Massless limit and conformal soft limit for celestial massive amplitudes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913661366304768 |
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| author | Fan, Wei |
| author_facet | Fan, Wei |
| contents | 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 $Γ(1-Δ)$ 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 $Δ\to 1$. The pole $1/(1-Δ)$ coming from $Γ(1-Δ)$ 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 $1/(Δ-1)$ arises and the leading contribution comes from the soft energy $ω\to 0$. The phase factors in the massless limit of massive conformal primary wave functions, dicussed 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 $Δ=1-n$ or $Δ= 2$, and we find that this consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators $Δ\in 2-\mathbb{Z}_{\geqslant 0}$ of massless bosons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Massless limit and conformal soft limit for celestial massive amplitudes Fan, Wei High Energy Physics - Theory 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 $Γ(1-Δ)$ 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 $Δ\to 1$. The pole $1/(1-Δ)$ coming from $Γ(1-Δ)$ 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 $1/(Δ-1)$ arises and the leading contribution comes from the soft energy $ω\to 0$. The phase factors in the massless limit of massive conformal primary wave functions, dicussed 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 $Δ=1-n$ or $Δ= 2$, and we find that this consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators $Δ\in 2-\mathbb{Z}_{\geqslant 0}$ of massless bosons. |
| title | Massless limit and conformal soft limit for celestial massive amplitudes |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2404.05137 |