Massless limit and conformal soft limit for celestial massive amplitudes

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 $$m\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for one of the massive scalars, during which process the gamma function $$\Gamma (1-\Delta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\Delta \rightarrow 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The pole $$1/(1-\Delta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> coming from $$\Gamma (1-\Delta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$1/(\Delta -1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> arises and the leading contribution comes from the soft energy $$\omega \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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 $$m^{2n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>m</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> can also contribute poles when the scaling dimension is analytically continued to $$\Delta =1-n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> or $$\Delta = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and we find that this consistent massless limit only exists for dimensions belonging to the generalized conformal primary operators $$\Delta \in 2-{\mathbb {Z}}_{\geqslant 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>∈</mml:mo> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> of massless bosons.