Soft and Collinear Limits in $\mathcal{N}=8$ Supergravity using Double Copy Formalism

Symmetries in a quantum field theory are their most important features. The difficulty of solving a theory, which means to compute the scattering amplitudes in terms of the correlation functions, is often dictated by the amount of symmetries the theory possesses. This is because symmetries in the theory are reflected in the amplitudes via the Ward identities and one can use these identities to constrain the correlation functions. This is called bootstrapping and has been a very effective tool in conformal field theory RevModPhys.91.015002 ; Poland:2016chs ; BELAVIN1984333 . This also goes in the reverse direction, that is knowledge about the nature of amplitudes can help us discover non-trivial symmetries of the theory. Important examples include soft and collinear limits of amplitudes. Soft limit of an amplitude is defined by taking the momenta of an external particle to be zero.¹¹1Of course the particle has to be massless for such a limit to make sense. In soft limit, the amplitude factorises into a universal soft factor which contains the divergence of the amplitude times the amplitude without the soft particle. The soft factor is universal in the sense that it does not depend on the intricate detailes of the theory, but only the helicity of the external particles. This universal factorisation can be extended to subleading order in electromagnetism and to sub-subleading order in gravity Sahoo:2020ryf ; Saha:2019tub ; PhysRev.135.B1049 ; PhysRev.140.B516 ; PhysRev.166.1287 ; PhysRev.168.1623 ; He:2014laa ; Bern:2014vva ; Campiglia:2014yka ; Laddha:2017ygw ; Klose:2015xoa ; Cachazo:2014fwa ; Casali:2014xpa . Soft limits of amplitudes at tree level provided important new insights about the symmetries of certain theories when they were interpreted as Ward identities of certain non-trivial symmetries PhysRevD.97.106019 ; PhysRevLett.120.201601 ; AtulBhatkar:2019vcb ; Campiglia:2019wxe ; Hijano:2020szl . For example, soft gluon theorem in Yang-Mills theory is related to large gauge transformations Catani:2000pi ; Mao:2017tey ; Klose:2015xoa ; Casali:2014xpa ; Strominger:2017zoo and soft graviton theorem in Einstein’s gravity is related to the so called Bondi-Metzner-Sachs (BMS) symmetries Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza ; Strominger:2017zoo . Thus studying soft limits of amplitudes even at tree level teach us more about the symmetries of the theory.

Another important limit in which one can study the amplitudes is the collinear limit, in which the momenta of two external particles is taken to be collinear. Again the amplitude factorises into a collinear factor containing the divergence times the amplitude with the collinear particles replaced by another particle (see Section 3 and Section 5 for precise details). Collinear limits have played important role in flat space holography Taylor:2017sph ; Fan:2019emx . The collinear limit of amplitude turns into an operator product expansion (OPE) of conformal operators of the celestial conformal field theory (CCFT) on the celestial sphere²²2In CCFT, one describes the four dimensional physics in terms of the conformal correlators of two dimensional CFT on the celestial sphere living at the null infinities of the Minkowski flat spacetime. The map from amplitudes in the bulk to conformal correlators on the boundary is the Mellin transform. on the boundary Schreiber:2017jsr ; Fan:2019emx ; Fan:2022vbz ; Mizera:2022sln ; Pasterski:2021raf ; Pasterski:2021rjz . These OPEs can be used to calculate the non-trivial asymptotic symmetries of the theory. The usual method of calculating asymptotic symmetries is by finding conformal Killing vectors and spinors becomes intractable in the presence of other fields in the theory. That is where CCFT becomes important. A recent proposal of Taylor et. al. asserts that one can calculate the asymptotic symmetries of gravity theories using soft and collinear limit of amplitude in the framework of CCFT. This has been confirmed to give consistent results in the few cases it has been implemented Fotopoulos:2020bqj ; Banerjee:2021uxe ; Fotopoulos:2019vac . Hence the study of soft and collinear limits in gravity theories becomes important from this perspective.

$\mathcal{N}=4$ supersymmetric Yang-Mills and $\mathcal{N}=8$ supergravity are maximally supersymmetric theories and are rich in symmetries. Due to enormous symmetries, one can compute higher and higher loop amplitudes and show that they are finite Bern:2006kd . In fact people argue that these are one of the simplest quantum field theories Arkani-Hamed:2008owk . One can then study the soft and collinear limits of amplitudes in these theories to learn more about their symmetries. The study of soft and collinear limits in $\mathcal{N}=4$ SYM has already been done Golden:2012hi ; Bourjaily:2011hi ; Nandan:2012rk and the corresponding CCFT was studied in Jiang:2021xzy . On the other hand, recent investigations into gravity and gauge theory amplitudes have resulted in non-trivial relationships between the two Bern:2002kj . Gravity tree level amplitudes can be expressed in terms of sums of products of gauge theory tree level amplitudes. This can be described by different double copy formalisms Kawai:1985xq ; Bern:2008qj ; Cachazo:2013gna ; Cachazo:2013hca . One can then naturally ask if it is possible to relate the soft and collinear limits in $\mathcal{N}=4$ SYM to soft and collinear limits in $\mathcal{N}=8$ supergravity. Indeed this can be done Liu:2014vva ; Bianchi:2008pu ; Bern:1998sv ; Bern:1998xc . The relevant double copy formalism are reviewed in Adamo:2022dcm which was originally formulated as a relation between open and closed string amplitudes Kawai:1985xq . The corresponding relation in the low energy effective theory gives a relation between gauge theory and gravity amplitudes. Thus one can explicitly calculate soft and collinear limits of amplitudes in gravity using the corresponding results in gauge theory.

In this paper we explicitly calculate the soft and collinear limits of all possible helicity combination in $\mathcal{N}=8$ supergravity using the known double copy relation to $\mathcal{N}=4$ SYM. General formulas for the double copy of amplitudes exists in literature Bern:1998xc ; Bern:1998sv but to our knowledge, they have not been worked out explicitly. These relations are explicitly derived and stated in this paper.

The paper is organised as follows. In Section 2, we set up the notations that we follow throughout the paper. In Section 3, we briefly review soft and collinear limits in $\mathcal{N}=4$ SYM which we use later in the paper. Double copy formalism and the relevant formula relating the amplitudes are reviewed in Section 4. In Section 5 we recall some basic facts about $\mathcal{N}=8$ supergravity and state our conventions for its factorisation into a pair of $\mathcal{N}=4$ SYM theories. Finally in Sections 6 and 7 we record the explicit soft and collinear limits of supergravity amplitudes. In the main body of the paper we have tabulated the collinear and soft limits of the amplitudes with the appropriate R-symmetry indices and the detailed calculations have been postponed to the appendices for reference. The appendices also include spinor-helicity formalism and a list of computational results.

The Minkowski space can be parameterized using the Bondi coordinates $(u,r,z,\bar{z})$ where $(z,\bar{z})$ parameterises the celestial sphere $\mathcal{CS}^{2}$ at null infinity. The Lorentz group $\mathrm{SL}(2,\mathbb{C})$ acts on $\mathcal{CS}^{2}$ as follows:

$$(z,\bar{z})\longmapsto\left(\frac{az+b}{cz+d},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\right),\quad\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{C}).$$

A general null momentum vector $p^{\mu}$ can be parametrized as

$$p^{\mu}=\omega q^{\mu},\quad q^{\mu}=\frac{1}{2}\left(1+|z|^{2},z+\bar{z},-i(z-\bar{z}),1-|z|^{2}\right),$$

where $q^{\mu}$ is a null vector, $\omega$ is identified with the light cone energy and all the particles momenta are taken to be outgoing. Under the Lorentz group the four momentum transforms as a Lorentz vector $p^{\mu}\mapsto\Lambda^{\mu}_{\leavevmode\nobreak\ \nu}p^{\nu}$. This induces the following transformation of $\omega$ and $q^{\mu}$ as

$$\omega\mapsto(cz+d)(\bar{c}\bar{z}+\bar{d})\omega,\quad q^{\mu}\mapsto q^{\prime\mu}=(cz+d)^{-1}(\bar{c}\bar{z}+\bar{d})^{-1}\Lambda_{\leavevmode\nobreak\ \nu}^{\mu}q^{\nu}.$$

It is useful to introduce the bispinor notation at this stage. We can write the basic null momentum vector $q^{\mu}$ as

$$q^{\alpha\dot{\alpha}}=\sigma_{\mu}^{\alpha\dot{\alpha}}q^{\mu}=\left(\begin{array}[]{cc}1&\;\bar{z}\\ z&\;z\bar{z}\end{array}\right)=\left(\begin{array}[]{l}1\\ z\end{array}\right)\left(\begin{array}[]{ll}1&\bar{z}\end{array}\right)$$

(1)

Here $\sigma_{\mu}^{\alpha\dot{\alpha}}=(1,\sigma_{x},\sigma_{y},\sigma_{z})$.We can thus introduce the familiar angle and square bracket spinor notation (see Appendix A for a brief review of spinor-helicity formalism) for the left and right-handed momentum spinors:

$$h^{\alpha}\equiv\langle p|^{\alpha}=\sqrt{\omega}\left(\begin{array}[]{l}1\\ z\end{array}\right)=\sqrt{\omega}\langle q|^{\alpha},\quad\tilde{h}^{\dot{\alpha}}\equiv|p]^{\dot{\alpha}}=\sqrt{\omega}\left(\begin{array}[]{l}1\\ \bar{z}\end{array}\right)=\sqrt{\omega}|q]^{\dot{\alpha}},$$

(2)

where we write

$$\langle q|^{\alpha}=\left(\begin{array}[]{l}1\\ z\end{array}\right),\quad|q]^{\dot{\alpha}}=\left(\begin{array}[]{c}1\\ \bar{z}\end{array}\right).$$

(3)

To shorten the notation, we denote the spinors for momenta $p_{i}$ by $\langle i|^{\alpha}$ and $|i]^{\dot{\alpha}}$ respectively. The inner product of momentas $p_{i}$ and $p_{j}$ can then be written in terms of the angle and square brackets of the corresponding spinors which are now given by

$$\langle ij\rangle=-\sqrt{\omega_{i}\omega_{j}}z_{ij},\quad[ij]=\sqrt{\omega_{i}\omega_{j}}\bar{z}_{ij}.$$

(4)

where $z_{ij}=z_{i}-z_{j}$ and similarly $\bar{z}_{ij}=\bar{z}_{i}-\bar{z}_{j}$.

As detailed in the introduction, in this paper we shall be studying the interesting limits of supergravity amplitudes using double copy relations. For this purpose we use $\mathcal{N}=4$ SYM as a machinary to find our desired results for gravity. Let us briefly recall some of the prime properties of $\mathcal{N}=4$ SYM. There are 16 different fields in $\mathcal{N}=4$ SYM, all of which can be packaged in a single superfield. Let $\{\eta_{a}\}_{a=1}^{4}$ be the Grassmann odd coordinates on the superspace. Then the superfield for $\mathcal{N}=4$ SYM can be written as

	$\displaystyle\Psi(p,\eta)=$	$\displaystyle G^{+}(p)+\eta_{a}\Gamma^{a}_{+}(p)+\frac{1}{2!}\eta_{a}\eta_{b}\Phi^{ab}(p)$		(5)
		$\displaystyle+\frac{1}{3!}\epsilon^{abcd}\eta_{a}\eta_{b}\eta_{c}\Gamma_{d}^{-}(p)+\frac{1}{4!}\epsilon^{abcd}\eta_{a}\eta_{b}\eta_{c}\eta_{d}G^{-}(p)$

where $G^{\pm}(p)$ denote positive and negative helicity gluons, $\Gamma^{a}_{+},\Gamma_{a}^{-}$ denote positive and negative helicity gluinos respectively and $\Phi^{ab}$ denotes the scalars. The superamplitude of $n$ such superfields is then given by the $n$-point correlation function

$$\mathcal{A}_{n}(\{p_{1},\eta^{1}\},\dots\{p_{n},\eta^{n}\})\equiv\langle\Psi_{1}(p_{1},\eta^{1})\dots\Psi_{n}(p_{n},\eta^{n})\rangle.$$

(6)

We sometimes suppress the momenta $p_{i}$ and superspace Grassmann coordinates $\eta^{i}$ and simply write $\mathcal{A}_{n}(1,2,\dots,n)$. Expanding both sides in $\eta$ and comparing, one gets the scattering amplitude of all the component fields. Next we find the soft and collinear limits of the superamplitude. We begin with the soft theorem following He:2014bga :

$$\mathcal{A}_{n}\left(\cdots,a,s,b,\cdots\right)\stackrel{{\scriptstyle p_{s}\rightarrow 0}}{{\longrightarrow}}\operatorname{Soft}^{\text{SYM}}\left(a,s,b\right)\mathcal{A}_{n-1}(\cdots,a,b,\cdots),$$

(7)

where $p_{s}$ is the momenta of the soft superfield and $a,b$ are the adjacent superfields. The soft factor $\operatorname{Soft}^{\text{SYM}}\left(a,s,b\right)$ is given by Liu:2014vva

$$\text{Soft}_{\mathrm{hol}}^{\mathrm{SYM}}(a,s,b)=\frac{1}{\varepsilon^{2}}\text{Soft(0)}_{\mathrm{hol}}^{\mathrm{SYM}}(a,s,b)+\frac{1}{\varepsilon}\text{Soft(1)}_{\mathrm{hol}}^{\mathrm{SYM}}(a,s,b).$$

(8)

where $(0)$ and $(1)$ indicate the leading and subleading terms. Let us explain the above notation. We associate a pair of spinors $\left(h_{s},\tilde{h}_{s}\right)$ with every soft momenta $p_{s}$. The limit $\left(h_{s},\tilde{h}_{s},\eta^{s}\right)\rightarrow\left(\varepsilon h_{s},\tilde{h}_{s},\eta^{s}\right)$ with $\varepsilon\to 0$ and $h_{s}$ some fixed spinor (namely $h_{s}\rightarrow 0$) is known as the holomorphic soft limit. The holomorphic soft factor is then given by Liu:2014vva

$$\begin{split}\text{Soft($k$)}_{\mathrm{hol}}^{\mathrm{SYM}}(a,s,b)=\frac{1}{k!}\frac{\langle ab\rangle}{\langle as\rangle\langle sb\rangle}\bigg{[}\frac{\langle sa\rangle}{\langle ba\rangle}\bigg{(}\tilde{h}_{s}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{h}_{b}^{\dot{\alpha}}}&+(\eta^{s})_{c}\frac{\partial}{\partial(\eta^{b})_{c}}\bigg{)}\\ &+\frac{\langle sb\rangle}{\langle ab\rangle}\left(\tilde{h}_{s}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{h}_{a}^{\dot{\alpha}}}+(\eta^{s})_{c}\frac{\partial}{\partial(\eta^{a})_{c}}\right)\bigg{]}^{k}.\end{split}$$

(9)

Similarly the limit $\left(h_{s},\tilde{h}_{s},\eta^{s}\right)\rightarrow\left(h_{s},\varepsilon\tilde{h}_{s},\eta^{s}\right)$ with $\tilde{h}_{s}$ a fixed spinor (namely $\tilde{h}_{s}\rightarrow 0$) is known as the anti-holomorphic soft limit. The anti-holomorphic soft factor is given by

$$\text{Soft($k$)}_{\text{anti-hol }}^{\mathrm{SYM}}(a,s,b)=\frac{1}{k!}\frac{[ab]}{[as][sb]}\delta^{4}\left(\eta^{s}+\frac{[as]}{[ab]}\eta^{b}+\frac{[sb]}{[ab]}\eta^{a}\right)\left[\frac{[sb]}{[ab]}h_{s}^{\alpha}\frac{\partial}{\partial h_{a}^{\alpha}}+\frac{[as]}{[ab]}h_{s}^{\alpha}\frac{\partial}{\partial h_{b}^{\alpha}}\right]^{k}$$

(10)

The physical soft limit $p_{s}\rightarrow 0$ is equivalent to considering both $h_{s},\tilde{h}_{s}\rightarrow 0$ simultaneously. Thus in the physical soft lomit, the soft factor splits as the sum of holomorphic as well as the anti-holomorphic soft factors. We use these results in Section 7 to compute soft limits in supergravity.

Next we discuss the collinear limits. In collinear limit, we take the momenta of two adjacent superfields $p_{1}$ and $p_{2}$ to be collinear. Under this limit, the two supefields can fuse to give another supefield with momentum $p_{12}=p_{1}+p_{2}$. We parametrize the momenta of the collinear superfields as

$$p_{1}=zp_{12},\quad p_{2}=(1-z)p_{12},$$

(11)

where $z$ corresponds to the combined momentum $p_{12}$ on the celestial sphere $\mathcal{CS}^{2}$. Since $p_{1}+p_{2}=p_{12}$, we see that, for massless fields, the collinear limit $p_{1}||p_{2}$ implies $p_{1}\cdot p_{2}\propto p_{1}^{2}=0$ which is equivalent to the condition $p_{12}^{2}\to 0$. Now the collinear limit in $\mathcal{N}=4$ SYM is given by Jiang:2021xzy ; Ferro:2020lgp

$$\mathcal{A}_{n}(1,2,3,\cdots,n)\stackrel{{\scriptstyle p_{12}^{2}\rightarrow 0}}{{\longrightarrow}}\sum_{l=1}^{2}\int d^{4}\eta^{p_{12}}\operatorname{Split}_{1-l}\left(1,2,p_{12}\right)\mathcal{A}_{n-1}\left(p_{12},3,\cdots,n\right).$$

(12)

The $l=1,2$ terms in the collinear limits are called the helicity-preserving and helicity-decreasing processes. The collinear singularity is contained in the split factors. The split factor of helicity preserving process is given by Ferro:2020lgp

$$\operatorname{Split}_{0}\left(z;\eta^{1},\eta^{2},\eta^{p_{12}}\right)=\frac{1}{\sqrt{z(1-z)}}\frac{1}{\langle 12\rangle}\prod_{a=1}^{4}\left(\eta^{p_{12}}_{a}-\sqrt{z}\eta^{1}_{a}-\sqrt{1-z}\eta^{2}_{a}\right).$$

(13)

Whereas for helicity-decreasing process, the split factor is given by Ferro:2020lgp

$$\text{Split}_{-1}\left(z;\eta^{1},\eta^{2},\eta^{p_{12}}\right)=\frac{1}{\sqrt{z(1-z)}}\frac{1}{[12]}\prod_{a=1}^{4}\left(\eta^{1}_{a}\eta^{2}_{a}-\sqrt{1-z}\eta^{1}_{a}\eta^{p_{12}}_{a}+\sqrt{z}\eta^{2}_{a}\eta^{p_{12}}_{a}\right).$$

(14)

The integral over $\eta^{p_{12}}$ can be performed using general results of Grassmann integration.³³3As an example for any function $f(\eta)$ we have Jiang:2021xzy $$\int d^{4}\eta^{p}\;\delta^{(4)}\left(\eta^{p}-\eta\right)f\left(\eta^{p}\right)=\int d^{4}\eta^{p}\prod_{a=1}^{4}\left(\eta^{p}_{a}-\eta_{a}\right)f\left(\eta^{p}\right)=f(\eta).$$ where, $\delta^{(4)}(\eta^{p}-\eta)=\prod_{a=1}^{4}\left(\eta^{p}_{a}-\eta_{a}\right).$ Here

$$\begin{split}\int d\eta^{p_{12}}_{a}\prod_{a=1}^{4}\big{(}\eta^{1}_{a}\eta^{2}_{a}-\sqrt{1-z}\eta^{1}_{a}\eta^{p_{12}}_{a}&+\sqrt{z}\eta^{2}_{a}\eta^{p_{12}}_{a}\big{)}f\left(\eta^{p_{12}}_{a}\right)\\ &=\delta^{(4)}\left(\sqrt{1-z}\eta^{1}_{a}-\sqrt{z}\eta^{2}_{a}\right)f\left(\frac{\eta^{2}_{a}}{\sqrt{1-z}}\right),\end{split}$$

(15)

Using these, we express the collinear limit (12) as

$$\begin{split}\mathcal{A}_{n}(1,2,3,\cdots,n)&\stackrel{{\scriptstyle p_{12}^{2}\rightarrow 0}}{{\longrightarrow}}\frac{1}{\sqrt{z(1-z)}}\frac{1}{[12]}\delta^{(4)}\left(\sqrt{1-z}\eta^{1}_{a}-\sqrt{z}\eta^{2}_{a}\right)\mathcal{A}_{n-1}\left(\{p_{12},\frac{\eta^{2}_{a}}{\sqrt{1-z}}\},3,\cdots,n\right)\\ &+\frac{1}{\sqrt{z(1-z)}}\frac{1}{\langle 12\rangle}\mathcal{A}_{n-1}(\{p_{12},\sqrt{z}\eta^{1}_{a}+\sqrt{1-z}\eta^{2}_{a}\},3,\cdots,n).\end{split}$$

(16)

Expanding both sides in $\eta^{1}$ and $\eta^{2}$, we can get collinear limits of the component fields. For collinear limit of component fields, we use the following notation:

$$A_{n}(1^{h_{1}},2^{h_{2}},\dots,n)\stackrel{{\scriptstyle 1||2}}{{\longrightarrow}}\sum_{h}\text{Split}_{-h}^{\text{SYM}}(z,1^{h_{1}},2^{h_{2}})A_{n-1}(p^{h},\dots,n),$$

(17)

where $A_{n}$ is the amplitude of $n$ different fields in the theory and the sum is over all helicities in the theory. Note that the split factor is trivial for helicities $h$ which does not have corresponding interaction with $h_{1}$ and $h_{2}$. The split also satisfies the conjugation relation Ferro:2020lgp

$$\begin{split}\text{Split}_{-h}\left(z;a^{h_{1}},b^{h_{2}}\right)=\text{Split}_{+h}\left(z;a^{-h_{1}},b^{-h_{2}}\right)|_{[ab]\leftrightarrow\langle ab\rangle}\end{split}$$

(18)

The split factor of component fields in SYM has two parts, the kinematic part and the index structure part. Kinematic part only depends on the momenta of the collinear particles while the index structure consists of the $\mathrm{SU}(4)$ R-symmetry indices of the component fields. As indicated earlier, one can compute the kinematic part of the split factors for various combinations of helicities by expanding both sides of (16) in $\eta^{1},\eta^{2}$ and then comparing the coefficients. This has been done using mathematica. The non-trivial split factor for collinear gluons are:

$$\begin{split}&\text{Split}_{+1}^{\text{SYM}}\left(z,a^{+1},b^{+1}\right)=0,\qquad\text{Split}_{-1}^{\text{SYM}}\left(z,a^{+1},b^{+1}\right)=\frac{1}{\sqrt{z(1-z)}}\frac{1}{\langle ab\rangle},\\ &\text{Split}_{+1}^{\text{SYM}}\left(z,a^{-1},b^{+1}\right)=\sqrt{\frac{z^{3}}{1-z}}\frac{1}{\langle ab\rangle},\qquad\text{Split}_{+1}^{\text{SYM}}\left(z,a^{+1},b^{-1}\right)=\frac{(1-z)^{2}}{\sqrt{z(1-z)}}\frac{1}{\langle ab\rangle}.\end{split}$$

(19)

The split factor for collinear gluinos and scalars are:

$$\begin{split}&\text{Split}_{0}^{\text{SYM}}\left(z,a^{+\frac{1}{2}},b^{+\frac{1}{2}}\right)=\frac{1}{\langle ab\rangle},\qquad\text{Split}_{+1}^{\text{SYM}}\left(z,a^{+\frac{1}{2}},b^{-\frac{1}{2}}\right)=\frac{(1-z)}{\langle ab\rangle},\\ &\text{Split}_{+1}^{\text{SYM}}\left(z,a^{-\frac{1}{2}},b^{+\frac{1}{2}}\right)=\frac{z}{\langle ab\rangle},\qquad\text{Split}_{1}^{\text{SYM}}\left(z,a^{0},b^{0}\right)=\sqrt{z(1-z)}\frac{1}{\langle ab\rangle}.\end{split}$$

(20)

Finally the split factor for mixed helicities are

$$\begin{split}&\text{Split}_{+\frac{1}{2}}^{\text{SYM}}\left(z,a^{-\frac{1}{2}},b^{+1}\right)=\frac{z}{\sqrt{(1-z)}}\frac{1}{\langle ab\rangle},\quad\text{Split}_{+\frac{1}{2}}^{\text{SYM}}\left(z,a^{+1},b^{-\frac{1}{2}}\right)=\frac{1-z}{\sqrt{z}}\frac{1}{\langle ab\rangle},\\ &\text{Split}_{-\frac{1}{2}}^{\text{SYM}}\left(z,a^{+\frac{1}{2}},b^{+1}\right)=\frac{1}{\sqrt{(1-z)}}\frac{1}{\langle ab\rangle},\quad\text{Split}_{-\frac{1}{2}}^{\text{SYM}}\left(z,a^{+1},b^{+\frac{1}{2}}\right)=\frac{1}{\sqrt{z}}\frac{1}{\langle ab\rangle},\\ &\text{Split}_{0}^{\text{SYM}}\left(z,a^{0},b^{+1}\right)=\sqrt{\frac{z}{1-z}}\frac{1}{\langle ab\rangle},\quad\text{Split}_{0}^{\text{SYM}}\left(z,a^{+1},b^{0}\right)=\sqrt{\frac{1-z}{z}}\frac{1}{\langle ab\rangle},\\ &\text{Split}_{\frac{1}{2}}^{\text{SYM}}\left(z,a^{0},b^{+\frac{1}{2}}\right)=\sqrt{z}\frac{1}{\langle ab\rangle},\quad\text{Split}_{\frac{1}{2}}^{\text{SYM}}\left(z,a^{+\frac{1}{2}},b^{0}\right)=\sqrt{(1-z)}\frac{1}{\langle ab\rangle}.\end{split}$$

(21)

All the other split factors can easily be obtained from Eq. (18). We now list the index structure part of the split factors for various component fields. We obtain it by expanding both sides of (16) in $\eta^{1}$ and $\eta^{2}$ and comparing the coefficients. Some of the index structures have been worked out in Jiang:2021xzy . We complete the list here. Note the index structure in the collinear limit of a gluon with any other component field is trivially determined, hence we omit them from Table 1.

Let us briefly review double copy (DC) technique that plays a crucial role in our analysis. It is a multiplicative bilinear operation to compute the amplitudes in one theory using amplitudes from other simpler theories. This is a method to express gravity tree level amplitudes in terms of sums of products of gauge theory tree level amplitudes. There are three different double copy formalisms for tree level amplitudes: KLT (named after Kawai, Lewellen, and Tye) Kawai:1985xq , BCJ (named after Bern, Carrasco, and Johansson) Bern:2008qj and CHY (named after Cachazo, He, and Yuan) Cachazo:2013gna ; Cachazo:2013hca formalism. We refer to Adamo:2022dcm for detailed review of these formalisms. Here we restrict our discussion to the application of double copy to soft and collinear limit of gravity amplitudes in terms of soft and collinear limits of gauge theory amplitudes.

In this section, we briefly review the field contents and basic properties of the theory and also establish notations that we follow in the remainder of the paper.

Let $\{\eta_{A}\}_{A=1}^{8}$ be the Grassmann coordinates on the $\mathcal{N}=8$ superspace. The degrees of $\mathcal{N}=8$ supergravity for an on-shell superfield is defined as

$$\begin{split}\Psi(p,\eta)&=H^{+}(p)+\eta_{A}\psi^{A}_{+}(p)+\eta_{AB}G^{AB}_{+}(p)+\eta_{ABC}\chi^{ABC}_{+}(p)\\ &+\eta_{ABCD}\Phi^{ABCD}(p)+\tilde{\eta}^{ABC}\chi_{ABC}^{-}(p)+\tilde{\eta}^{AB}G_{AB}^{-}(p)+\tilde{\eta}^{A}\psi_{A}^{-}(p)+\tilde{\eta}H^{-}(p),\end{split}$$

(30)

where we have introduced the notation

$$\begin{split}&\eta_{A_{1}\ldots A_{n}}\equiv\frac{1}{n!}\eta_{A_{1}}\ldots\eta_{A_{2}}\\ &\tilde{\eta}^{A_{1}\ldots A_{n}}\equiv\epsilon^{A_{1}\ldots A_{n}B_{1}\ldots B_{8-n}}\eta_{B^{1}\ldots B^{8-n}}\\ &\tilde{\eta}\equiv\prod_{A=1}^{8}\eta_{A}.\end{split}$$

(31)

The fields $H^{\pm}$ represent graviton, $G^{AB}_{+}$ and $G_{AB}^{-}$ represent gluons, $\psi^{A}_{+}$ and $\psi_{A}^{-}$ represent gravitinos, $\chi^{ABC}_{+}$ and $\chi_{ABC}^{-}$ represent gluinos and finally $\Phi^{ABCD}$ represent the real scalars. The (sub)super scripts ${\pm}$ denote positive and negative helicity of various fields. The superamplitude is then defined by

$$\mathcal{M}_{n}(\{p_{1},\eta^{1}\},\dots\{p_{n},\eta^{n}\})=\langle\Psi_{1}(p_{1},\eta^{1})\dots\Psi_{n}(p_{n},\eta^{n})\rangle.$$

(32)

We now explain the factorisation of states in supergravity into tensor product of states in super Yang-Mills. We begin by counting the degrees of freedom in the two theories. It is summarised in Table 2 below.