  In celestial conformal field theory (CCFT), the 4d massless scalars are
represented by 2d conformal operators with conformal dimensions
$h=\bar{h}=(1+i\lambda)/2$. The Mellin transform of 4d massless scalar
amplitudes gives the conformal correlators of CCFT. We study the conformal
block decomposition of these celestial correlators in CCFT and obtain the
explicit blocks. This conformal block decomposition is highly nontrivial, even
for the simplest 4d massless scalar amplitude. We use the analytic continuation
of the Appell hypergeometric function $F_1$ and the method of monodromy
projection of conformal blocks, to achieve this block decomposition. This
procedure is consistent with the crossing symmetry, in both the
correlator-level and each explicit block-level. We also investigate its
behavior in the conformal soft limit and find that the Appell hypergeometric
function $F_1$ does not reduce to the Gauss hypergeometric function. This is
different from the block decomposition of celestial gluons we studied before,
where the Appell hypergeometric function $F_1$ reduces to the Gauss
hypergeometric function. This difference comes from the shift of conformal
dimensions and is the reason why we adopt the new method here for the block
decomposition of celestial massless scalars.
