Efficient calculation of two-neutrino double-beta-decay nuclear matrix elements

Reliable nuclear matrix elements (NMEs) are essential for interpreting double-beta-decay experiments and for connecting measured or constrained half-lives to the underlying weak-interaction physics. The two-neutrino mode ($2\nu\beta\beta$) is allowed by the Standard Model and has been observed in several nuclei, whereas the neutrinoless mode ($0\nu\beta\beta$) remains the key experimental signature of lepton-number violation and Majorana neutrino masses. Recent statistical shell-model studies indicate a strong correlation between the $2\nu\beta\beta$ and $0\nu\beta\beta$ NMEs, making accurate and efficient calculations of the former especially useful for assessing the latter. Direct evaluations of $2\nu\beta\beta$ NMEs usually require summing over many $1^+$ states in the intermediate odd-odd nucleus, a procedure that becomes expensive and may converge slowly in large model spaces. We present and test an improved strength-function method based on Lanczos iterations that avoids full diagonalization while preserving the accuracy of explicit summation where such benchmarks are possible. The method is applied to several experimentally important emitters and to different effective Hamiltonians. We also show that the same framework can be used for the higher-order NMEs entering Taylor-expanded phase-space treatments of $2\nu\beta\beta$ and related decay modes.