Search for two neutrino double electron capture of ${}^{124}$Xe and ${}^{126}$Xe in the full exposure of the LUX detector

Two-neutrino double electron capture is a process allowed in the Standard Model of Particle Physics. This rare decay has been observed in ${}^{78}$Kr, ${}^{130}$Ba and more recently in ${}^{124}$Xe. In this publication we report on the search for this process in ${}^{124}$Xe and ${}^{126}$Xe using the full exposure of the Large Underground Xenon (LUX) experiment, in a total of of 27769.5~kg-days. No evidence of a signal was observed, allowing us to set 90% C.L. lower limits for the half-lives of these decays of $2.0\times {10}^{21}$~years for ${}^{124}$Xe and $1.9\times {10}^{21}$~years for ${}^{126}$Xe.