than those in @takiwaki12, in which the cooling by heavy-lepton neutrinos was not taken into account. Due to the inclusion of the $\nu_x$ cooling, the (angle-averaged) $\bar{\nu}_e$ luminosity decreases more quickly after bounce (compare Figure \[fig3\] and Figure 14 of [@takiwaki12]), which leads to the less energetic shock expansion in this study. It should be mentioned that by comparing our $\nu_x$ luminosity estimated by the leakage scheme with that obtained by the work by @buras06 with detailed neutrino transport, the peak luminosity is more than 20 % smaller in our case. Such underestimation of cooling by heavy-lepton neutrinos should lead to artificially larger maximum shock extent ($R_{\rm max} \sim 260$ km, blue line in the right panel of Figure \[fig2\]) compared to $R_{\rm max} \sim170$ km in @buras06. We have to emphasize that the use of the leakage scheme together with the omission of inelastic neutrino scattering on electrons and GR effects in the present scheme is likely to facilitate artificially easier explosions. Regarding our 2D models, the relatively earlier shock revival ($\sim 100$ ms postbounce) coincides with the decline of the mass accretion rate onto the central PNS. This could be the reason that the timescale is similar to 