Massless limit and conformal soft limit for celestial massive amplitudes

In celestial holography, the massive and massless scalars in 4d space-time are represented by the Fourier transform of the bulk-to-boundary propagators and the Mellin transform of plane waves respectively. Recently, the 3pt celestial amplitude of one massive scalar and two massless scalars was discussed in arXiv:2312.08597. In this paper, we compute the 3pt celestial amplitude of two massive scalars and one massless scalar. Then we take the massless limit \(m\rightarrow 0\) for one of the massive scalars, during which process the gamma function \(\Gamma (1-\Delta )\) appears. By requiring the resulting amplitude to be well-defined, that is it goes to the 3pt amplitude of arXiv:2312.08597, the scaling dimension of this massive scalar has to be conformally soft \(\Delta \rightarrow 1\). The pole \(1/(1-\Delta )\) coming from \(\Gamma (1-\Delta )\) is crucial for this massless limit. Without it the resulting amplitude would be zero. This can be compared with the conformal soft limit in celestial gluon amplitudes, where a singularity \(1/(\Delta -1)\) arises and the leading contribution comes from the soft energy \(\omega \rightarrow 0\). The phase factors in the massless limit of massive conformal primary wave functions, discussed in arXiv:1705.01027, plays an import and consistent role in the celestial massive amplitudes. Furthermore, the subleading orders \(m^{2n}\) can also contribute poles when the scaling dimension is analytically continued to \(\Delta =1-n\) or \(\Delta = 2\), and we find that this consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators \(\Delta \in 2-{\mathbb {Z}}_{\geqslant 0}\) of massless bosons.