Investigation of $2\nu$ECEC in ${}^{132}$Ba and revisited for ${}^{78}$Kr

We present a theoretical investigation of two-neutrino double electron capture ($2\nu$ECEC) in ${}^{132}$Ba and ${}^{78}$Kr based on large-scale shell-model calculations. The nuclear matrix elements (NMEs) for the $2\nu$ECEC process in ${}^{132}$Ba and ${}^{78}$Kr have been calculated using the SN100PN and GWBXG effective interactions, respectively. The reliability of these interactions is first assessed by reproducing the spectroscopic properties of the parent, intermediate, and granddaughter nuclei of the decay. Two different formulations are employed to estimate the half-lives. We also examine the cumulative contribution of $1^{+}$ and $0^{+}$ states in the intermediate nuclei to the NMEs associated with the Gamow-Teller (GT) and Fermi transitions, respectively. Our results provide an improved prediction of the NME and half-life for $2\nu$ECEC in ${}^{78}$Kr compared to previous studies and offer a precise theoretical benchmark for future experimental investigations in ${}^{132}$Ba.