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 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 B(GT) strength under a unique quenched $g_A$. For $^{136}$Xe, we find that the flip of the sign for the decay strength will lead the spectra to go beyond HSD. These results call for future high precision measurement of charge-exchange reaction.