Celestial conformal blocks of massless scalars and analytic continuation of the Appell function $F_1$

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.