Eur. Phys. J. C (2025) 85: 55
https://doi.org/10.1140/epjc/s10052-025-13762-5

Regular Article - Theoretical Physics

Massless limit and conformal soft limit for celestial massive amplitudes

Department of Physics, School of Science, Jiangsu University of Science and Technology, 212114, Zhenjiang, China

^a fanwei@just.edu.cn

Received: 1 October 2024
Accepted: 27 December 2024
Published online: 22 January 2025

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 $m\to 0$ for one of the massive scalars, during which process the gamma function $\mathrm{\Gamma }(1-\mathrm{\Delta })$ 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 $\mathrm{\Delta }\to 1$. The pole $1/(1-\mathrm{\Delta })$ coming from $\mathrm{\Gamma }(1-\mathrm{\Delta })$ 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 $1/(\mathrm{\Delta }-1)$ arises and the leading contribution comes from the soft energy $\omega \to 0$. 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 $\mathrm{\Delta }=1-n$ or $\mathrm{\Delta }=2$, and we find that this consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators $\mathrm{\Delta }\in 2-{\mathbb{Z}}_{⩾0}$ of massless bosons.