11institutetext: Institute of Physics,
Sachivalaya Marg, Bhubaneshwar, India-75100522institutetext: Okinawa Institute
of Science and Technology, 1919-1 Tancha, Onna-son, Okinawa 904-0495,
Japan33institutetext: Harish-Chandra Research Institute,
Chhatnag Road, Jhunsi, Allahabad, India-21101944institutetext: Homi Bhabha
National Institute, Anushakti Nagar, Mumbai, India-400085

# Modified celestial amplitude in Einstein gravity

Shamik Banerjee 2 Sudip Ghosh 1,4 Pranjal Pandey 3,4 Arnab Priya Saha
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

In this paper we evaluate the modified celestial amplitude for gravitons and
gluons, as defined in arXiv:1801.10171[hep-th]. We find that the modified
(tree) amplitude is finite for gravitons in Einstein gravity. The modified
amplitude behaves like correlation function of operators inserted at various
points of null-infinity in the Minkowski space-time. Therefore, unlike the
standard celestial amplitudes, these are three dimensional objects. We also
show that this amplitude admits conformal soft factorization recently studied
in the literature.

##  1 Introduction

In flat space holography the natural observables are the $S$-matrix elements.
They are the analogues of the correlation functions of the boundary CFT in the
AdS/CFT correspondence. In a holographic description it is expected that the
global symmetries of the dual theory should match the asymptotic symmetries of
the bulk theory of quantum gravity. When the bulk is four dimensional
asymptotically flat space-time, the asymptotic symmetry group is the extended
BMS group Bondi:1962px ; Sachs:1962zza ; Strominger:2013jfa ; Barnich:2009se ;
Kapec:2014opa ; Campiglia:2015kxa  which, besides supertranslation, also
contains superrotation. Superrotations are local conformal transformations
acting on the two dimensional celestial sphere. This is an extension of the
Lorentz group $SL(2,\mathbb{C})$ which acts on the celestial sphere as the
group of global conformal transformations. Therefore, from a holographic
perspective, it is desirable to have a (complete) set of observables which
transform naturally under the (Lorentz) conformal group. In order to achieve
this Pasterski:2016qvg ; Pasterski:2017kqt ; Pasterski:2017ylz  has put
forward a very interesting proposal in which, instead of plane-waves, one uses
the conformal primary wave-functions to describe the states of the incoming
and outgoing particles in an $S$-matrix element. To be more precise, for
massless particles, the change of basis is given by Pasterski:2016qvg ;
Pasterski:2017kqt ; Pasterski:2017ylz ,

$\tilde{S}\big{(}\\{z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{i\lambda_{i}}S\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\big{)}$
(1)

Here $\sigma_{i}$ denotes the helicity of the $i$-th particle 111In a later
part we will denote the simplex variables Pasterski:2017ylz  also by $\sigma$.
This should be clear from the context. and the on-shell momenta are
parametrized as,

$p_{i}=\omega_{i}(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i}),\quad
p_{i}^{2}=0$ (2)

Under $SL(2,\mathbb{C})$ (Lorentz) transformation $\omega$ and $z$ transforms
as,

$\omega\rightarrow\omega|cz+d|^{2},\quad
z\rightarrow\frac{az+b}{cz+d},\quad\bar{z}\rightarrow
c.c,\qquad\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL(2,\mathbb{C})$ (3)

The new amplitude
$\tilde{S}\big{(}\\{z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}$ is a
Mellin transformation of the standard $S$-matrix element
$S\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\big{)}$ and is known as
the celestial amplitude because $(z_{i},\bar{z}_{i})$ can be thought of as
complex coordinates of points on the celestial sphere. Under Lorentz
transformation $\tilde{S}$ transforms covariantly,

$\tilde{S}\big{(}\\{z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}=\prod_{i=1}^{n}\frac{1}{(cz_{i}+d)^{2h_{i}}}\frac{1}{(\bar{c}\bar{z}_{i}+\bar{d})^{2\bar{h}_{i}}}\tilde{S}\bigg{(}\frac{az_{i}+b}{cz_{i}+d}\
,\frac{\bar{a}\bar{z}_{i}+\bar{b}}{\bar{c}\bar{z}_{i}+\bar{d}}\ ,\lambda_{i}\
,\sigma_{i}\bigg{)}$ (4)

where,

$h=\frac{1+i\lambda-\sigma}{2},\quad\bar{h}=\frac{1+i\lambda+\sigma}{2}$ (5)

Equation (4) shows that $\tilde{S}$ transforms like the correlation function
of conformal (quasi) primaries of weights $(h_{i},\bar{h}_{i})$, inserted at
the points $(z_{i},\bar{z}_{i})$ on the celestial sphere. This transformation
law is natural because the Lorentz group acts on the celestial sphere as the
group of global conformal transformations. The action of global space-time
translations on the Mellin-amplitude $\tilde{S}$ was studied in
Stieberger:2018onx .

Now, a modified form of Mellin amplitude was defined in Banerjee:2018gce ;
Banerjee:2018fgd ,

$\boxed{\mathcal{A}\big{(}\\{u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{i\lambda_{i}}e^{-i\varepsilon_{i}\omega_{i}u_{i}}S\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\big{)}}$
(6)

where $\varepsilon_{i}=+1$ for an outgoing particle and $\varepsilon_{i}=-1$
for an incoming particle. The $u$ coordinate has the interpretation of
(retarded) time and $(u,z,\bar{z})$ can be thought of as coordinates at null-
infinity in asymptotically flat space. This interpretation is further
confirmed by the transformation property of $(u,z,\bar{z})$ under Lorentz
transformation Banerjee:2018gce ; Banerjee:2018fgd ,

$u\rightarrow\frac{u}{|cz+d|^{2}},\quad
z\rightarrow\frac{az+b}{cz+d},\quad\bar{z}\rightarrow c.c$ (7)

The modified Mellin transformation (6) can also be inverted in the standard
way to recover the standard $S$-matrix element,

$\boxed{S\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\big{)}=e^{i\sum_{i=1}^{n}\varepsilon_{i}\omega_{i}u_{i}}\prod_{i=1}^{n}\int_{-\infty}^{\infty}\frac{d\lambda_{i}}{2\pi}\
\omega_{i}^{-i\lambda_{i}-1}\mathcal{A}(\\{u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\})}$
(8)

Under (Lorentz) conformal transformation the Mellin amplitude $\mathcal{A}$
transforms as,

$\mathcal{A}\big{(}\\{u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}=\prod_{i=1}^{n}\frac{1}{(cz_{i}+d)^{2h_{i}}}\frac{1}{(\bar{c}\bar{z}_{i}+\bar{d})^{2\bar{h}_{i}}}\mathcal{A}\bigg{(}\frac{u_{i}}{|cz_{i}+d|^{2}}\
,\frac{az_{i}+b}{cz_{i}+d}\
,\frac{\bar{a}\bar{z}_{i}+\bar{b}}{\bar{c}\bar{z}_{i}+\bar{d}}\ ,\lambda_{i}\
,\sigma_{i}\bigg{)}$ (9)

and under global space-time translation,

$\mathcal{A}\big{(}\\{u_{i}+f(z_{i},\bar{z}_{i}),z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}=\mathcal{A}\big{(}\\{u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}$
(10)

where

$f(z,\bar{z})=a+bz+\bar{b}\bar{z}+cz\bar{z}$ (11)

For the sake of completeness, let us now briefly explain the origin of the
modified transformation equation (6). In the usual $S$-matrix the asymptotic
states are described by the direct product of the single particle Wigner
states $\ket{p,\sigma}$ where $p$ is an on-shell momentum and $\sigma$ is the
helicity. Here $p$ is parametrized as in (2). Now, in the Mellin amplitude
$\tilde{S}\big{(}\\{z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}$ the
asymptotic states are described as the direct product of the single particle
states $\ket{z,\bar{z},\lambda,\sigma}$ which are the Mellin transform of the
$\ket{p,\sigma}$ states Pasterski:2016qvg ; Pasterski:2017kqt ;
Pasterski:2017ylz ; Banerjee:2018gce , i.e,

$\ket{z,\bar{z},\lambda,\sigma}=N\int_{0}^{\infty}d\omega\omega^{i\lambda}\ket{p(\omega,z,\bar{z}),\sigma}$
(12)

where $N$ is a normalization constant. Now, given these states, one can
calculate the quantum mechanical transition amplitude given by
Banerjee:2018gce  ,

$\displaystyle\bra{z,\bar{z},\lambda,\sigma}e^{-iH(U-U^{\prime})}\ket{z^{\prime},\bar{z}^{\prime},\lambda^{\prime},\sigma^{\prime}}$
$\displaystyle=\bra{u,z,\bar{z},\lambda,\sigma}\ket{u^{\prime},z^{\prime},\bar{z}^{\prime},\lambda^{\prime},\sigma^{\prime}}$
(13) $\displaystyle=\lim_{\delta\rightarrow
0+}\frac{\delta_{\sigma\sigma^{\prime}}}{2\pi}\frac{\Gamma\big{(}i(\lambda^{\prime}-\lambda)\big{)}\delta^{2}(z^{\prime}-z)}{\big{(}-i(u^{\prime}-u+i\delta)\big{)}^{i(\lambda^{\prime}-\lambda)}}$

where $u=(1+z\bar{z})U$ and we have also defined the Heisenberg picture basis
state,

$\ket{u,z,\bar{z},\lambda,\sigma}=e^{iHU}\ket{z,\bar{z},\lambda,\sigma}=N\int_{0}^{\infty}d\omega\omega^{i\lambda}e^{i\omega
u}\ket{p(\omega,z,\bar{z}),\sigma}$ (14)

The action of the Poincare group on the states
$\ket{u,z,\bar{z},\lambda,\sigma}$ is simply given by,

$U(\Lambda)\ket{u,z,\bar{z},\lambda,\sigma}=\frac{1}{(cz+d)^{2h}}\frac{1}{(\bar{c}\bar{z}+\bar{d})^{2\bar{h}}}\ket{\frac{u}{|cz+d|^{2}},\frac{az+b}{cz+d},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}},\lambda,\sigma}$
(15)

and

$e^{-iP.a}\ket{u,z,\bar{z},\lambda,\sigma}=\ket{u+f(z,\bar{z}),z,\bar{z}}$
(16)

where
$f(z,\bar{z})=(a^{0}-a^{3})-(a^{1}-ia^{2})z-(a^{1}+ia^{2})\bar{z}+(a^{0}+a^{3})z\bar{z}$.

This suggests a picture in which the particle is moving in a three dimensional
space-time with coordinates $(u,z,\bar{z})$ on which the Poincare group acts
geometrically, i.e,

$(u,z,\bar{z})\xrightarrow{L.T}\bigg{(}\frac{u}{|cz+d|^{2}},\frac{az+b}{cz+d},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\bigg{)}$
(17)
$(u,z,\bar{z})\xrightarrow{Translation}(u+(a^{0}-a^{3})-(a^{1}-ia^{2})z-(a^{1}+ia^{2})\bar{z}+(a^{0}+a^{3})z\bar{z},z,\bar{z})$
(18)

In fact, this is exactly the way in which the Poincare group acts on null
infinity in Minkowski space parametrized by the Bondi coordinates
$(u,z,\bar{z})$. One can also check that as expected, the transition amplitude
(13) is (covariant) invariant under the Poincare transformation.

Now, _the modified Mellin amplitude
$\mathcal{A}\big{(}\\{u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}$
is essentially the $S$-matrix element when the asymptotic in and out states
are described by free particles moving in this three dimensional space with
coordinates $(u,z,\bar{z})$_. To be more precise, the states of the incoming
or outgoing particles are now given by the direct product of the states,

$\ket{u,z,\bar{z},\lambda,\sigma,in/out}=N\int_{0}^{\infty}d\omega\omega^{i\lambda}e^{i\omega
u}\ket{p(\omega,z,\bar{z}),\sigma,in/out}$ (19)

The main difference between the Mellin transforms $\tilde{S}$ and
$\mathcal{A}$ is that in $\tilde{S}$ all the incoming and outgoing particles
are constrained to lie on the same equal-$u$ hyper surface, whereas in
$\mathcal{A}$ the particles are separated in time. Let us now explain the
difference between $\tilde{S}$ and $\mathcal{A}$ from the point of view of the
(Minkowski space) wave functions of the asymptotic particles. For simplicity
we consider the external particles to be scalars.

Now, as we have already discussed, the Mellin amplitude $\tilde{S}$ describes
the scattering process when the asymptotic particles are described by the
conformal primary wave functions Pasterski:2016qvg ; Pasterski:2017kqt ;
Pasterski:2017ylz ,

$\Phi_{\Delta}^{\pm}(x^{\mu}|z,\bar{z})=\frac{(\mp
i)^{\Delta}\Gamma(\Delta)}{(-q(z,\bar{z})\cdot x\mp i\delta)^{\Delta}}\
,\quad\Delta=1+i\lambda,\quad\delta\rightarrow 0+$ (20)

where $q^{\mu}(z,\bar{z})=(1+z\bar{z},z+\bar{z},-i(z-\bar{z}),1-z\bar{z})$ is
a unit null-vector. The superscript $\pm$ refers to the Mellin transformation
of positive and negative energy plane waves, respectively. Now, the wave
function (20) is distinguished by the fact that it is singular along the null-
hyperplane, $x\cdot q(z,\bar{z})=0$, which passes through the origin of the
Minkowski space. In other words, one can say that the particle described by
(20) is localised on the null-hyperplane, $x\cdot q(z,\bar{z})=0$. Now, since
the Klein-Gordon equation is Poincare invariant, it is natural to consider
other solutions in which the particle is localised on different null-
hyperplanes in Minkowski space. The complete set of such solutions can be
parametrized as Banerjee:2018gce ,

$\Phi_{\Delta}^{\pm}(x^{\mu}|u,z,\bar{z})=\frac{(\mp
i)^{\Delta}\Gamma(\Delta)}{(-q(z,\bar{z})\cdot x+u\mp
i\delta)^{\Delta}},\quad-\infty<u<\infty,\quad\delta\rightarrow 0+$ (21)

The particle described by the wave function (21) is now localized along the
null-hyperplane, $x\cdot q(z,\bar{z})-u=0$ and by varying $(u,z,\bar{z})$ we
can generate all the null-hyperplanes in the Minkowski space. From the
Poincare transformation property of the wave function (21) one can conclude
that the three parameters $(u,z,\bar{z})$ have the interpretation of Bondi
coordinates at null-infinity Banerjee:2018gce . The modified Mellin amplitude
$\mathcal{A}$ describes the scattering process when the asymptotic particles
are described by wave functions (21). To be more precise, for the $\tilde{S}$
amplitude the wave functions are
$\big{\\{}\Phi_{\Delta_{i}}^{\pm}(x^{\mu}|u,z_{i},\bar{z}_{i})\big{\\}}$.
Since the retarded time $u$ is the _same for all the particles_ , we can set
$u=0$ by time translation invariance. On the other hand, the $\mathcal{A}$
amplitude is computed with the external particle wave functions
$\big{\\{}\Phi_{\Delta_{i}}^{\pm}(x^{\mu}|u_{i},z_{i},\bar{z}_{i})\big{\\}}$
and so _different particles are now localized at different retarded times_.
The _separation in retarded time is the main difference between the
$\mathcal{A}$ and $\tilde{S}$ amplitudes_.

The rest of the paper is organized as follows. In section-(2) we discuss the
role of $i\delta$ in regularizing the modified amplitude $\mathcal{A}$. In
section-(3) we evaluate $\mathcal{A}$ corresponding to $4$ \- graviton and $4$
\- gluon MHV amplitudes. We find that for gravitons $\mathcal{A}$ is finite
even in Einstein gravity. This is a welcome feature from a holographic
perspective. We then go on to study the conformal soft factorization property
of $\mathcal{A}$. Soft factorization is non-trivial in the Mellin basis
because one integrates over the energies of the asymptotic particles. It has
been proposed and studied in Fan:2019emx ; Nandan:2019jas ; Pate:2019mfs ;
Donnay:2018neh ; Adamo:2019ipt ; Puhm:2019zbl ; Guevara:2019ypd  that soft
limit in the Mellin amplitude corresponds to taking $\lambda_{p}\rightarrow 0$
if the $p$-th particle is going soft. To be more precise, it has been shown
that the limiting value of
$i\lambda_{p}S\big{(}\\{z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}$,
as $\lambda_{p}\rightarrow 0$, factorizes. We show in section-(4) that
analogous (conformal) soft factorization is also true for
$i\lambda_{p}\mathcal{A}\big{(}\\{u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i}\\}\big{)}$,
as $\lambda_{p}\rightarrow 0$.

##  2 Regularization

The modified Mellin amplitude $\mathcal{A}$ has the form,

$\mathcal{A}\big{(}\\{u_{p},z_{p},\bar{z}_{p},\lambda_{p},\sigma_{p}\\}\big{)}=\prod_{p=1}^{n}\int_{0}^{\infty}d\omega_{p}\
\omega_{p}^{i\lambda_{p}}e^{-i\varepsilon_{p}\omega_{p}u_{p}}S\big{(}\\{\omega_{p},z_{p},\bar{z}_{p},\sigma_{p}\\}\big{)}$
(22)

For large values of energy the exponential factors $e^{\pm i\omega u}$
oscillate rapidly. These can be regulated by adding a small imaginary part to
$u$. To be more precise, we make the following shift,

$u_{p}\rightarrow u_{p}-i\varepsilon_{p}\delta,\qquad\delta\rightarrow 0+$
(23)

The origin of the convergence factor can also be traced back to the conformal
primary wave-functions (21),

$\Phi_{\Delta}^{\pm}(x^{\mu}|u,z,\bar{z})=\frac{(\mp
i)^{\Delta}\Gamma(\Delta)}{(-q(z,\bar{z})\cdot x+u\mp i\delta)^{\Delta}}$ (24)

We can see that effectively $u$ has a small imaginary part $\propto\delta$,
whose sign depends on the sign of the energy.

Therefore the amplitude $\mathcal{A}$ should be understood as,

$\begin{gathered}\mathcal{A}\big{(}\\{u_{p},z_{p},\bar{z}_{p},\lambda_{p},\sigma_{p}\\}\big{)}=\prod_{p=1}^{n}\int_{0}^{\infty}d\omega_{p}\
\omega_{p}^{i\lambda_{p}}e^{-i\varepsilon_{p}\omega_{p}(u_{p}-i\varepsilon_{p}\delta)}S\big{(}\\{\omega_{p},z_{p},\bar{z}_{p},\sigma_{p}\\}\big{)}\\\
=\prod_{p=1}^{n}\int_{0}^{\infty}d\omega_{p}\
\omega_{p}^{i\lambda_{p}}e^{-i\varepsilon_{p}\omega_{p}u_{p}}e^{-\delta
s}S\big{(}\\{\omega_{p},z_{p},\bar{z}_{p},\sigma_{p}\\}\big{)},\quad\delta\rightarrow
0+\end{gathered}$ (25)

where

$s=\sum_{p}\omega_{p}>0$ (26)

This provides the necessary damping for the integrals to be well defined and
plays a crucial role in the rest of the paper. In most of the formulas below,
we will not explicitly write down the convergence factor but it should always
be kept in mind.

##  3 Modified Mellin Amplitudes

Mellin transform of scattering amplitudes in the absence of the retarded time
has been studied in detail in Pasterski:2016qvg ; Pasterski:2017ylz ;
Schreiber:2017jsr ; Cardona:2017keg ; Lam:2017ofc ; Banerjee:2017jeg ;
Stieberger:2018edy . In this section we calculate the modified Mellin
transform of some low-point tree level amplitudes for gravitons and gluons.

###  3.1 Four particle graviton tree amplitude in Einstein gravity

It is known that the tree-level Mellin amplitude $\tilde{S}$ for gravitons is
divergent in Einstein gravity Stieberger:2018edy . An interesting resolution
suggested by Stieberger:2018edy  is that instead of Einstein gravity one
should consider graviton scattering in string theory and this indeed makes the
amplitude $\tilde{S}$ finite. However, it seems that there is no limit in
which one can recover the classical Einstein gravity from the $\tilde{S}$
amplitude computed in string theory. In this paper we show that the modified
Mellin amplitude $\mathcal{A}$ is in fact finite for tree level graviton
scattering amplitude in Einstein gravity. It turns out that separating the
asymptotic particles in retarded time naturally regularizes the $\mathcal{A}$
amplitudes in Einstein gravity. Let us now describe the results.

With gravitons $1,2$ of negative helicities and $3,4$ of positive helicities
the four graviton tree amplitude in spinor-helicity variables is

$S(1^{--}2^{--}3^{++}4^{++})(\omega_{i},z_{i},\bar{z}_{i})=\frac{\kappa^{2}}{4}\frac{\langle
12\rangle^{4}[34]^{4}}{stu}\delta^{4}(\sum_{i}\varepsilon_{i}\omega_{i}q_{i})~{}~{}$
(27)

where $\kappa=\sqrt{8\pi G_{N}}$ and $G_{N}$ is the $4$-dimensional Newton’s
constant.

We define the kinematic variables $s,t,u$ as

$s=\langle 12\rangle[12]~{}~{};~{}~{}t=\langle
13\rangle[13]~{}~{};~{}~{}u=\langle 14\rangle[14]$ (28)

Using the definition of spinor helicity variables Pasterski:2017ylz  and the
parametrisation of null momentum as in (2), we have,

$\langle
ij\rangle=-\varepsilon_{i}\varepsilon_{j}2\sqrt{\omega_{i}\omega_{j}}z_{ij}~{}~{};~{}~{}[ij]=2\sqrt{\omega_{i}\omega_{j}}\bar{z}_{ij}\quad;\quad
z_{ij}=z_{i}-z_{j}$ (29)

So in terms of $\omega,z,\bar{z}$ the $S$-matrix (27) becomes,

$S(1^{--}2^{--}3^{++}4^{++})(\omega_{i},z_{i},\bar{z}_{i})=\kappa^{2}\frac{\omega_{2}\omega_{3}\omega_{4}z_{12}^{4}\bar{z}^{4}_{34}}{\omega_{1}z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{14}\bar{z}_{14}}\delta^{4}(\sum_{i}\varepsilon_{i}\omega_{i}q_{i})$
(30)

Now we want to find out the modified amplitude $\mathcal{A}$ corresponding to
(30). Before doing this it is convenient to change $\omega_{i}$’s to the
simplex variables Pasterski:2017ylz  : $\sigma_{i}=s^{-1}\omega_{i}$ ,
$\sum_{i}\omega_{i}=s$. In terms of simplex variables $\mathcal{A}$ can be
written as,

$\displaystyle\mathcal{A}(1^{--}2^{--}3^{++}4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})=\prod_{i=1}^{4}\int_{0}^{\infty}d\omega_{i}\omega^{i\lambda_{i}}e^{-i\varepsilon_{i}\omega_{i}u_{i}}[...]$
$\displaystyle=\int_{0}^{\infty}ds~{}s^{4-1+i\sum_{i}\lambda_{i}}\prod_{i=1}^{4}\int_{0}^{1}d\sigma_{i}\sigma_{i}^{i\lambda_{i}}e^{-i\varepsilon_{i}s\sigma_{i}u_{i}}\delta(\sum_{i}\sigma_{i}-1)[...]$
(31)

where,

$\displaystyle[...]=S(1^{--}2^{--}3^{++}4^{++})(s,\sigma_{i},z_{i},\bar{z}_{i})=s^{-2}\kappa^{2}\frac{\sigma_{2}\sigma_{3}\sigma_{4}z_{12}^{3}\bar{z}^{4}_{34}}{\sigma_{1}z_{13}\bar{z}_{13}z_{14}\bar{z}_{14}\bar{z}_{12}}\delta^{4}(\sum_{i}\varepsilon_{i}\sigma_{i}q_{i})$
(32)

We can rewrite the delta functions as Pasterski:2017ylz ,

$\displaystyle\delta^{4}(\sum_{i}\varepsilon_{i}\sigma_{i}q_{i})\delta(\sum_{i}\sigma_{i}-1)=C(z_{i},\bar{z}_{i})\prod_{i=1}^{4}\delta(\sigma_{i}-\sigma_{i}^{\ast})$
$\displaystyle
C(z_{i},\bar{z}_{i})=\frac{1}{4}\delta\left(|z_{12}z_{34}\bar{z}_{13}\bar{z}_{24}-\bar{z}_{12}\bar{z}_{34}z_{13}z_{24}|\right)=\frac{\delta(|z-\bar{z}|)}{4z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}$
$\displaystyle\sigma_{1}^{\ast}=-\frac{\varepsilon_{1}\varepsilon_{4}}{D}\frac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}~{}~{},~{}~{}\sigma_{2}^{\ast}=\frac{\varepsilon_{2}\varepsilon_{4}}{D}\frac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}$
$\displaystyle\sigma_{3}^{\ast}=-\frac{\varepsilon_{3}\varepsilon_{4}}{D}\frac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}~{}~{},~{}~{}\sigma_{4}^{\ast}=\frac{1}{D}$
$\displaystyle
D=(1-\varepsilon_{1}\varepsilon_{4})\frac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}+(\varepsilon_{2}\varepsilon_{4}-1)\frac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}+(1-\varepsilon_{3}\varepsilon_{4})\frac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}$
(33)

In terms of these the modified Mellin amplitude $\mathcal{A}$ becomes,

$\displaystyle\mathcal{A}(1^{--}2^{--}3^{++}4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})$
$\displaystyle=\int_{0}^{\infty}ds~{}s^{4-1+i\sum_{i}\lambda_{i}}\prod_{i=1}^{4}\int_{0}^{1}d\sigma_{i}\sigma_{i}^{i\lambda_{i}}e^{-i\varepsilon_{i}s\sigma_{i}u_{i}}\delta(\sum_{i}\sigma_{i}-1)S(1^{--}2^{--}3^{++}4^{++})(\sigma_{i},z_{i},\bar{z}_{i})$
$\displaystyle=\frac{\kappa^{2}}{4}\left(\int_{0}^{\infty}ds~{}s^{1+i\sum_{i}\lambda_{i}}e^{-i\sum_{i}\varepsilon_{i}s\sigma_{i}^{\ast}u_{i}}\right)\left(\prod_{i}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{2}^{\ast}\sigma_{3}^{\ast}\sigma_{4}^{\ast}z_{12}^{3}\bar{z}^{4}_{34}}{\sigma_{1}^{\ast}z_{13}\bar{z}_{13}z_{14}\bar{z}_{14}\bar{z}_{12}}\frac{\delta(|z-\bar{z}|)}{z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$
$\displaystyle=\lim_{\delta\rightarrow
0+}\frac{\kappa^{2}}{4}\left(\int_{0}^{\infty}ds~{}s^{1+i\sum_{i}\lambda_{i}}e^{-i\sum_{i}\varepsilon_{i}s\sigma_{i}^{\ast}u_{i}}e^{-\delta
s}\right)\left(\prod_{i}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{2}^{\ast}\sigma_{3}^{\ast}\sigma_{4}^{\ast}z_{12}^{3}\bar{z}^{4}_{34}}{\sigma_{1}^{\ast}z_{13}\bar{z}_{13}z_{14}\bar{z}_{14}\bar{z}_{12}}\frac{\delta(|z-\bar{z}|)}{z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$
(34)

Here we have regularized the integral as described in section-(2). The
$\sigma_{i}$ integrals are done by simply using the delta functions. Here the
term $\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$ ensures that
$\sigma_{i}^{\ast}$ are in range $[0,1]$.

$\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})=\begin{cases}1,&\sigma_{i}^{\ast}~{}\epsilon~{}[0,1]\\\
0,&otherwise\end{cases}$ (35)

Now using

$\lim_{\delta\rightarrow
0+}\int_{0}^{\infty}ds~{}s^{1+i\sum_{i}\lambda_{i}}e^{-i\sum_{i}\varepsilon_{i}s\sigma_{i}^{\ast}u_{i}}e^{-\delta
s}=\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\sum\lambda_{i})}{(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta)^{2+i\sum\lambda_{i}}}$
(36)

the final result for the modified Mellin amplitude becomes,

$\begin{gathered}\mathcal{A}(1^{--}2^{--}3^{++}4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
=\frac{\kappa^{2}}{4}\bigg{[}\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\sum\lambda_{i})}{(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta)^{2+i\sum\lambda_{i}}}\bigg{]}\left(\prod_{i}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{2}^{\ast}\sigma_{3}^{\ast}\sigma_{4}^{\ast}z_{12}^{3}\bar{z}^{4}_{34}}{\sigma_{1}^{\ast}z_{13}\bar{z}_{13}z_{14}\bar{z}_{14}\bar{z}_{12}}\frac{\delta(|z-\bar{z}|)}{z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})\end{gathered}$
(37)

One can check that the amplitude (37) is translationally invariant. Under
global space-time translation, $u_{i}\rightarrow
u_{i}+A+Bz_{i}+\bar{B}\bar{z}_{i}+Cz_{i}\bar{z}_{i}$, and the change in the
term $\sum_{i=1}^{4}\varepsilon_{i}\sigma_{i}^{\ast}u_{i}$ is always
proportional to $(z-\bar{z})$. Hence the amplitude is invariant due to the
$\delta(|z-\bar{z}|)$ constraint.

###  3.2 Four particle gluon tree amplitude in Yang-Mills

Four particle gluon amplitude for particles $1,2$ with negative helicity and
$3,4$ with positive helicity is given as,

$S(1^{-}2^{-}3^{+}4^{+})(\omega_{i},z_{i},\bar{z}_{i})=\frac{\langle
12\rangle^{3}}{\langle 23\rangle\langle 34\rangle\langle
41\rangle}\delta^{4}(\sum_{i=1}^{4}\varepsilon_{i}\omega_{i}q_{i})$ (38)

Writing this amplitude in terms of $\omega,z,\bar{z}$ we get,

$S(1^{-}2^{-}3^{+}4^{+})(\omega_{i},z_{i},\bar{z}_{i})=\frac{\omega_{1}\omega_{2}}{\omega_{3}\omega_{4}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\delta^{4}(\sum_{i=1}^{4}\varepsilon_{i}\omega_{i}q_{i})$
(39)

Rewriting this amplitude in terms of simplex variables,

$S(1^{-}2^{-}3^{+}4^{+})(\sigma_{i},z_{i},\bar{z}_{i})=s^{-4}\frac{\sigma_{1}\sigma_{2}}{\sigma_{3}\sigma_{4}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\delta^{4}(\sum_{i=1}^{4}\varepsilon_{i}\sigma_{i}q_{i})$
(40)

Using (33) and (31) we find the modified Mellin amplitude for four gluons to
be,

$\displaystyle\mathcal{A}(1^{-}2^{-}3^{+}4^{+})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})$
$\displaystyle=\int_{0}^{\infty}ds~{}s^{4-1+i\sum_{i}\lambda_{i}}\prod_{i=1}^{4}\int_{0}^{1}d\sigma_{i}\sigma_{i}^{i\lambda_{i}}e^{-i\varepsilon_{i}s\sigma_{i}u_{i}}\delta(\sum_{i}\sigma_{i}-1)S(1^{-}2^{-}3^{+}4^{+})(\sigma_{i},z_{i},\bar{z}_{i})$
$\displaystyle=\frac{1}{4}\left(\int_{0}^{\infty}dss^{-1+i\sum_{i=1}^{4}\lambda_{i}}e^{-i\sum_{i=1}^{4}\varepsilon_{i}s\sigma_{i}^{\ast}u_{i}}\right)\left(\prod_{i}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{1}^{\ast}\sigma_{2}^{\ast}}{\sigma_{3}^{\ast}\sigma_{4}^{\ast}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\frac{\delta(|z-\bar{z}|)}{z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$
(41)

The $u_{i}$ dependence of the amplitude is given by,

$\int_{0}^{\infty}dss^{-1+i\sum_{i=1}^{4}\lambda_{i}}e^{-i\sum_{i=1}^{4}\varepsilon_{i}s\sigma_{i}^{\ast}u_{i}}=\lim_{\delta\rightarrow
0+}\frac{\Gamma(i\sum_{i}\lambda_{i})}{\left(i\sum_{i=1}^{4}\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta\right)^{i\sum_{i}\lambda_{i}}}$
(42)

The final result is given by,

$\displaystyle\mathcal{A}(1^{-}2^{-}3^{+}4^{+})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})$
$\displaystyle=$ $\displaystyle\frac{1}{4}\bigg{[}\lim_{\delta\rightarrow
0+}\frac{\Gamma(i\sum_{i}\lambda_{i})}{\left(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta\right)^{i\sum_{i}\lambda_{i}}}\bigg{]}\left(\prod_{i}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{1}^{\ast}\sigma_{2}^{\ast}}{\sigma_{3}^{\ast}\sigma_{4}^{\ast}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\frac{\delta(|z-\bar{z}|)}{z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$

Just like in the graviton case this amplitude is also translationally
invariant due to the Dirac delta function constraint on the cross ratios.

##  4 Conformal Soft Factorization

Let us now show that the modified amplitudes admit _conformal_ soft
factorization Fan:2019emx ; Nandan:2019jas ; Pate:2019mfs ; Donnay:2018neh ;
Adamo:2019ipt ; Puhm:2019zbl ; Guevara:2019ypd .

###  4.1 $n$-point graviton amplitudes

Consider a $n$-point graviton scattering amplitude
$S_{n}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)$. Now if we take
the limit where the momentum of the $n$-th particle goes to zero, then
according to Weinberg’s soft graviton theorem Weinberg:1965nx  the $n$-point
amplitude factorizes into a $(n-1)$ point amplitude as follows

$\begin{split}S_{n}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)=\mathcal{F}^{(0)}S_{n-1}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)+\ldots\end{split}$
(44)

where $\mathcal{F}^{(0)}$ is the universal leading Weinberg soft factor. The
ellipses in equation 44 denote subleading contributions in the soft momentum.
However, in this section we will only concern ourselves with the leading soft
factor.

Now if the soft graviton has positive helicity, the soft factor can be
expressed in terms of spinor helicity variables as Cachazo:2014fwa

$\begin{split}\mathcal{F}^{(0)}_{+}=-\sum_{i=1}^{n-1}\frac{[ni]}{\langle
ni\rangle}\frac{\langle xi\rangle\langle yi\rangle}{\langle xn\rangle\langle
yn\rangle}\end{split}$ (45)

Here we have introduced a subscript $(+)$ in the soft factor to explicitly
denote the case that the soft graviton here has positive helicity. Now using
the parametrisation of null momenta in equation 2 this becomes

$\begin{split}\mathcal{F}^{(0)}_{+}=-\sum_{i=1}^{n-1}\frac{\varepsilon_{i}\omega_{i}}{\varepsilon_{n}\omega_{n}}\frac{\bar{z}_{ni}z_{xi}z_{yi}}{z_{ni}z_{xn}z_{yn}}\end{split}$
(46)

Similarly in the case where the soft graviton has negative helicity, we have

$\begin{split}\mathcal{F}^{(0)}_{-}=-\sum_{i=1}^{n-1}\frac{\langle
ni\rangle}{[ni]}\frac{[xi][yi]}{[xn][yn]}=-\sum_{i=1}^{n-1}\frac{\varepsilon_{i}\omega_{i}}{\varepsilon_{n}\omega_{n}}\frac{z_{ni}}{\bar{z}_{ni}}\frac{\bar{z}_{xi}\bar{z}_{yi}}{\bar{z}_{xn}\bar{z}_{yn}}\end{split}$
(47)

Let us now consider the conformal soft limit of the modified Mellin transform
of the $n$-point graviton amplitude. Following Fan:2019emx ; Nandan:2019jas ;
Pate:2019mfs , we define this as,

$\begin{split}&\lim_{\lambda_{n}\to 0}i\lambda_{n}\hskip
1.13791pt\mathcal{A}_{n}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})\end{split}$
(48)

Recently in Puhm:2019zbl  the conformal soft limit of graviton amplitudes has
been studied. It is important to note that the main difference between our
analysis and the analysis in Puhm:2019zbl  is that we are using the
prescription (6) of Banerjee:2018gce ; Banerjee:2018fgd  for defining the
Mellin transform of scattering amplitudes which is a modification of the
definition proposed in Pasterski:2017ylz  and subsequently used in
Puhm:2019zbl . Now, using our definition for the Mellin transform in equation
(6) we get

$\begin{split}&\lim_{\lambda_{n}\to 0}i\lambda_{n}\hskip
1.13791pt\mathcal{A}_{n}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})\\\
&=\int\prod_{k=1}^{n-1}d\omega_{k}\hskip
1.42271pt\omega_{k}^{i\lambda_{k}}\hskip
2.84544pte^{-i\sum\limits_{k=1}^{n-1}\varepsilon_{k}\omega_{k}u_{k}}\int_{0}^{\infty}d\omega_{n}\left(\lim_{\lambda_{n}\to
0}i\lambda_{n}\hskip 1.13791pt\omega_{n}^{i\lambda_{n}-1}\right)\hskip
1.13791pt\omega_{n}\hskip
1.13791pte^{-i\varepsilon_{n}\omega_{n}u_{n}}S_{n}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)\\\
&=\int\prod_{k=1}^{n-1}d\omega_{k}\hskip
1.42271pt\omega_{k}^{i\lambda_{k}}\hskip
2.84544pte^{-i\sum\limits_{k=1}^{n-1}\varepsilon_{k}\omega_{k}u_{k}}\int_{0}^{\infty}d\omega_{n}\hskip
1.13791pt\delta(\omega_{n})\hskip 1.13791pt\omega_{n}\hskip
1.13791pte^{-i\varepsilon_{n}\omega_{n}u_{n}}S_{n}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)\\\
\end{split}$ (49)

where in the last line above we have used the identity

$\begin{split}&\delta(x)=\frac{1}{2}\lim_{\epsilon\to 0}\epsilon\hskip
1.13791pt|x|^{\epsilon-1}\end{split}$ (50)

Using this limit representation of the delta function inside the $\omega_{n}$
integral is well defined only if remaining integrand does not diverge for
large $\omega_{n}$. In our case this is a valid operation since as we argued
in section (2), the factor $e^{-i\varepsilon_{n}\omega_{n}u_{n}}$ guarantees
convergence of the integral for large $\omega_{n}$ upon analytically
continuing $u_{n}$ to $u_{n}-i\varepsilon_{n}\delta$. Here $\delta\rightarrow
0^{+}$ and $\varepsilon_{n}=\pm 1$ if the $n$-th particle is outgoing
(incoming).

Now using (44) in equation (49) we get

$\begin{split}\lim_{\lambda_{n}\to 0}i\lambda_{n}\hskip
1.13791pt\mathcal{A}_{n}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})&=-\int\prod_{k=1}^{n-1}d\omega_{k}\hskip
1.42271pt\omega_{k}^{i\lambda_{k}}\hskip
2.84544pte^{-i\sum\limits_{k=1}^{n-1}\varepsilon_{k}\omega_{k}u_{k}}\hskip
1.70709pt\sum_{i=1}^{n-1}\left(\frac{\varepsilon_{i}\omega_{i}}{\varepsilon_{n}}\frac{\bar{z}_{ni}z_{xi}z_{yi}}{z_{ni}z_{xn}z_{yn}}\right)S_{n-1}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)\\\
&=-i\varepsilon_{n}\sum_{j=1}^{n-1}\frac{\bar{z}_{ni}z_{xi}z_{yi}}{z_{ni}z_{xn}z_{yn}}\frac{\partial}{\partial
u_{j}}\int\prod_{k=1}^{n-1}d\omega_{k}\hskip
1.42271pt\omega_{k}^{i\lambda_{k}}\hskip
2.84544pte^{-i\sum\limits_{k=1}^{n-1}\varepsilon_{k}\omega_{k}u_{k}}\hskip
1.70709ptS_{n-1}\left(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\right)\\\
&=-\left(i\varepsilon_{n}\sum_{i=1}^{n-1}\frac{\bar{z}_{ni}}{z_{ni}}\frac{z_{xi}z_{yi}}{z_{xn}z_{yn}}\frac{\partial}{\partial
u_{i}}\right)\mathcal{A}_{n-1}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})\end{split}$
(51)

where in the second line above we have used,

$\begin{split}\omega_{j}e^{-i\varepsilon_{j}\omega_{j}u_{j}}=i\varepsilon_{j}\frac{\partial}{\partial
u_{j}}\left(e^{-i\varepsilon_{j}\omega_{j}u_{j}}\right)\end{split}$ (52)

Thus in the conformal soft limit the Mellin transformed graviton amplitude
takes the form

$\begin{split}\lim_{\lambda_{n}\to 0}i\lambda_{n}\hskip
1.13791pt\mathcal{A}_{n}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})&=\tilde{\mathcal{F}}^{(0)}_{+}\mathcal{A}_{n-1}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})\end{split}$
(53)

where we have defined the conformal soft factor
$\tilde{\mathcal{F}}^{(0)}_{+}$ to be

$\begin{split}\tilde{\mathcal{F}}^{(0)}_{+}=-i\varepsilon_{n}\sum_{i=1}^{n-1}\frac{\bar{z}_{ni}}{z_{ni}}\frac{z_{xi}z_{yi}}{z_{xn}z_{yn}}\frac{\partial}{\partial
u_{i}}\end{split}$ (54)

Similarly for a negative helicity soft graviton, we obtain

$\begin{split}\lim_{\lambda_{n}\to 0}i\lambda_{n}\hskip
1.13791pt\mathcal{A}_{n}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})&=\tilde{\mathcal{F}}^{(0)}_{-}\mathcal{A}_{n-1}(u_{i},z_{i},\bar{z}_{i},\lambda_{i},\sigma_{i})\end{split}$
(55)

where

$\begin{split}\tilde{\mathcal{F}}^{(0)}_{-}=-i\varepsilon_{n}\sum_{i=1}^{n-1}\frac{z_{ni}}{\bar{z}_{ni}}\frac{\bar{z}_{xi}\bar{z}_{yi}}{\bar{z}_{xn}\bar{z}_{yn}}\frac{\partial}{\partial
u_{i}}\end{split}$ (56)

Now, supertranslation acts naturally on the modified amplitude
$\mathcal{A}(u_{i},z_{i},\bar{z}_{i},\sigma_{i})$ by shifting $u_{i}$ to
$u_{i}+f(z_{i},\bar{z}_{i})$. The appearance of the derivative
$\partial/\partial u$ is related to the fact that the leading soft graviton
theorem is essentially the Ward identity for supertranslation
Strominger:2013jfa . In Puhm:2019zbl  the analog of $\partial/\partial u$,
acting on the lower point amplitude, is the shift of each $\lambda$ to
$\lambda-i$ in the lower point amplitude appearing on the R.H.S of the
conformal soft theorem for gravitons.

In the next subsection we will explicitly check this general result in the
case of $4$-point MHV graviton amplitudes. For this let us first compute the
modified Mellin transform of $3$-point graviton amplitudes which will now
appear on the R.H.S. of the soft theorem.

###  4.2 $3$-point MHV amplitude

Since $3$-point graviton amplitudes vanish in ordinary Minkowski signature, in
this section we will work with the mixed signature spacetime metric
$(-,+,-,+)$. In this case the parametrisation of null momenta becomes

$\begin{split}p_{i}=\omega_{i}(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},z_{i}-\bar{z}_{i},1-z_{i}\bar{z}_{i})\end{split}$
(57)

where $z_{i},\bar{z}_{i}$ are now independent real numbers. The $3$-point
graviton MHV amplitude is then given by

$\begin{split}S(1^{--},2^{--},3^{++})(\omega_{i},z_{i},\bar{z}_{i})&=\frac{\kappa}{2}\frac{\langle
12\rangle^{6}}{\langle 23\rangle^{2}\langle
31\rangle^{2}}\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}p_{i}^{\mu}\right)\\\
&=2\kappa\hskip
1.70709pt\frac{\omega_{1}^{2}\omega_{2}^{2}}{\omega_{3}^{2}}\frac{z_{12}^{6}}{z_{23}^{2}z_{31}^{2}}\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\omega_{i}q_{i}^{\mu}\right)\end{split}$
(58)

As in the case of $4$-point amplitudes, it is convenient to first define the
simplex variables, $\sigma_{i}=s^{-1}\omega_{i},s=\sum_{i}\omega_{i}$ before
computing the Mellin transform of the amplitude. Then we can write the
momentum conservation imposing delta function as Pasterski:2017ylz

$\begin{split}\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\omega_{i}q_{i}^{\mu}\right)=s^{-4}\hskip
1.42271pt\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\sigma_{i}q_{i}^{\mu}\right)\delta\left(\sum_{i=1}^{3}\sigma_{i}-1\right)=\frac{\delta(z_{12})\delta(z_{13})}{4\sigma_{1}\sigma_{2}\sigma_{3}D_{3}^{2}}\prod_{i=1}^{3}\delta(\sigma_{i}-\sigma^{*}_{i})\end{split}$
(59)

where

$\begin{split}&\sigma_{1}^{*}=\frac{z_{23}}{D_{3}},\quad\sigma_{2}^{*}=-\varepsilon_{1}\varepsilon_{2}\frac{z_{13}}{D_{3}},\quad\sigma_{3}^{*}=\varepsilon_{1}\varepsilon_{3}\frac{z_{12}}{D_{3}}\\\
&D_{3}=(1-\varepsilon_{1}\varepsilon_{2})z_{13}+(\varepsilon_{1}\varepsilon_{3}-1)z_{12}\end{split}$
(60)

Now the modified Mellin transform of the $3$-point MHV amplitude is

$\begin{split}&\mathcal{A}(1^{--},2^{--},3^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\prod_{i=1}^{3}\int d\omega_{i}\hskip
1.42271pt\omega_{i}^{i\lambda_{i}}\hskip
1.42271pte^{-i\epsilon_{i}\omega_{i}u_{i}}S(1^{--},2^{--},3^{++})(\omega_{i},z_{i},\bar{z}_{i})\\\
&=2\hskip 1.13791pt\kappa\int\prod_{i=1}^{3}d\sigma_{i}\hskip
1.42271pt\sigma_{i}^{i\lambda_{i}}\hskip 1.42271pt\int ds\hskip
1.42271pts^{i\sum\limits_{i}\lambda_{i}}e^{-is\sum\limits_{i}\epsilon_{i}\sigma_{i}u_{i}}\hskip
1.42271pt\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\sigma_{3}^{2}}\frac{z_{12}^{6}}{z_{23}^{2}z_{31}^{2}}\hskip
1.42271pt\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\sigma_{i}q_{i}^{\mu}\right)\delta\left(\sum_{i=1}^{3}\sigma_{i}-1\right)\end{split}$
(61)

The integral over the energy scale $s$ yields,

$\int ds\hskip
1.42271pts^{i\sum_{i}\lambda_{i}}e^{-is\sum_{i}\epsilon_{i}\sigma_{i}u_{i}}=\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(1+i\Lambda)}{\left(i\sum\varepsilon_{i}\sigma_{i}u_{i}+\delta\right)^{1+i\Lambda}}\right]$
(62)

where $\Lambda=\sum\limits_{i=1}^{3}\lambda_{i}$. Then performing the
$\sigma_{i}$ integrals using the representation of the delta functions in
equation (59) we get

$\begin{split}&\mathcal{A}(1^{--},2^{--},3^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\hskip 1.13791pt\kappa\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(1+i\Lambda)}{\left(i\sum\varepsilon_{i}\sigma^{*}_{i}u_{i}+\delta\right)^{1+i\Lambda}}\right]\prod_{i=1}^{3}(\sigma_{i}^{*})^{i\lambda_{i}}\left(\frac{\sigma_{1}^{*}\sigma_{2}^{*}}{\sigma_{3}^{*}}\frac{z_{12}^{3}}{z_{23}z_{31}}\right)^{2}\frac{\delta(\bar{z}_{12})\delta(\bar{z}_{13})}{2\sigma_{1}^{*}\sigma_{2}^{*}\sigma_{3}^{*}D_{3}^{2}}\prod_{i=1}^{3}\mathbf{1}_{[0,1]}\left(\sigma^{*}_{i}\right)\\\
\end{split}$ (63)

where the indicator function
$\prod\limits_{i=1}^{3}\mathbf{1}_{[0,1]}\left(\sigma^{*}_{i}\right)$ imposes
the constraint that $\sigma_{i}^{*}\in[0,1]$.

###  4.3 $3$-point $\overline{\mathrm{MHV}}$ amplitude

Here we will compute the modified Mellin transform of the $3$-point graviton
$\overline{\mathrm{MHV}}$ amplitude. In momentum space this is given by

$\begin{split}S(1^{++},2^{++},3^{--})(\omega_{i},z_{i},\bar{z}_{i})&=\frac{\kappa}{2}\frac{[12]^{6}}{[23]^{2}[31]^{2}}\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}p_{i}^{\mu}\right)\\\
&=2\hskip 1.13791pt\kappa\hskip
1.70709pt\frac{\omega_{1}^{2}\omega_{2}^{2}}{\omega_{3}^{2}}\frac{\bar{z}_{12}^{6}}{\bar{z}_{23}^{2}\bar{z}_{31}^{2}}\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\omega_{i}q_{i}^{\mu}\right)\end{split}$
(64)

Now the modified Mellin transform can be obtained following essentially
identical steps as for the MHV amplitude in the previous section. The final
result is simply given by equation (LABEL:mellin3gravmhv) with the
subsitiution $z_{ij}\rightarrow\bar{z}_{ij}$ as follows

$\begin{split}&\mathcal{A}(1^{++},2^{++},3^{--})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\hskip 1.13791pt\kappa\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(1+i\Lambda)}{\left(i\sum\varepsilon_{i}\sigma^{*}_{i}u_{i}+\delta\right)^{1+i\Lambda}}\right]\prod_{i=1}^{3}(\sigma_{i}^{*})^{i\lambda_{i}}\left(\frac{\sigma_{1}^{*}\sigma_{2}^{*}}{\sigma_{3}^{*}}\frac{\bar{z}_{12}^{3}}{\bar{z}_{23}{z}_{31}}\right)^{2}\frac{\delta(z_{12})\delta(z_{13})}{2\sigma_{1}^{*}\sigma_{2}^{*}\sigma_{3}^{*}D_{3}^{2}}\prod_{i=1}^{3}\mathbf{1}_{[0,1]}\left(\sigma^{*}_{i}\right)\\\
\end{split}$ (65)

where now we have

$\begin{split}&\sigma_{1}^{*}=\frac{\bar{z}_{23}}{D_{3}},\quad\sigma_{2}^{*}=-\varepsilon_{1}\varepsilon_{2}\frac{\bar{z}_{13}}{D_{3}},\quad\sigma_{3}^{*}=\varepsilon_{1}\varepsilon_{3}\frac{\bar{z}_{12}}{D_{3}}\\\
&D_{3}=(1-\varepsilon_{1}\varepsilon_{2})\bar{z}_{13}+(\varepsilon_{1}\varepsilon_{3}-1)\bar{z}_{12}\end{split}$
(66)

###  4.4 Conformal soft limit of MHV graviton amplitudes

We can now verify the conformal soft factorization result obtained in
equations (53) and (55) for $4$-point tree-level MHV graviton amplitudes. Our
analysis in this section will be similar to the one carried out in
Puhm:2019zbl . Here again we will take the signature of the spacetime metric
to be $(-,+,-,+)$.

The modified Mellin transform of the $4$-point MHV amplitude was computed in
section (3.1) and is given by

$\begin{split}&\mathcal{A}(1^{--},2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\frac{\kappa^{2}}{4}\bigg{[}\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\Lambda)}{(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta)^{2+i\Lambda}}\bigg{]}\left(\prod_{i=1}^{4}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{2}^{\ast}\sigma_{3}^{\ast}\sigma_{4}^{\ast}}{\sigma_{1}^{\ast}}f(z_{i},\bar{z}_{i})\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})\end{split}$
(67)

where we have defined

$\begin{split}f(z_{i},\bar{z}_{i})=\frac{z_{12}^{3}\bar{z}_{34}^{4}}{\bar{z}_{12}z_{13}\bar{z}_{13}z_{14}\bar{z}_{14}}\delta\left(z_{12}z_{34}\bar{z}_{13}\bar{z}_{24}-z_{13}z_{24}\bar{z}_{12}\bar{z}_{34}\right),\quad\quad\Lambda=\sum_{i=1}^{4}\lambda_{i}\end{split}$
(68)

with $z_{i},\bar{z}_{i}$ being independent real quantities. Now let us
consider the conformal soft limit of the Mellin amplitude in equation (67) by
taking the $1$-st negative helicity particle to be soft. Thus we have

$\begin{split}&\lim_{\lambda_{1}\to 0}i\lambda_{1}\hskip
1.42271pt\mathcal{A}(1^{--},2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\frac{\kappa^{2}}{4}\left(\lim_{\lambda_{1}\to
0}i\lambda_{1}\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\Lambda)}{\left(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta\right)^{2+i\Lambda}}\right]\sigma_{1}^{\ast
i\lambda_{1}-1}\right)\left(\prod_{i=2}^{4}\sigma_{i}^{\ast
1+i\lambda_{i}}\right)f(z_{i},\bar{z}_{i})\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})\end{split}$
(69)

Then using the following identity222Note that there is no factor of $1/2$ here
as opposed to equation (50). This is because the indicator function
$\textbf{1}_{[0,1]}(\sigma_{1}^{\ast})$ ensures that $\sigma_{1}\in[0,1]$.

$\begin{split}&\delta(x)=\lim_{\epsilon\to 0}\epsilon\hskip
1.13791ptx^{\epsilon-1},\quad\quad(0\leq x\leq 1)\end{split}$ (70)

we get,

$\begin{split}\lim_{\lambda_{1}\to 0}i\lambda_{1}\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\Lambda)}{\left(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta\right)^{2+i\Lambda}}\right]\sigma_{1}^{\ast
i\lambda_{1}-1}&=\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\Lambda^{\prime})}{\left(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta\right)^{2+i\Lambda^{\prime}}}\right]\delta(\sigma_{1}^{*})\\\
\end{split}$ (71)

where $\Lambda^{\prime}=\sum\limits_{i=2}^{4}\lambda_{i}$. Assuming
$\bar{z}_{34}\neq 0$ the delta function in the above expression gives

$\begin{split}\delta(\sigma_{1}^{*})=\delta\left(-\frac{\varepsilon_{1}\varepsilon_{4}}{D}\frac{\bar{z}_{34}z_{24}}{\bar{z}_{13}z_{12}}\right)=\left|\frac{z_{12}\bar{z}_{13}}{\bar{z}_{34}}D\right|\delta(z_{24})\end{split}$
(72)

Further on the support of $\delta\left(z_{24}\right)$, we have

$\begin{split}f(z_{i},\bar{z}_{i})&\rightarrow\frac{z_{12}^{3}\bar{z}_{34}^{4}}{z_{13}z_{14}\bar{z}_{12}\bar{z}_{13}\bar{z}_{14}}\left|\frac{1}{z_{12}\bar{z}_{13}\bar{z}_{24}}\right|\delta\left(z_{34}\right)\\\
&=\frac{\bar{z}_{34}^{4}}{\bar{z}_{12}\bar{z}_{13}\bar{z}_{14}}\left|\frac{1}{\bar{z}_{13}\bar{z}_{24}}\right|\mathrm{sgn}(z_{12})\delta(z_{34})\end{split}$
(73)

Now due to the delta functions $\delta\left(z_{24}\right)$ and
$\delta\left(z_{34}\right)$, the remaining simplex variables
$\sigma^{*}_{2},\sigma^{*}_{3}$ and $\sigma^{*}_{4}$ become

$\begin{split}&\sigma_{2}^{*}\rightarrow\rho^{*}_{2}=\varepsilon_{2}\varepsilon_{4}\frac{\bar{z}_{43}}{D_{3}},\quad\sigma_{3}^{*}\rightarrow\rho^{*}_{3}=-\varepsilon_{3}\varepsilon_{4}\frac{\bar{z}_{42}}{D_{3}},\quad\sigma_{4}^{*}\rightarrow\rho^{*}_{4}=\frac{\bar{z}_{32}}{D_{3}}\\\
\end{split}$ (74)

where

$\begin{split}D\rightarrow\frac{D_{3}}{\bar{z}_{23}},\quad
D_{3}=(1-\varepsilon_{4}\varepsilon_{3})\bar{z}_{42}+(\varepsilon_{4}\varepsilon_{2}-1)\bar{z}_{43}\end{split}$
(75)

Combining the above results we then obtain

$\begin{split}\left(\prod_{i=2}^{4}\sigma_{i}^{\ast
1+i\lambda_{i}}\right)f(z_{i},\bar{z}_{i})\delta(\sigma_{1}^{*})&\rightarrow\left(\prod_{i=2}^{4}\rho_{i}^{\ast
i\lambda_{i}}\right)\frac{z_{12}}{\bar{z}_{12}}\frac{\bar{z}_{43}^{4}}{\bar{z}_{13}\bar{z}_{14}D_{3}^{2}}\hskip
1.42271pt\mathrm{sgn}(\varepsilon_{2}\varepsilon_{3}\bar{z}_{34}\bar{z}_{42}\bar{z}_{32}D_{3})\delta(z_{24})\delta(z_{34})\\\
&=\left(\prod_{i=2}^{4}\rho_{i}^{\ast
i\lambda_{i}}\right)\frac{z_{12}}{\bar{z}_{12}}\frac{\bar{z}_{43}^{4}}{\bar{z}_{13}\bar{z}_{14}D_{3}^{2}}\hskip
1.42271pt\delta(z_{24})\delta(z_{34})\end{split}$ (76)

Note that the last line in the above expression follows from the fact that due
to the $3$-particle indicator function
$\prod_{i=2}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$, we get
$\mathrm{sgn}(\varepsilon_{2}\varepsilon_{3}\bar{z}_{34}\bar{z}_{42}\bar{z}_{32}D_{3})=\mathrm{sgn}\left(\rho^{*}_{2}\rho^{*}_{3}\rho^{*}_{4}\right)=1$.

Now equation (69) takes the form

$\begin{split}&\lim_{\lambda_{1}\to 0}i\lambda_{1}\hskip
1.42271pt\mathcal{A}(1^{--},2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\frac{\kappa^{2}}{4}\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(2+i\Lambda^{\prime})}{\left(i\sum\varepsilon_{i}\rho_{i}^{\ast}u_{i}+\delta\right)^{2+i\Lambda^{\prime}}}\right]\left(\prod_{i=2}^{4}\rho_{i}^{\ast
i\lambda_{i}}\right)\frac{z_{12}}{\bar{z}_{12}}\frac{\bar{z}_{43}^{4}}{\bar{z}_{13}\bar{z}_{14}D_{3}^{2}}\hskip
1.42271pt\delta(z_{24})\delta(z_{34})\prod_{i=2}^{4}\textbf{1}_{[0,1]}(\rho_{i}^{\ast})\\\
&=-\frac{\kappa}{2}\hskip
1.13791pt\varepsilon_{3}\varepsilon_{4}\left[\lim_{\delta\rightarrow
0+}\frac{1+i\Lambda^{\prime}}{\left(i\sum\varepsilon_{i}\rho_{i}^{\ast}u_{i}+\delta\right)}\right]\rho^{*}_{2}\left(\frac{z_{12}}{\bar{z}_{12}}\frac{\bar{z}_{32}\bar{z}_{42}}{\bar{z}_{31}\bar{z}_{41}}\right)\hskip
1.42271pt\mathcal{A}(2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\end{split}$
(77)

where $\mathcal{A}(2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})$
is the modified Mellin transform of the $3$-point $\overline{\mathrm{MHV}}$
graviton amplitude
$S(2^{--},3^{++},4^{++})(\omega_{i},z_{i},\bar{z}_{i})$.This was evaluated in
section (4.3) and is given by

$\begin{split}&\mathcal{A}(2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\kappa\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma(1+i\Lambda^{\prime})}{\left(i\sum\varepsilon_{i}\rho_{i}^{\ast}u_{i}+\delta\right)^{1+i\Lambda^{\prime}}}\right]\left(\prod_{i=2}^{4}\rho_{i}^{\ast
i\lambda_{i}}\right)\left(\frac{\rho^{*}_{4}\rho^{*}_{3}}{\rho^{*}_{2}}\frac{\bar{z}_{43}^{3}}{\bar{z}_{32}\bar{z}_{24}}\right)^{2}\frac{\delta(z_{24})\delta(z_{34})}{2\rho^{*}_{4}\rho^{*}_{3}\rho^{*}_{2}D_{3}^{2}}\prod_{i=2}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})\end{split}$
(78)

Let us now differentiate the above amplitude with respect to $u_{2}$. This
yields,

$\begin{split}&\frac{\partial}{\partial
u_{2}}\mathcal{A}(2^{--}3^{++}4^{++})=-\left[\lim_{\delta\rightarrow
0+}\frac{1+i\Lambda^{\prime}}{\left(i\sum\varepsilon_{i}\rho_{i}^{\ast}u_{i}+\delta\right)}\right]i\epsilon_{2}\rho^{*}_{2}\hskip
1.42271pt\mathcal{A}(2^{--},3^{++},4^{++})\\\ \end{split}$ (79)

Using this in equation (77) we finally get

$\begin{split}&\lim_{\lambda_{1}\to 0}i\lambda_{1}\hskip
1.42271pt\mathcal{A}(1^{--},2^{--},3^{++},4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\\\
&=\frac{\kappa}{2}\left(-i\varepsilon_{2}\varepsilon_{3}\varepsilon_{4}\frac{z_{12}}{\bar{z}_{12}}\frac{\bar{z}_{32}\bar{z}_{42}}{\bar{z}_{31}\bar{z}_{41}}\right)\frac{\partial}{\partial
u_{2}}\left(\mathcal{A}(2^{--}3^{++}4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\right)\\\
&=\frac{\kappa}{2}\left(-i\varepsilon_{1}\frac{z_{12}}{\bar{z}_{12}}\frac{\bar{z}_{32}\bar{z}_{42}}{\bar{z}_{31}\bar{z}_{41}}\right)\frac{\partial}{\partial
u_{2}}\left(\mathcal{A}(2^{--}3^{++}4^{++})(u_{i},z_{i},\bar{z}_{i},\lambda_{i})\right)\end{split}$
(80)

In going from the second to the last line in the above we have used the fact
$\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\varepsilon_{4}=1$ when either
two particles are incoming and two particles are outgoing or all $4$ particles
are incoming (outgoing).

Evidently the result in (80) agrees with the general case considered in
equation (55) if we choose the reference spinors to be $x=3,y=4$.

###  4.5 Conformal soft limit in gluon four point amplitude

Conformal soft limit of celestial amplitudes in gauge theory has been studied
in Pate:2019mfs . In this section we show how to take soft limit in modified
Mellin transformed gauge amplitude. The 4-point modified Mellin MHV amplitude
in Yang-Mills theory in $\left(-,+,+,+\right)$ spacetime signature obtained in
Eq.(3.2) can be expressed as

$\displaystyle\mathcal{A}\left(1^{-}2^{-}3^{+}4^{+}\right))(u_{i},z_{i},\bar{z}_{i},\lambda_{i})$
$\displaystyle=$ $\displaystyle\frac{1}{4}\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma\left(i\sum\limits_{i=1}^{4}\lambda_{i}\right)}{\left(i\sum\varepsilon_{i}\sigma_{i}^{\ast}u_{i}+\delta\right)^{i\sum\limits_{i=1}^{4}\lambda_{i}}}\right]\left(\prod\limits_{i=1}^{4}\sigma_{i}^{\ast
i\lambda_{i}}\right)\frac{\sigma_{1}^{\ast}\sigma_{2}^{\ast}}{\sigma_{3}^{\ast}\sigma_{4}^{\ast}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\frac{\delta(|z-\bar{z}|)}{z_{13}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})$
$\displaystyle=$ $\displaystyle\frac{1}{4}\left[\lim_{\delta\rightarrow
0+}\frac{\Gamma\left(i\Lambda\right)}{\left(i\varepsilon_{4}\bigl{\\{}-\frac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}u_{1}+\frac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}u_{2}-\frac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{12}}u_{3}+u_{4}\bigr{\\}}+\delta\right)^{i\Lambda}}\right]\left(\frac{\varepsilon_{3}}{\varepsilon_{1}}\frac{z_{23}\bar{z}_{34}}{z_{12}\bar{z}_{14}}\right)^{1+i\lambda_{1}}\left(\frac{\varepsilon_{3}}{\varepsilon_{2}}\frac{z_{13}\bar{z}_{34}}{z_{12}\bar{z}_{42}}\right)^{1+i\lambda_{2}}$
$\displaystyle\times\left(\frac{\varepsilon_{3}}{\varepsilon_{4}}\frac{\bar{z}_{13}z_{23}}{\bar{z}_{14}z_{42}}\right)^{-1+i\lambda_{4}-i\Lambda}\delta\left(|z_{12}z_{34}\bar{z}_{13}\bar{z}_{24}-z_{13}z_{24}\bar{z}_{12}\bar{z}_{34}|\right)\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast}).$

Here $\Lambda=\sum\limits_{i=1}^{4}\lambda_{i}$. The indicator functions
signify that $\sigma_{i}^{\ast}$ should be positive otherwise the amplitude
vanishes. For two-to-two scattering process, $ij\rightleftarrows kl$ with
$\varepsilon_{i}=\varepsilon_{j}=-\varepsilon_{k}=-\varepsilon_{l}$ the
indicator functions imply

$\displaystyle 12\rightleftarrows 34$ $\displaystyle\Rightarrow$
$\displaystyle z_{1}>z_{3}>z_{2}>z_{4}\quad\text{or}\quad
z_{1}<z_{3}<z_{2}<z_{4},$ $\displaystyle 13\rightleftarrows 24$
$\displaystyle\Rightarrow$ $\displaystyle
z_{1}>z_{2}>z_{3}>z_{4}\quad\text{or}\quad z_{1}<z_{2}<z_{3}<z_{4},$
$\displaystyle 14\rightleftarrows 23$ $\displaystyle\Rightarrow$
$\displaystyle z_{1}>z_{3}>z_{4}>z_{2}\quad\text{or}\quad
z_{1}<z_{3}<z_{4}<z_{2}.$ (82)

Similar orderings exist for $\bar{z}_{ij}$. It can be checked that above
conditions are consistent with replacing the product of indicator functions by
step functions as

$\prod_{i=1}^{4}\textbf{1}_{[0,1]}(\sigma_{i}^{\ast})=\Theta\left(\frac{\varepsilon_{3}}{\varepsilon_{1}}\frac{z_{23}\bar{z}_{34}}{z_{12}\bar{z}_{14}}\right)\Theta\left(\frac{\varepsilon_{3}}{\varepsilon_{2}}\frac{z_{13}\bar{z}_{34}}{z_{12}\bar{z}_{42}}\right)\Theta\left(\frac{\varepsilon_{3}}{\varepsilon_{4}}\frac{\bar{z}_{13}z_{23}}{\bar{z}_{14}z_{42}}\right).$
(83)

Including the factor
$\left(\frac{\varepsilon_{3}}{\varepsilon_{4}}\frac{\bar{z}_{13}z_{23}}{\bar{z}_{14}z_{42}}\right)^{-i\Lambda}$
denominator of Eq.(4.5) becomes

$\lim_{\delta\rightarrow
0+}\left[i\varepsilon_{3}\left(\frac{z_{23}\bar{z}_{34}}{z_{12}\bar{z}_{14}}u_{1}+\frac{z_{13}\bar{z_{34}}}{z_{12}\bar{z}_{24}}u_{2}+\frac{\bar{z}_{13}}{\bar{z}_{12}}u_{3}+\frac{\bar{z}_{13}z_{23}}{\bar{z}_{14}z_{42}}u_{4}\right)+\delta\right]^{i\Lambda}.$
(84)

To take soft limit we multiply both sides of Eq.(4.5) by
$\lim_{\lambda_{4}\rightarrow 0}i\lambda_{4}$. Following Eq.(50) we get

$\lim_{\lambda_{4}\rightarrow
0}i\lambda_{4}\left(\frac{\varepsilon_{3}}{\varepsilon_{4}}\frac{\bar{z}_{13}z_{23}}{\bar{z}_{14}z_{42}}\right)^{-1+i\lambda_{4}}=2\delta\left(\frac{\varepsilon_{3}}{\varepsilon_{4}}\frac{\bar{z}_{13}z_{23}}{\bar{z}_{14}z_{42}}\right)=\text{sgn}\left(\bar{z}_{14}z_{42}z_{23}\right)2\frac{\bar{z}_{14}z_{42}}{z_{23}}\delta\left(\bar{z}_{13}\right).$
(85)

Using the above relation delta function appearing in Eq.(4.5) becomes

$\delta\left(|z_{12}z_{34}\bar{z}_{13}\bar{z}_{24}-z_{13}z_{24}\bar{z}_{12}\bar{z}_{34}|\right)=\frac{\text{sgn}\left(z_{13}z_{24}\bar{z}_{34}\right)}{z_{13}z_{24}\bar{z}_{34}}\delta\left(\bar{z}_{12}\right).$
(86)

These two constraints also imply $\bar{z}_{12}=\bar{z}_{13}=\bar{z}_{23}=0$.
Then in the soft limit Eq.(84) takes the form

$\lim_{\delta\rightarrow
0+}\left[i\varepsilon_{3}z_{12}^{-1}\left(z_{23}u_{1}+z_{13}u_{2}+z_{12}u_{3}\right)+\delta\right]^{i\sum\limits_{i=1}^{3}\lambda_{i}}.$
(87)

We note that in the soft limit argument inside the last $\Theta$ function in
Eq.(83) vanishes which gives $\Theta\left(0\right)=\frac{1}{2}$. Then
combining the delta functions and other $z$-dependent factors we get

$\displaystyle\lim_{\lambda_{4}\rightarrow
0}i\lambda_{4}\mathcal{A}\left(1^{-}2^{-}3^{+}4^{+}\right)(u_{i},z_{i},\bar{z}_{i},\lambda_{i})$
(88) $\displaystyle=$
$\displaystyle\text{sgn}\left(z_{23}z_{31}\right)\frac{z_{31}}{z_{34}z_{41}}\delta\left(\bar{z}_{12}\right)\delta\left(\bar{z}_{13}\right)\varepsilon_{1}\varepsilon_{2}\varepsilon_{1}^{i\lambda_{1}}\varepsilon_{2}^{i\lambda_{2}}\varepsilon_{3}^{i\lambda_{3}}z_{12}^{1+i\lambda_{3}}z_{23}^{-1+i\lambda_{1}}z_{31}^{-1+i\lambda_{2}}$
$\displaystyle\frac{1}{4}\lim_{\delta\rightarrow
0}\frac{\Gamma\left(i\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)\right)}{\left[i\left(u_{1}z_{23}+u_{2}z_{31}+u_{3}z_{12}\right)+\delta\right]^{i\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)}}\Theta\left(\frac{\varepsilon_{3}}{\varepsilon_{1}}\frac{z_{23}}{z_{12}}\right)\Theta\left(\frac{\varepsilon_{3}}{\varepsilon_{2}}\frac{z_{31}}{z_{12}}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{z_{31}}{z_{34}z_{41}}\mathcal{A}\left(1^{-}2^{-}3^{+}\right).$

Here the 3-point gluon amplitude is expressed in spacetime with split
signature $\left(-,+,-,+\right)$. In this case 3-point gluon amplitude can be
worked out in analogous way to the graviton 3-point amplitude.

## Acknowledgement

We would like to thank Alok Laddha and Ashoke Sen for valuable discussions.
This research was supported in part by the International Centre for
Theoretical Sciences (ICTS) during a visit for participating in the program -
AdS/CFT at 20 and Beyond (Code: ICTS/adscft20/05). SG would like to thank
Yasha Neiman, Vyacheslav Lysov and Lin-Qing Chen for useful discussions and
acknowledges the hospitality of the Korea Institute for Advanced Study (KIAS),
Seoul, South Korea while this work was being completed. SG is supported by the
Quantum Gravity Unit of the Okinawa Institute of Science and Technology
Graduate University (OIST).

## References

  * (1) S. Pasterski, S. H. Shao and A. Strominger, “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,” Phys. Rev. D 96, no. 6, 065026 (2017) doi:10.1103/PhysRevD.96.065026 [arXiv:1701.00049 [hep-th]].
  * (2) S. Pasterski and S. H. Shao, “Conformal basis for flat space amplitudes,” Phys. Rev. D 96, no. 6, 065022 (2017) doi:10.1103/PhysRevD.96.065022 [arXiv:1705.01027 [hep-th]].
  * (3) S. Pasterski, S. H. Shao and A. Strominger, “Gluon Amplitudes as 2d Conformal Correlators,” Phys. Rev. D 96, no. 8, 085006 (2017) doi:10.1103/PhysRevD.96.085006 [arXiv:1706.03917 [hep-th]].
  * (4) S. Banerjee, “Null Infinity and Unitary Representation of The Poincare Group,” JHEP 1901, 205 (2019) doi:10.1007/JHEP01(2019)205 [arXiv:1801.10171 [hep-th]].
  * (5) S. Banerjee, “Symmetries of free massless particles and soft theorems,” arXiv:1804.06646 [hep-th].
  * (6) A. Schreiber, A. Volovich and M. Zlotnikov, “Tree-level gluon amplitudes on the celestial sphere,” arXiv:1711.08435 [hep-th].
  * (7) C. Cardona and Y. t. Huang, “S-matrix singularities and CFT correlation functions,” JHEP 1708, 133 (2017) doi:10.1007/JHEP08(2017)133 [arXiv:1702.03283 [hep-th]].
  * (8) H. T. Lam and S. H. Shao, “Conformal Basis, Optical Theorem, and the Bulk Point Singularity,” arXiv:1711.06138 [hep-th].
  * (9) N. Banerjee, S. Banerjee, S. Atul Bhatkar and S. Jain, “Conformal Structure of Massless Scalar Amplitudes Beyond Tree level,” arXiv:1711.06690 [hep-th].
  * (10) S. Stieberger and T. R. Taylor, “Strings on Celestial Sphere,” Nucl. Phys. B 935, 388 (2018) doi:10.1016/j.nuclphysb.2018.08.019 [arXiv:1806.05688 [hep-th]].
  * (11) S. Stieberger and T. R. Taylor, “Symmetries of Celestial Amplitudes,” Phys. Lett. B 793, 141 (2019) doi:10.1016/j.physletb.2019.03.063 [arXiv:1812.01080 [hep-th]].
  * (12) W. Fan, A. Fotopoulos and T. R. Taylor, “Soft Limits of Yang-Mills Amplitudes and Conformal Correlators,” JHEP 1905, 121 (2019) doi:10.1007/JHEP05(2019)121 [arXiv:1903.01676 [hep-th]].
  * (13) M. Pate, A. M. Raclariu and A. Strominger, “Conformally Soft Theorem in Gauge Theory,” arXiv:1904.10831 [hep-th].
  * (14) D. Nandan, A. Schreiber, A. Volovich and M. Zlotnikov, “Celestial Amplitudes: Conformal Partial Waves and Soft Limits,” arXiv:1904.10940 [hep-th].
  * (15) L. Donnay, A. Puhm and A. Strominger, “Conformally Soft Photons and Gravitons,” JHEP 1901, 184 (2019) doi:10.1007/JHEP01(2019)184 [arXiv:1810.05219 [hep-th]].
  * (16) T. Adamo, L. Mason and A. Sharma, “Celestial amplitudes and conformal soft theorems,” arXiv:1905.09224 [hep-th].
  * (17) A. Puhm, “Conformally Soft Theorem In Gravity,” arXiv:1905.09799 [hep-th].
  * (18) A. Guevara, “Notes on Conformal Soft Theorems and Recursion Relations in Gravity,” arXiv:1906.07810 [hep-th].
  * (19) S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516 (1965). doi:10.1103/PhysRev.140.B516
  * (20) F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” arXiv:1404.4091 [hep-th].
  * (21) H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A 269, 21 (1962). doi:10.1098/rspa.1962.0161 R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270, 103 (1962). doi:10.1098/rspa.1962.0206
  * (22) R. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128, 2851 (1962). doi:10.1103/PhysRev.128.2851
  * (23) A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 1407, 152 (2014) doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]]. T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg?s soft graviton theorem,” JHEP 1505, 151 (2015) doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]]. A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 1601, 086 (2016) doi:10.1007/JHEP01(2016)086 [arXiv:1411.5745 [hep-th]].
  * (24) G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105, 111103 (2010) doi:10.1103/PhysRevLett.105.111103 [arXiv:0909.2617 [gr-qc]]. G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS CNCFG 2010, 010 (2010) [Ann. U. Craiova Phys. 21, S11 (2011)] [arXiv:1102.4632 [gr-qc]].
  * (25) D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix,” JHEP 1408, 058 (2014) doi:10.1007/JHEP08(2014)058 [arXiv:1406.3312 [hep-th]]. D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “2D Stress Tensor for 4D Gravity,” Phys. Rev. Lett. 119, no. 12, 121601 (2017) doi:10.1103/PhysRevLett.119.121601 [arXiv:1609.00282 [hep-th]]. T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP 1708, 050 (2017) doi:10.1007/JHEP08(2017)050 [arXiv:1701.00496 [hep-th]].
  * (26) M. Campiglia and A. Laddha, “Asymptotic symmetries of gravity and soft theorems for massive particles,” JHEP 1512, 094 (2015) doi:10.1007/JHEP12(2015)094 [arXiv:1509.01406 [hep-th]].