Review of recent achievements on the two neutrino double beta decay

A critical analysis of recently developed approaches for the calculation of the two-neutrino double beta decay (2νββ-decay) nuclear matrix elements is presented. We discuss the Intermediate Nucleus Approach (INA), Operator Expansion Method (OEM) and the S-matrix approach in detail. Special attention is paid to the approximation schemes used in these types of calculations. The introduction of the time integral representation form of nuclear matrix elements, which is derived within the field theory, allows an analytical study of the validity of the different approximations. It is shown that the results obtained within the INA approaches, which is based on the quasi-boson approximation (QBA) and the renormalized QBA are questionable from physics point of view and that higher order approximations are necessary. We have reconsidered the OEM and its approximations which go beyond the QBA, and we have derived the OEM transitions operators in a consistent way. We have found that one of the basic aspects of the 2νββ-decay process is the energy difference between the initial and final nuclei. In addition, we show that the summation over the complete set of intermediate nuclear states in the present 2νββ-decay INA studies corresponds to a summation over a class of meson exchange diagrams. We conclude that the S-matrix approaches gives a more qualitative insight into the problem. Finally, we show that the two-nucleon mechanism considered presently does not describe accurately enough the two neutrino double beta decay amplitude and thus we propose additional new mechanisms for this process. (authors)