Recursive method for computing matrix elements for two-body interactions

A recursive method for the efficient computation of two-body matrix elements is presented. The method consists of a set of recursion relations for the computationally demanding radial integral and adds one more tool to the set of computational methods introduced by Horie and Sasaki [H. Horie and K. Sasaki, Prog. Theor. Phys. 25, 475 (1961)]. The neutrinoless double-$\beta$ decay will serve as the primary application and example, but the method is general and can be applied equally well to other kinds of nuclear structure calculations involving matrix elements of two-body interactions.

Figure 1

CPU time comparison between our old and new $0\nu \beta \beta$ codes for computing the nuclear matrix elements $M_{\mathrm{F}}^{\left(0\nu \right)}$ and $M_{\mathrm{GT}}^{\left(0\nu \right)}$. Toy calculation uses just four oscillator basis states, realistic calculation uses ten. Speed-up factors for these calculations are 26 (toy) and 35 (realistic). These particular computations were done for the nucleus ${}^{76}\mathrm{Ge}$.

Figure 2

CPU time consumption for the computation of the two-body matrix elements with our new $0\nu \beta \beta$ code. These particular computations were done for the nucleus ${}^{76}\mathrm{Ge}$.