Massless limit and conformal soft limit for celestial massive amplitudes

Celestial conformal field theory (CCFT) is a flat holography [1, 2, 3, 4] of Minkowski spacetime. The bulk QFT in Minkowski spacetime $\mathbb{R}^{1,3}$ can be recast as a boundary CFT in the celestial sphere. The amplitudes are converted from the momentum basis to the boost basis (the so-called conformal primary wavefunction) [5] and then become the CFT correlation functions. The consequence of 4d soft theorem to celestial holography is one of the import problems. For example, the soft energy $\omega\to 0$ generates the leading contribution to the conformal soft limit $\Delta\to 1$ in celestial gluon amplitudes [6, 7, 8, 9].

A natural question to ask is what would be the corresponding physics related with the conformal soft limit of celestial massive amplitudes? We address this question in this paper using the 3pt celestial amplitude of two massive scalars and one massless scalar. We find that the massless limit $m\to 0$ of this amplitude leads to the conformal soft limit $\Delta\to 1$, to make the resulting amplitude well-defined. On the basis level, the massless limit of the conformal primary wave functions has a phase factor $m^{-\Delta}$ [5]. After going to the celestial amplitudes, the massless limit is accompanied by the gamma function $\Gamma(1-\Delta)$, which contributes a pole in the conformal soft limit $\Delta\to 1$. This resembles, in spirit, to the conformal soft limit of celestial gluons, which appears in their amplitudes rather than conformal primary waves.

The celestial massive amplitudes are hard to compute in 4d space-time. In the original paper of massive conformal primary waves [10], the 3pt celestial amplitudes of three massive scalars was computed in the near-extremal limit $m_{1}=2(1+\epsilon)m,m_{2}=m_{3}=m$. The result is the tree-level 3pt Witten diagram in the leading order of $\sqrt{\epsilon}$ when $\epsilon\to 0$. The 3pt celestial amplitude of one massive scalar and two massless scalars $\langle\Phi_{\Delta_{1}}^{m_{1}}\phi_{\Delta_{2}}\phi_{\Delta_{3}}\rangle$ was computed recently by [11]¹¹1This kind of celestial amplitudes was firstly discussed by  [12] in 3d space-time, for the implications of the 4d optical theorem on celestial holography., where the non-local behavior of massive states on celestial sphere was discussed. Since massive states are highly nontrivial in celestial holography, it is meaningful to investigate the physics related with their conformal soft limit. Here we continue to compute the 3pt celestial amplitude of two massive scalars and one massless scalar $\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle$. Using the standard method of hyperbolic coordinates and Feynman parametrization, the two masses $m_{1}$ and $m_{2}$ appear only in the finite integral of the Feynman parameters. Then we can safely take the massless limit $m_{2}\to 0$. Combined with the conformal soft limit $\Delta_{2}\to 1$, the 3pt massive amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle$ has the desired limit $\langle\Phi_{\Delta_{1}}^{m_{1}}\phi_{\Delta_{2}}\phi_{\Delta_{3}}\rangle$.

On the basis level, the massless limit of the massive conformal primary wave function leads to two massless fields $\Phi_{\Delta_{2}}^{m}\to\phi_{\Delta_{2}},\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\phi_{\tilde{\Delta}_{2}}$}{\stretchto{\scaleto{\SavedStyle\mkern 0.2mu\sim}{.5467}}{0.5}}{O}{c}{F}{T}{S}}$ [5], which are the same for $\Delta_{2}=1$. They are different for general values of $\Delta_{2}$. By studying subleading orders $m_{2}^{2n}$ of the massless limit, we find that it leads to the 3pt amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\phi_{\Delta_{2}}\phi_{\Delta_{3}}\rangle$ for the analytically continued dimension $\Delta_{2}=1-n$, and to the 3pt amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\phi_{\tilde{\Delta}_{2}}$}{\stretchto{\scaleto{\SavedStyle\mkern 0.2mu\sim}{.5467}}{0.5}}{O}{c}{F}{T}{S}}\phi_{\Delta_{3}}\rangle$ for the analytically continued dimension $\Delta_{2}=2$. For analytically continued dimension $\Delta_{2}\geq 3$, there is a massless limit but it is neither $\langle\Phi_{\Delta_{1}}^{m_{1}}\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\phi_{\tilde{\Delta}_{2}}$}{\stretchto{\scaleto{\SavedStyle\mkern 0.2mu\sim}{.5467}}{0.5}}{O}{c}{F}{T}{S}}\phi_{\Delta_{3}}\rangle$ nor $\langle\Phi_{\Delta_{1}}^{m_{1}}\phi_{\Delta_{2}}\phi_{\Delta_{3}}\rangle$. So it is only for dimensions $\Delta_{2}\leq 2$ that the massless limit of this amplitude is consistent. These dimensions happen to belong to the range of generalized conformal primary operators $\Delta\in 2-\mathbb{Z}_{\geqslant 0}$ of massless bosons [13]. So the consistent massless limit of $\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle$ picks up these generalized conformal primary operators. Note that this amplitude was also discussed recently in [14], whose results and physics content have no overlap with us.

The paper is organized as follows. In Section 2, we give a short account of the notation and briefly review the 3pt celestial amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\phi_{\Delta_{2}}\phi_{\Delta_{3}}\rangle$. Then we compute the shadowed 3pt amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\phi_{\tilde{\Delta}_{2}}$}{\stretchto{\scaleto{\SavedStyle\mkern 0.2mu\sim}{.5467}}{0.5}}{O}{c}{F}{T}{S}}\phi_{\Delta_{3}}\rangle$. In Section 3, we compute the 3pt celestial amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle$ using the Feynman parametrization. The complete result of this amplitude is put to the appendix. In Section 4, we investigate the massless limit $m_{2}\to 0$ and the conformal soft limit $\Delta_{2}\to 1$. Then we discuss the subleading terms $m_{2}^{2n}$ and the analytic continuation to $\Delta_{2}=1-n$ and $\Delta_{2}=2$. We conclude with a discussion of open questions in Section 5.

In CCFT, the massless four-momentum $p_{i}^{\mu}=\epsilon_{i}\omega_{i}\hat{q}_{i}^{\mu}$ is parameterized by the energy $\omega_{i}$ and the point $(w_{i},\bar{w}_{i})$ of the celestial sphere via the formula [5]

$\displaystyle\hat{q}_{i}=(1+w_{i}\bar{w}_{i},w_{i}+\bar{w}_{i},-i(w_{i}-\bar{w}_{i}),1-w_{i}\bar{w}_{i}),$

(II.1)

where $\epsilon_{i}=\pm 1$ represents outgoing/incoming momentum. During the computation, we use the coordinate $\vec{w}_{i}=(\operatorname{Re}(w_{i}),\operatorname{Im}(w_{i}))$ for the celestial sphere, in order to save space. Each 4d massless scalar corresponds to a scalar conformal primary wave function $\phi_{\Delta}(X^{\mu};\vec{w})$ (the so-called boost basis) via the Mellin transform

$\displaystyle\phi_{\Delta}^{(\pm)}(X^{\mu};\vec{w})\coloneqq\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{\pm i\omega\hat{q}(\vec{w})\cdot X-\varepsilon\omega},$

(II.2)

where $\varepsilon>0$ is a regularization parameter and the conformal dimensions are $h_{i}=\bar{h}_{i}=\Delta_{i}/2=(1+i\lambda_{i})/2,\lambda_{i}\in\mathbb{R}$.

The massive four-momentum $p_{i}^{\mu}=\epsilon_{i}m_{i}\hat{p}^{\mu}$ is parameterized by the mass of the particle $m_{i}$ and the hyperbolic coordinate $(y_{i},\vec{z}_{i})$ via the formula [10]

$\displaystyle\hat{p}_{i}^{\mu}(y_{i},\vec{z}_{i})=\left(\frac{1+y_{i}^{2}+|z_{i}|^{2}}{2y_{i}},\frac{\vec{z}_{i}}{y_{i}},\frac{1-y_{i}^{2}-|z_{i}|^{2}}{2y_{i}}\right).$

(II.3)

The massive scalar corresponds to a scalar conformal primary wave $\Phi_{\Delta}^{m}(X^{\mu};\vec{w})$ via the Fourier transform

$\displaystyle\Phi_{\Delta}^{m(\pm)}(X^{\mu};\vec{w})\coloneqq\int_{0}^{\infty}\frac{dy}{y^{3}}\int d^{2}zG_{\Delta}(y,\vec{z};\vec{w})e^{\pm im\hat{p}(y,\vec{z})\cdot X},$

(II.4)

where $G_{\Delta}(y,\vec{z};\vec{w})$ is the bulk-to-boundary propagator with scaling dimension $\Delta$

$\displaystyle G_{\Delta}(\hat{p}(y,\vec{z});\hat{q}(\vec{w}))=\frac{1}{(-\hat{p}\cdot\hat{q})^{\Delta}}=\left(\frac{y}{y^{2}+|\vec{z}-\vec{w}|^{2}}\right)^{\Delta}.$

(II.5)

The Lorentz transformation $\Lambda^{\mu}_{\nu}\in SO(1,3)$ acts non-linearly on the coordinates $\vec{w}\to\vec{w}^{\prime}(\vec{w}),\vec{z}\to\vec{z}^{\prime}(y,\vec{z}),y\to y^{\prime}(y,\vec{z})$, but the four-momenta transform linearly

$\displaystyle\hat{p}^{\mu}\left(y^{\prime},\vec{z}^{\prime}\right)=\Lambda^{\mu}{}_{\nu}\hat{p}^{\nu},\,q^{\mu}\left(\vec{w}^{\prime}\right)=\left|\frac{\partial\vec{w}^{\prime}}{\partial\vec{w}}\right|^{1/2}\Lambda_{\nu}^{\mu}q^{\nu}(\vec{w}).$

(II.6)

Then the conformal primary waves have the conformal symmetry on the celestial sphere

$\displaystyle\Big{\{}\Phi_{\Delta}^{m},\phi_{\Delta}\Big{\}}(\Lambda_{\nu}^{\mu}X^{\nu};\vec{w}^{\prime}(\vec{w}))=\left|\frac{\partial\vec{w}^{\prime}}{\partial\vec{w}}\right|^{-\Delta/2}\Big{\{}\Phi_{\Delta}^{m},\phi_{\Delta}\Big{\}}(X^{\mu};\vec{w}),$

(II.7)

where we write $\Phi_{\Delta}^{m}(X;\vec{w})$ and $\phi_{\Delta}(X;\vec{w})$ together to save space.

In the massless limit, the massive conformal primary wavefunction $\Phi_{\Delta}^{m(\pm)}(X^{\mu};\vec{w})$ has the following behavior [5]

$\displaystyle\Phi_{\Delta}^{m}(X;\vec{w})\overset{m\rightarrow 0}{\longrightarrow}\left(\frac{m}{2}\right)^{-\Delta}\frac{\pi\Gamma(\Delta-1)}{\Gamma\left(\Delta\right)}\phi_{\Delta}(X^{\mu};\vec{w})+\left(\frac{m}{2}\right)^{-\tilde{\Delta}}\frac{\pi\Gamma(\tilde{\Delta}-1)}{\Gamma\left(\tilde{\Delta}\right)}\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\phi_{\tilde{\Delta}}$}{\stretchto{\scaleto{\SavedStyle\mkern 0.2mu\sim}{.5467}}{0.5}}{O}{c}{F}{T}{S}}(X^{\mu};\vec{w})$

where $\tilde{\Delta}=2-\Delta$ is the shadow dimension and $\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\phi_{\tilde{\Delta}}$}{\stretchto{\scaleto{\SavedStyle\mkern 0.2mu\sim}{.5467}}{0.5}}{O}{c}{F}{T}{S}}$ is the shadow operator of $\phi_{\tilde{\Delta}}$. In the basis level, the massless limit is not well-defined, due to this phase factor $m^{-\Delta}$. In the amplitude level, however, things are different. The physical interaction between particles make the massless limit well-defined, provided that the conformal soft limit $\Delta\to 1$ is together with the massless limit $m\to 0$. This is what we will show in this paper. It turns out that the phase factor $m^{-\Delta}$ is crucial for the massless limit of celestial amplitudes.

For convenience we choose the 1st particle to be incoming, the 2nd and the 3rd to be outgoing. The 3pt celestial amplitude $\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle$ is given by the following integral

$\displaystyle\langle\Phi_{\Delta_{1}}^{m_{1}}(\vec{w}_{1})\Phi_{\Delta_{2}}^{m_{2}}(\vec{w}_{2})\phi_{\Delta_{3}}(\vec{w}_{3})\rangle=i\lambda\int d^{4}X\,\Phi_{\Delta_{1}}^{m_{1}(-)}\left(X;\vec{w}_{1}\right)\Phi_{\Delta_{2}}^{m_{2}(+)}\left(X;\vec{w}_{2}\right)\phi_{\Delta_{3}}^{(+)}\left(X;\vec{w}_{3}\right).$

(III.1)

After the integration over $X$ it becomes

$\displaystyle i\lambda(2\pi)^{4}\displaystyle{\int\left(\prod_{i=1}^{2}\frac{dy_{i}d^{2}z_{i}}{y_{i}^{3}}\left(\frac{y_{i}}{y_{i}^{2}+|\vec{z}_{i}-\vec{w}_{i}|^{2}}\right)^{\Delta_{i}}\right)\int\frac{d\omega_{3}}{\omega_{3}}\omega_{3}^{\Delta_{3}}\delta^{4}(-m_{1}\hat{p}_{1}+m_{2}\hat{p}_{2}+\omega_{3}\hat{q}_{3})}.$

(III.2)

The momentum conservating delta function can be evaluated as

$\displaystyle\delta^{4}(-m_{1}\hat{p}_{1}+m_{2}\hat{p}_{2}+\omega_{3}\hat{q}_{3})=\frac{y_{1}^{4}(m_{1}^{2}y_{2}^{2}+m_{2}^{2}(\vec{z}_{2}-\vec{w}_{3})^{2})}{m_{1}^{3}m_{2}^{2}(y_{2}^{2}+(\vec{z}_{2}-\vec{w}_{3})^{2})^{2}}\delta(y_{1}-y_{1}^{*})\delta^{2}(\vec{z}_{1}-\vec{z}_{1}^{*})\delta(\omega_{3}-\omega_{3}^{*}),$

(III.3)

where the factor is the Jacobian coming from solving the momentum conservation using hyperbolic coordinates. The fixed integration variables are as following

	$\displaystyle y_{1}^{*}$	$\displaystyle=\frac{m_{1}m_{2}y_{2}(y_{2}^{2}+(\vec{z}_{2}-\vec{w}_{3})^{2})}{m_{1}^{2}y_{2}^{2}+m_{2}^{2}(\vec{z}_{2}-\vec{w}_{3})^{2}}$
	$\displaystyle\vec{z}_{1}^{*}$	$\displaystyle=\frac{m_{2}^{2}(y_{2}^{2}+(\vec{z}_{2}-\vec{w}_{3})^{2})\vec{z}_{2}+y_{2}^{2}(m_{1}^{2}-m_{2}^{2})\vec{w}_{3}}{m_{1}^{2}y_{2}^{2}+m_{2}^{2}(\vec{z}_{2}-\vec{w}_{3})^{2}}$
	$\displaystyle\omega_{3}^{*}$	$\displaystyle=\frac{y_{2}(m_{1}^{2}-m_{2}^{2})}{2m_{2}(y_{2}^{2}+(\vec{z}_{2}-\vec{w}_{3})^{2})}.$		(III.4)

Since $y\geq 0,\omega\geq 0$, the kinematics constrains the masses to be $m_{1}\geq m_{2}$.

This integral can be simplified by the conformal symmetry (II.7). This conformal symmetry fixes the $\vec{w}$ dependence of the integral to be

$\displaystyle\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle\propto\frac{1}{|\vec{w}_{12}|^{\Delta_{1}+\Delta_{2}-\Delta_{3}}|\vec{w}_{13}|^{\Delta_{1}+\Delta_{3}-\Delta_{2}}|\vec{w}_{23}|^{\Delta_{2}+\Delta_{3}-\Delta_{1}}}.$

(III.5)

The exact factor of this proportion can be computed by using the famous linear fractional transformation to fix the three points on the celestial sphere

$\displaystyle\vec{w}_{1}\to\infty,\,\vec{w}_{2}\to\vec{1}=(1,0),\,\vec{w}_{3}\to 0.$

(III.6)

Plugging this fixed points into the integral, we obtain the factor of proportion ²²2The term $1/(\infty^{2})^{\Delta_{1}}$ coming from $\vec{w}_{1}\to\infty$ is cancelled by the Jacobian of this linear fractional transformation. as following

	$\displaystyle i\lambda(2\pi)^{4}\displaystyle{\int\left(\prod_{i=1}^{2}\frac{dy_{i}d^{2}z_{i}}{y_{i}^{3}}\right)\frac{y_{1}^{\Delta_{1}}y_{2}^{\Delta_{2}}}{(y_{2}^{2}+|\vec{z}_{2}-\vec{1}|^{2})^{\Delta_{2}}}\int\frac{d\omega_{3}}{\omega_{3}}\,\omega_{3}^{\Delta_{3}}\,\frac{y_{1}^{4}(m_{1}^{2}y_{2}^{2}+m_{2}^{2}\vec{z}_{2}^{2})}{m_{1}^{3}m_{2}^{2}(y_{2}^{2}+\vec{z}_{2}^{2})^{2}}}$
	$\displaystyle\times\delta(y_{1}-\frac{m_{1}m_{2}y_{2}(y_{2}^{2}+\vec{z}_{2}^{2})}{m_{1}^{2}y_{2}^{2}+m_{2}^{2}\vec{z}_{2}^{2}})\delta^{2}(\vec{z}_{1}-\frac{m_{2}^{2}(y_{2}^{2}+\vec{z}_{2}^{2})\vec{z}_{2}}{m_{1}^{2}y_{2}^{2}+m_{2}^{2}\vec{z}_{2}^{2}})\delta(\omega_{3}-\frac{y_{2}(m_{1}^{2}-m_{2}^{2})}{2m_{2}(y_{2}^{2}+\vec{z}_{2}^{2})}).$		(III.7)

Eliminating the delta functions we are left with a three-fold integral in a single hyperbolic space

$\displaystyle\frac{i\lambda(2\pi)^{4}(m_{1}^{2}-m_{2}^{2})^{\Delta_{3}-1}}{2^{\Delta_{3}-1}m_{1}^{2-\Delta_{1}}m_{2}^{\Delta_{1}+\Delta_{3}}}\displaystyle{\int dy_{2}d^{2}z_{2}\frac{y_{2}^{\Delta_{1}+\Delta_{2}+\Delta_{3}-3}}{(y_{2}^{2}+|\vec{z}_{2}-\vec{1}|^{2})^{\Delta_{2}}\,(y_{2}^{2}+\vec{z}_{2}^{2})^{\Delta_{3}-\Delta_{1}}\,(\vec{z}_{2}^{2}+\frac{m_{1}^{2}}{m_{2}^{2}}y_{2}^{2})^{\Delta_{1}}}}.$

(III.8)

This integral can be computed using the Feynman parametrization. The denominator under the Feynman parametrization becomes

		$\displaystyle\frac{1}{(y_{2}^{2}+|\vec{z}_{2}-\vec{1}|^{2})^{\Delta_{2}}\,(y_{2}^{2}+\vec{z}_{2}^{2})^{\Delta_{3}-\Delta_{1}}\,(\vec{z}_{2}^{2}+\frac{m_{1}^{2}}{m_{2}^{2}}y_{2}^{2})^{\Delta_{1}}}=\frac{\Gamma(\Delta_{2}+\Delta_{3})}{\Gamma(\Delta_{2})\Gamma(\Delta_{3}-\Delta_{1})\Gamma(\Delta_{1})}$
		$\displaystyle\times\int_{0}^{1}dt_{1}\int_{0}^{1-t_{1}}dt_{2}\frac{t_{1}^{\Delta_{2}-1}t_{2}^{\Delta_{3}-\Delta_{1}-1}(1-t_{1}-t_{2})^{\Delta_{1}-1}}{\big{[}(\vec{z}_{2}-t_{1}\vec{1})^{2}+t_{1}(1-t_{1})+y_{2}^{2}\big{(}t_{1}+t_{2}+(1-t_{1}-t_{2})\frac{m_{1}^{2}}{m_{2}^{2}}\big{)}\big{]}^{\Delta_{2}+\Delta_{3}}}.$		(III.9)

Then the integration over $\vec{z}_{2}$ is the standard Euclidean integral encountered in dimensional regularization

	$\displaystyle\int d^{2}z_{2}$	$\displaystyle\frac{1}{\big{[}(\vec{z}_{2}-t_{1}\vec{1})^{2}+t_{1}(1-t_{1})+y_{2}^{2}\big{(}t_{1}+t_{2}+(1-t_{1}-t_{2})\frac{m_{1}^{2}}{m_{2}^{2}}\big{)}\big{]}^{\Delta_{2}+\Delta_{3}}}$
		$\displaystyle=\frac{\pi\Gamma(\Delta_{2}+\Delta_{3}-1)}{\Gamma(\Delta_{2}+\Delta_{3})}\frac{1}{\big{[}t_{1}(1-t_{1})+y_{2}^{2}\big{(}t_{1}+t_{2}+(1-t_{1}-t_{2})\frac{m_{1}^{2}}{m_{2}^{2}}\big{)}\big{]}^{\Delta_{2}+\Delta_{3}-1}}.$		(III.10)

The integration over $y_{2}$ can be computed using the Euler beta function

	$\displaystyle\int_{0}^{\infty}dy_{2}$	$\displaystyle\frac{y_{2}^{\Delta_{1}+\Delta_{2}+\Delta_{3}-3}}{\big{[}t_{1}(1-t_{1})+y_{2}^{2}\big{(}t_{1}+t_{2}+(1-t_{1}-t_{2})\frac{m_{1}^{2}}{m_{2}^{2}}\big{)}\big{]}^{\Delta_{2}+\Delta_{3}-1}}$
		$\displaystyle=\frac{\Gamma(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}-1)\Gamma(\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2})}{2\Gamma(\Delta_{2}+\Delta_{3}-1)}\frac{\big{[}t_{1}(1-t_{1})\big{]}^{\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}}}{\big{[}t_{1}+t_{2}+(1-t_{1}-t_{2})\frac{m_{1}^{2}}{m_{2}^{2}}\big{]}^{\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}-1}}.$		(III.11)

Finally only the integration of Feynman parameters are left

$\displaystyle\int_{0}^{1}dt_{1}\int_{0}^{1-t_{1}}dt_{2}\frac{t_{1}^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}-1}t_{2}^{\Delta_{3}-\Delta_{1}-1}(1-t_{1}-t_{2})^{\Delta_{1}-1}(1-t_{1})^{\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}}}{\big{[}t_{1}+t_{2}+(1-t_{1}-t_{2})\frac{m_{1}^{2}}{m_{2}^{2}}\big{]}^{\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}-1}}.$

(III.12)

Using the famous Mellin-Barnes method, this integral can be computed into an infinite series of the mass ratio $m_{2}/m_{1}$ as following

	$\displaystyle\frac{\Gamma(\Delta_{1})\Gamma(\Delta_{3}-\Delta_{1})}{\Gamma(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}-1)\Gamma(\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}+1)\Gamma(\Delta_{3})}\sum_{n=0}^{\infty}\,\Bigg{\{}\frac{(-1)^{n}}{n!}\left(\frac{m_{2}}{m_{1}}\right)^{\Delta_{1}+\Delta_{2}+\Delta_{3}-2+2n}$
	$\displaystyle\times\Gamma(1-\Delta_{2}-n)\Gamma(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}+n)\Gamma(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}-1+n)$
	$\displaystyle\times{}_{2}F_{1}\left(\begin{array}[]{c}\Delta_{3}-\Delta_{1},\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}-1+n\\ \Delta_{3}\end{array};1-\frac{m_{2}^{2}}{m_{1}^{2}}\right)+(\Delta_{2}\rightarrow\tilde{\Delta}_{2})\Bigg{\}},$		(III.15)

To keep focused on the main physics, we put the computation of this series into the appendix ³³3However, it is hard to carry out the summation of this series into a concise and clean expression. In the appendix, we show that taking the massless limit $m_{2}\to 0$ of this infinite series is equivalent to the massless limit of this integral. The extra information obtained from this infinite series is the contribution of subleading terms, as discussed in section IV.1. and return to this complete result when discuss subleading terms of the massless limit in section IV.1.

Let’s focus on the physics here. Because this is a well-defined finite integral, when the mass $m_{2}$ is decreasing, it does not touch any singularity nor branch cut. So it is safe to take the massless limit $m_{2}\to 0$ in this finite integral (III.12). Under this limit, the integral simplifies as

		$\displaystyle\left(\frac{m_{2}}{m_{1}}\right)^{\Delta_{1}+\Delta_{2}+\Delta_{3}-2}\int_{0}^{1}dt_{1}\int_{0}^{1-t_{1}}dt_{2}\,t_{1}^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}-1}t_{2}^{\Delta_{3}-\Delta_{1}-1}(1-t_{1}-t_{2})^{\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}}$
		$\displaystyle{\quad}\times(1-t_{1})^{\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}}=\left(\frac{m_{2}}{m_{1}}\right)^{\Delta_{1}+\Delta_{2}+\Delta_{3}-2}\frac{\Gamma(1-\Delta_{2})\Gamma(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2})\Gamma(\Delta_{3}-\Delta_{1})}{\Gamma(1+\frac{\Delta_{3}-\Delta_{1}-\Delta_{2}}{2})},$		(III.16)

where only the leading term is kept because all subleading terms vanish when combining with the phase factor of  (II) and the limit $\Delta_{2}=1$, as will be shown in the next section. We see that a special gamma function $\Gamma(1-\Delta_{2})$ appears in the massless limit $m_{2}\to 0$. It has a pole in the conformal soft limit $\Delta_{2}\to 1$. This pole $1/(1-\Delta_{2})$ is the key result of our paper. It is crucial for the massless limit of this 3pt amplitude. Without it, the massless limit of $\langle\Phi_{\Delta_{1}}^{m_{1}}\Phi_{\Delta_{2}}^{m_{2}}\phi_{\Delta_{3}}\rangle$ would be zero, which is unphysical because $\langle\Phi_{\Delta_{1}}^{m_{1}}\phi_{\Delta_{2}}\phi_{\Delta_{3}}\rangle$ exists for sure.