What can we learn from recent $2\nu\beta\beta$ experiments?

With recent measurements of the two neutrino double beta decay high precision electron spectra, combining with charge exchange or $\beta$-decay experimental data, we give severe constraints over the current nuclear many body calculations. Our calculation shows that Quasi-particle Random Phase Approximation (QRPA) approach can well reproduce the measured spectra for the two open shell nuclei, $^{82}$Se and $^{100}$Mo. For the closed shell nucleus $^{136}$Xe, QRPA can also reproduce the spectra with proper treatments. We also find that considering the high-lying state reduction, Nuclear Shell Model can also well reproduce the spectra as well as Gamow-Teller transition strength under a unique quenched Axial-vector coupling constant $g_A$. For $^{136}$Xe, we find that the flip of the sign for the decay strength will lead the spectra to go beyond the so-called High-lying state dominance hypothesis. These results call for future high precision measurement of charge-exchange reaction.