cases a) b) c) d) listed above by

#math62#

#math63#

#math64#

#math65#

#math66#

#math67#

#math68#

#math69#

#math70#

#math71#

#math72#

#math73#

<#434#>B) Non relativistic case:<#434#> we evaluate matrix elements of the #math74##tex2html_wrap_inline1465#
(#tex2html_wrap_inline1466#
+ #tex2html_wrap_inline1467#
)/2M operator between non relativistic states

#math75#

The derivation of this matrix elements requires a bit more algebra than the non relativistic case. With the help of some useful formulas from the appendix of ref. [#13##1###] we obtain the following result

#math76#

#math77#

#math78#

#math79#

with #math80#λ + l + l' an odd number, where the last function is given by

#math81#

#math82#

#math83#

#math84#

#math85#

#math86#

#math87#