  We present new two-dimensional (2D) axisymmetric neutrino
radiation/hydrodynamic models of core-collapse supernova (CCSN) cores. We use
the CASTRO code, which incorporates truly multi-dimensional, multi-group,
flux-limited diffusion (MGFLD) neutrino transport, including all relevant
$\mathcal{O}(v/c)$ terms. Our main motivation for carrying out this study is to
compare with recent 2D models produced by other groups who have obtained
explosions for some progenitor stars and with recent 2D VULCAN results that did
not incorporate $\mathcal{O}(v/c)$ terms. We follow the evolution of 12, 15,
20, and 25 solar-mass progenitors to approximately 600 milliseconds after
bounce and do not obtain an explosion in any of these models. Though the reason
for the qualitative disagreement among the groups engaged in CCSN modeling
remains unclear, we speculate that the simplifying ``ray-by-ray' approach
employed by all other groups may be compromising their results. We show that
``ray-by-ray' calculations greatly exaggerate the angular and temporal
variations of the neutrino fluxes, which we argue are better captured by our
multi-dimensional MGFLD approach. On the other hand, our 2D models also make
approximations, making it difficult to draw definitive conclusions concerning
the root of the differences between groups. We discuss some of the diagnostics
often employed in the analyses of CCSN simulations and highlight the intimate
relationship between the various explosion conditions that have been proposed.
Finally, we explore the ingredients that may be missing in current calculations
that may be important in reproducing the properties of the average CCSNe,
should the delayed neutrino-heating mechanism be the correct mechanism of
explosion.
