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Description
Here we discuss the 3pt celestial amplitude of two massive scalars and one massless scalar. In the massless limit $m\to 0$ for one of the massive scalars, 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 $\Delta \to 1$. The pole $1/(1-\Delta)$ coming from $\Gamma(1-\Delta)$ is crucial for this massless limit. Without it the resulting amplitude would be zero. 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 $\Delta = 2$. This consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators $\Delta \in 2-\mathbb{Z}_{\geq 0}$ of massless bosons.
