Future researches

One of the main subject of the future researches is to generalize existing procedures and develop new approximations used in description of many-body nuclear wave-functions which will than be applied in searches of the neutrinoless double beta decay, cold dark matter and the double charge exchange with pions. In other words, one expects to achieve reasonable description of finite many-body nuclear system via some extension of the Quasiparticle Random Phase Approximation in a way which will help to enhance the understanding of such complex system considerably.

The following topics are potential subjects of interest:

$\rightarrow$ The application of the usual QRPA to charge-changing modes was used by several authors to investigate beta+ decay in heavier nuclei, to double charge exchange reactions and double beta decay. Most of such calculations have shown that nuclear matrix elements of the above mentioned processes are sensitive to the particle-particle interaction. On the other hand, comparison between the pnQRPA and exact shell-model indicates that usual QRPA could not reproduce shell-model results without adjusting the parameters in the pnQRPA equation. Attempts to improve the QRPA results by developing projected QRPA or extend it by involving proton-neutron pairing could be of some importance in this context. Also inclusion of the so-called higher order corrections to the QRPA or new modes of excitations (B-modes) could offer a satisfactory solution of difficulties connected with sensitivity of QRPA upon the particle-particle interaction strength. But applying all such corrections and approximation one should keep in mind that many particle excitations are obviously missing in the normal pnQRPA. Also the Pauli exclusion principle is violated more and more with increasing ground state correlations. Thus the so-called renormalized version of the QRPA method should be applied. It has been shown recently that it reduces the dependence of the calculated matrix elements or cross sections on the strength of the particle-particle interaction. On the other hand the self-consistency between the description of the ground state and the excited states is broken in this approach. One should develop a self-consistent version of the QRPA taking also the ground-state correlations into account. It is clear as well that the QRPA equation of motion corresponds exactly to the full many-body Schrödinger equation if and only if the ground state in the equation of motion is the true ground state and a phonon creation operator exhausts the whole Hilbert space, e.g. all excitations: 1p(article)-1h(ole), 2p-2h, 3p-3h, and so on. Till now only one-particle-one-hole creation and destruction operators are used.

$\rightarrow$ One is going to use such tested nuclear many-body wave functions to calculate nuclear matrix elements in the neutrinoless double beta decay and SUSY/CDM searches and to deduce limits on nonstandard physics parameters: the mass of Majorana light electron neutrino, the mass of right-handed vector boson, the mixing angle between left- and right-handed vector bosons, the R-parity breaking constant and the strength of neutralino interaction with nuclear matter.

$\rightarrow$ These wave functions can be also used in description of the DCX reaction with pions. Improvements of the model of the reaction can involve:

1.

The full and consistent treatment of the neutral pion propagation between reaction vertices in the s-wave sequential mechanism. The formalism, based on the S-matrix approach has been already developed.

2.

Taking into account distortion and absorption effects of pions, both incoming/outgoing and intermediate. The wave functions of the charged pions should be taken as a solutions of the Klein-Gordon equation with the pion-nucleus optical potential with phenomenological parameters fitted to pionic single charge exchange and scattering data. The necessary modifications of the model have been outlined in and have been extended and applied in some cases. Medium modifications of neutral pion propagation between two nucleons has been studied -- accommodation of these results is not straightforward and some approximations should be developed. As previously these effects should result in reduction of the cross-section above $T_\pi=50\,$MeV giving more resonance-like shape of the cross-section vs. energy curve.

3.

The answer to the still opened question how far the DCX processes are influenced by the isobaric degrees of freedom of the nucleon. Delta isobars are supposed to pre-exist in the ground state of the nucleus or can be virtually excited by an incoming pion, giving rise to other mechanisms than the conventional two-nucleon route. Although they start playing the main role for the pion kinetic energies around 100 MeV, they can give some contribution in the low-energy region too.

$\rightarrow$ We plan the extension of the neutrinoless double beta decay description to the framework of MSSM with supersymmetry breaking mediated by gravitational interactions (SUGRA). When confronted with last communications of Super-Kamiokande and LSND collaborations, it makes this version of MSSM the extremely interesting framework for studies of the neutrinoless double beta decay. Such an approach allows to obtain limits on the supersymmetry breaking parameters, using standard (motivated by supergravity theories) technique of unification of proper supersymmetric parameters at the GUT scale, followed by renormalization group (RGE) evolution to the weak scale.

$\rightarrow$ The next task is connected with recently developed idea of supersymmetry breaking mediated by gauge interactions. In these models, the technique of finding low-scale SUSY parameters is different: the mechanism of gauge mediation sets the values of scalar masses at the so called Messenger scale M, reducing huge in the case of gravity-mediated MSSM number of free parameters, and, after RGE evolution, effecting with interesting phenomenology of low-energy world. Moreover, it has been found that limits on R-parity breaking by trilinear (not bilinear LH) terms in gauge mediated model coming from non-observability of the neutrinoless double beta decay in ⁷⁶Ge are much stronger than in the usual MSSM. We propose to extend the studies to the case of bilinear term, with bigger sample of nuclei and improved nuclear part of calculations.