Uncertainties in nuclear transition matrix elements for ${\beta }^{+}{\beta }^{+}$ and $\epsilon {\beta }^{+}$ modes of neutrinoless positron double-$\beta$ decay within PHFB model

Uncertainties in the nuclear transition matrix elements $M^{(0\nu )}$ and $M^{(0N)}$ of the double-positron emission $({\beta }^{+}{\beta }^{+}{)}_{0\nu }$ and electron-positron conversion $(\epsilon {\beta }^{+}{)}_{0\nu }$ modes due to the exchange of light and heavy Majorana neutrinos, respectively, are calculated for ${}^{96}$Ru, ${}^{102}$Pd, ${}^{106}$Cd, ${}^{124}$Xe, ${}^{130}$Ba and ${}^{156}$Dy isotopes by employing the PHFB model with four different parameterization of the pairing plus multipolar two-body interactions and three different parameterizations of the Jastrow short range correlations. In all cases but for ${}^{130}$Ba, the uncertainties are smaller than 14% for light Majorana neutrino exchange and 35% for the exchange of a heavy Majorana neutrino.