Improved description of the $2νββ$-decay and a possibility to determine the effective axial-vector coupling constant

An improved formalism of the two-neutrino double-beta decay ($2νββ$-decay) rate is presented, which takes into account the dependence of energy denominators on lepton energies via the Taylor expansion. Till now, only the leading term in this expansion has been considered. The revised $2νββ$-decay rate and differential characteristics depend on additional phase-space factors weighted by the ratios of $2νββ$-decay nuclear matrix elements with different powers of the energy denominator. For nuclei of experimental interest all phase-space factors are calculated by using exact Dirac wave functions with finite nuclear size and electron screening. For isotopes with measured $2νββ$-decay half-life the involved nuclear matrix elements are determined within the quasiparticle random phase approximation with partial isospin restoration. The importance of correction terms to the $2νββ$-decay rate due to Taylor expansion is established and the modification of shape of single and summed electron energy distributions is discussed. It is found that the improved calculation of the $2νββ$-decay predicts slightly suppressed $2νββ$-decay background to the neutrinoless double-beta decay signal. Further, a novel approach to determine the value of effective weak-coupling constant in nuclear medium $g^{\rm eff}_{\rm A}$ is proposed.