  We consider the Bose-Einstein transition of homogeneous weakly interacting
spin-0 particles based on the normal-state Phi-derivable approximation.
Self-consistent calculations of Green's function and the chemical potential
with several approximate Phi's are performed numerically as a function of
temperature near Tc, which exhibit qualitatively different results. The ladder
approximation apparently shows a continuous transition with the prefactor
c=2.94 for the transition-temperature shift given in terms of the scattering
length a and density n. In contrast, the second-order, particle-hole, and
fluctuation-exchange approximations yield a first-order transition. The fact
that some standard Phi's predict a first-order transition challenges us to
clarify whether or not the transition is really continuous.
