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We present a precision analysis of the $^{136}$Xe two-neutrino $\beta\beta$electron spectrum above 0.8 MeV, based on high-statistics data obtained withthe KamLAND-Zen experiment. An improved formalism for the two-neutrino$\beta\beta$ rate allows us to measure the ratio of the leading and subleading$2\nu\beta\beta$ nuclear matrix elements (NMEs), $\xi^{2\nu}_{31} =-0.26^{+0.31}_{-0.25}$. Theoretical predictions from the nuclear shell modeland the majority of the quasiparticle random-phase approximation (QRPA)calculations are consistent with the experimental limit. However, part of the$\xi^{2\nu}_{31}$ range allowed by the QRPA is excluded by the presentmeasurement at the 90% confidence level. Our analysis reveals that predicted$\xi^{2\nu}_{31}$ values are sensitive to the quenching of NMEs and thecompeting contributions from low- and high-energy states in the intermediatenucleus. Since these aspects are also at play in neutrinoless $\beta\beta$decay, $\xi^{2\nu}_{31}$ provides new insights towards reliable neutrinoless$\beta\beta$ NMEs.

Precision Analysis of the $^{136}$Xe Two-Neutrino $\beta\beta$ Spectrum in KamLAND-Zen and Its Impact on the Quenching of Nuclear Matrix Elements