Massless limit and conformal soft limit for celestial massive amplitudes

Abstract

Abstract 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 $$m\rightarrow 0$$ m → 0 for one of the massive scalars, during which process the gamma function $$\Gamma (1-\Delta )$$ Γ ( 1 - Δ ) 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 $$\Delta \rightarrow 1$$ Δ → 1 . The pole $$1/(1-\Delta )$$ 1 / ( 1 - Δ ) coming from $$\Gamma (1-\Delta )$$ Γ ( 1 - Δ ) 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 $$1/(\Delta -1)$$ 1 / ( Δ - 1 ) arises and the leading contribution comes from the soft energy $$\omega \rightarrow 0$$ ω → 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 $$m^{2n}$$ m 2 n can also contribute poles when the scaling dimension is analytically continued to $$\Delta =1-n$$ Δ = 1 - n or $$\Delta = 2$$ Δ = 2 , and we find that this consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators $$\Delta \in 2-{\mathbb {Z}}_{\geqslant 0}$$ Δ ∈ 2 - Z ⩾ 0 of massless bosons.