\section{Introduction}

Scattering theory plays essential role in understanding fundamental principles of particles. For past decades, there is huge progress on scattering theory in flat-space, which not only successfully predicts and explains many exciting discoveries made by collider \cite{Weinberg:1995mt}, but also remarkably reveals hidden structures linked to the entity of local quantum field theories, such as Bern-Carrasco-Johansson (BCJ) relations \cite{Bern:2008qj} and the double copy structures \cite{KAWAI19861,Bern:2010ue,Bern:2019prr}. 

However, our universe is not flat. Although our universe is generally curved, local scattering of particles in a small regime compared to the curvature can still be approximated by flat-space physics, as verified by scattering experiments. It is then natural and crucial to ask does those beautiful structures of scattering still remain for curved spacetime and how good the locally flat scattering experiments can say about curved spacetime. To answer these questions, natural starting point would be studying other maximally symmetric spacetimes such as Anti de-Sitter (AdS) space and de-Sitter (dS) space. 

AdS scattering is studied extensively due to its correspondence with conformal field theories (CFT) \cite{Maldacena:1997re}, and the unitary AdS physics can be explored by using unitary large-$N$ CFT \footnote{For dS, although it turns out its essential structures are similar to AdS and analytic continuation exists to go from one to another \cite{Sleight:2020obc}, the unitarity of dS scattering is not standard concept from CFT perspective \cite{Hogervorst:2021uvp}.} that is highly constrained by conformal symmetry and crossing symmetry (see \cite{Hartman:2022zik,Bissi:2022mrs} for quick reviews). As expected, appropriate limit of conformal correlators and conformal data corresponds to flat-space limit of local AdS scattering, giving back the flat-space scattering data (see, e.g., \cite{Okuda:2010ym,Penedones:2010ue,Fitzpatrick:2011hu,Maldacena:2015iua,Raju:2012zr,Paulos:2016fap,Komatsu:2020sag,Hijano:2019qmi,Hijano:2019qmi,Li:2021snj} and references therein) \footnote{However, there exists exemptions for special analytic regime that are not well-understood yet, see \cite{Komatsu:2020sag} for recent explorations.}. However, previously it is usually not expected to reconstruct full AdS amplitudes/CFT correlators from flat-space, in this sense the flat-space limit of AdS/CFT is crucially different from ``flat holography'' \cite{Bagchi:2016bcd}. Recent progress was made in \cite{Diwakar:2021juk,Herderschee:2022ntr,Cheung:2022pdk}, which surprisingly found the differential representation of scalar (A)dS amplitudes by universally writing (A)dS amplitudes as conformal generators acting on scalar contact Witten diagrams. This differential representation does not only make flat-space limit manifest, but also allows one to uplift the flat-space amplitudes to (A)dS in a universal way.

In this paper, we aim to make progress toward extending the differential representation to spinning correlators by focusing on massless gluon and graviton in AdS from Yang-Mills (YM) theory and Einstein gravity. Massless spinning particles in flat-space are constrained by stringent consistency conditions and encode hidden structures such as double copy \cite{KAWAI19861,Bern:2010ue,Bern:2019prr}. We report the differential representation for YM and graviton amplitudes in AdS, and we are convinced by our results that gluon and graviton amplitudes in AdS can be uplifted from flat-space up to extra structures that are expected to be fixed by conservation of operators. So do those hidden structures like BCJ relations and double copy, which are now in a differential format. Although we start our exploration in AdS, our results should be readily translated to dS.

The rest of this paper is organized as follows. In section \ref{sec: building blocks}, we introduce our differential operators for representing spinning correlators and comment on power counting principles in CFT. In section \ref{sec: diff rep}, we construct the differential representation for three-point and four-point YM amplitudes and graviton amplitudes; for three-point amplitudes we show the double copy is straightforward. In section \ref{sec: BCJ}, we propose another differential representation for YM amplitudes, which allow us to uplift flat-space BCJ numerators and prove the differential BCJ relations. We summarize and point out future directions in section \ref{sec: summary}. In appendix \ref{app: vertex} and \ref{app: ddh rep and id} we record detailed ingredients in our derivations.

\noindent {\bf Note:} During the preparation of this work, \cite{Lee:2022fgr} appeared which has partial overlap with the idea of using weight-shifting operators and the discussions on three-point double copy in section \ref{sec: building blocks} and subsection \ref{subsec: three-pt}.

\section{Building blocks for differential representation}
\label{sec: building blocks}

In this section, we introduce our notations and building blocks for differentially representing conserved current and stress-tensor correlators. 

For scalars, it is easy to find the differential representation for any contact diagrams up to any points by using the coset construction \cite{Cheung:2022pdk}. The spirit is that the contracted bulk derivatives can be replaced by the contractions of conformal generators, as guaranteed by the conformal symmetry. For spinning objects, the tensor structures appear, which can either contract among themselves or with bulk derivatives. From the perspective of CFTs, the spinning indices shall be captured by the spin-up weight-shifting operator \cite{Karateev:2017jgd} (written in terms of embedding formalism \cite{Costa:2011mg})
\begin{equation}
\mathcal{D}^{0+}_{\mu}= (J+\Delta)Z_\mu+X_\mu Z\cdot\partial_X\,.
\end{equation}
On the other hand, we speculate and show that bulk derivatives can be replaced by dimension-up weight-shifting operator $\mathcal{D}^{+0}$ (that raises up the scaling dimensions \cite{Karateev:2017jgd}) modulo bulk coordinates
\begin{align}
 \mathcal{D}^{+0}_{\mu}& = c_1 \partial_{X^\mu}+c_2 X_\mu \partial_X^2+c_3 Z_\mu \partial_Z\cdot\partial_X+c_4 Z\cdot\partial_X \partial_{Z^\mu}\nonumber\\
& 
+c_5 X_\mu  Z\cdot \partial_X \partial_Z\cdot\partial_X + c_6 Z_\mu Z\cdot\partial_X \partial_Z^2
\nonumber\\
& +c_7 X_\mu (Z\cdot\partial_X)^2 \partial_Z^2\,,
\end{align}
where the coefficients can be found in \cite{Karateev:2017jgd}. An intuitive way to convince ourselves the dimension-up weight-shifting operator plays a role like momentum in flat-space is that the flat-space momentum is actually $i\partial/\partial x$ which also increases the ``scaling dimensions''. In this paper, we find that it is instructive to define following differential operators proportional to weight-shifting operators with state dependent normalizations
\begin{align}
& \mathcal{E} = -\fft{\big(X\cdot \partial_X+Z\cdot\partial_Z\big)}{X\cdot\partial_X \big(X\cdot\partial_X+1\big)}\mathcal{D}_{\mu}^{0+}\,,\nonumber\\
& \mathcal{P} = \frac{2}{(X\cdot \partial_X+1) (d +X\cdot \partial_X-2) (d +2 X\cdot \partial_X-2)}\mathcal{D}_{\mu}^{+0}\,,\label{eq: ops}
\end{align}
where $-X\cdot\partial_X$ gives $\Delta$ for operators with scaling dimension $\Delta$, and $Z\cdot\partial_Z$ gives $J$ for spin-$J$ operator.

Before ending this section, we would like to comment, in general, on how \eqref{eq: ops} can be served as fundamental ingredients for large-$N$ CFT with natural power counting rules. We will show YM and graviton amplitudes can be uplifted from flat-space to AdS by using \eqref{eq: ops}. In addition to these examples, we actually claim that using \eqref{eq: ops} can uplift flat-space amplitudes of effective field theories (EFT) to AdS as general large-$N$ conformal correlators, where Wilson coefficients depend on details of conformal theory. The reason is that we find bulk derivatives can be traded by $\mathcal{P}$ modulo bulk coordinates, and thus the unexpected terms always have fewer number of $\mathcal{P}$. This claim implies that CFT correlators at large-$N$ limit enjoy the same power counting rules as EFTs in flat-space, which is consistent with counting maps between conformal correlators and flat-space amplitudes \cite{Kravchuk:2016qvl}. Remarkably, in this way, different OPE structures can be easily distinguished. For example, three-point functions of conserved currents in generic CFT have two parity-even structures corresponding to $F^2$ and $F^3$ in the bulk respectively, and there was no obvious way to construct three-point structures precisely corresponding to them by using embedding formalism or spin-up operators \cite{Costa:2011mg}. This is the main reason that spinning bootstrap is so hard to perform since the OPE matrix might be messy in an inappropriate basis \cite{Karateev:2018oml}. The helicity basis provides a clean way to organize the OPE matrix \cite{Caron-Huot:2021kjy}, however, it is still not easy to distinguish between two structures by simply staring. Now \eqref{eq: ops} makes the distinction manifest as for flat-space amplitudes! In this way, we believe that the differential representation in terms of \eqref{eq: ops} with power counting rules encoded could be very useful for clean spinning bootstrap in the future.

\section{Construction of the differential representation}
\label{sec: diff rep}

We consider the following action for Yang-Mills theory and Einstein gravity
\begin{equation}
S= \int d^{d+1}x\sqrt{g}\Big(\fft{1}{16\pi G}(R-2\Lambda)-\fft{1}{4g_{\rm YM}^2} F^a_{\mu\nu}F^{a\mu\nu}\Big)\,,\label{eq: action}
\end{equation}
where $\Lambda=-(d-1)(d-2)/(2R_{\rm AdS}^2)$. In this paper, we usually set $R_{\rm AdS}=1$ unless we emphasize. Our goal is to compute the four-point function for conserved currents and stress-tensors in holographic CFT that is effectively described by \eqref{eq: action}. These ``amplitudes'' can be computed by using the holographic dictionary \cite{Witten:1998qj,Gubser:1998bc}
\begin{equation}
\mathcal{M}:=\langle \mathcal{O}_1\cdots \mathcal{O}_n\rangle = \big(\prod_i \fft{\delta}{\delta \mathcal{J}_i^{(0)}}\big)\left\langle e^{-S_{\rm bulk}}\right\rangle_{\rm bulk}\,,\label{eq: def AdS amp}
\end{equation}
where $\mathcal{J}_i^{(0)}$ denotes the source as the non-normalizable mode of bulk fields, and we keep the spinning indices implicit. The variations produce the bulk-to-boundary propagators and the remaining fields are wick contracted by the bulk expectation value.

\subsection{Warm-up: three-point amplitudes}
\label{subsec: three-pt}

\subsubsection{Yang-Mills}

We start by looking into three-point functions as warm-up. For YM theory, it is rather straightforward to evaluate the three-point function
\begin{equation}
\mathcal{M}_{3,{\rm YM}}=g_{\rm YM}\int D^{d+1}Y f^{abc} V_{g,12}^{\mu,ab}(Y)\delta_3 A_{\mu}^c\,,
\end{equation}
where we are using the embedding AdS coordinate $Y$ \cite{Costa:2014kfa} and the shorthand notation $D^{d+1}Y:=d^{d+2}Y \delta(Y^2+1)$. For latter convenience, we explicitly write the three-point vertex function $V_{ij}^{\nu,ab}$
\begin{align}
&V_{g,12}^{\nu,ab}= \big(\nabla_\mu \delta_1 A^{\nu a} \delta_2 A^{\mu b}- \delta_1 A^{\mu a} \nabla_\mu \delta_2 A^{\nu b}\big) \nonumber\\
& +\fft{1}{2}\big(\delta_1 A^{\mu a}\nabla^\nu\delta_2 A_\mu^b-\nabla^\nu\delta_1 A_\mu^a\delta_2 A^{\mu b}\big) \,,\label{eq: 3-pt YM}
\end{align}
where $\delta_i A$ denotes the bulk-to-boundary propagator (in terms of the embedding space formalism \cite{Costa:2014kfa})
\begin{equation}
\delta_i A^{\mu a}=  \mathcal{C}_{d-1,1}\fft{2\big(X_i^\mu\, Y\cdot Z_i - Z_i^\mu \,Y\cdot X_i\big)}{\big(-2X_i\cdot Y\big)^d}\,,
\end{equation}
where $X^2=Z^2=X\cdot Z\equiv 0$. We use the standard normalization 
\begin{equation}
\mathcal{C}_{\Delta,J}=\frac{\pi ^{-\frac{d}{2}} \Gamma (\Delta ) (\Delta +J-1)}{2 (\Delta -1) \Gamma \left(-\frac{d}{2}+\Delta +1\right)}\,.
\end{equation}
Usually, it is also instructive to introduce the bulk embedding polarizations $W$ that contract the bulk indices. The bulk indices can be recovered by appropriate differential operators  \cite{Costa:2014kfa}. It is easy to explicitly evaluate three-point functions like \eqref{eq: 3-pt YM}, nevertheless, in this paper, we aim to provide a differential representation that rewrites \eqref{eq: 3-pt YM} in terms of differential operators acting on scalar seeds with zero derivatives. As we claim in the last section, the bulk-to-boundary propagators shall be represented by weight-shifting operators modulo bulk coordinates. In our conventions, we find
\begin{align}
& \delta_i A_\mu=\mathcal{E}_{i,\mu}\delta_i\phi_{d-1}\,,\nonumber\\
& \nabla_\mu\delta_i A _\nu =\fft{d-1}{2} \Big(\mathcal{E}_{i,\nu} \mathcal{P}_{i,\mu} \delta_i\phi_{d-2}  - Y_{(\mu} \mathcal{E}_{i,\nu)}\delta_i\phi_{d-1}\big)\,.\label{eq: dA rep}
\end{align}
Aside of the second part of the second line in \eqref{eq: dA rep}, we have already observed flat-space structure by identifying $\mathcal{E}\rightarrow\epsilon$, $\mathcal{P}\rightarrow p$, and the transverse property also remains
\begin{equation}
\mathcal{E}_i\cdot \mathcal{P}_i\, \delta_i \phi_{d-2} =0\,.
\end{equation}
The overall coefficient $(d-1)/2$ seems to ruin the precise flat-space structure, but we claim this is the normalization factor that can be absorbed into the plane wave in the flat-space limit. For AdS$_4$/CFT$_3$, this normalization is precisely unit. We can then readily show \eqref{eq: 3-pt YM} can be rewritten by
\begin{align}
& \mathcal{M}_{3,{\rm YM}}= - \fft{d-1}{2}g_{\rm YM}\mathcal{T}_{\mathcal{O}}\Big((\mathcal{E}_2\cdot \mathcal{E}_3) (\mathcal{E}_1\cdot \mathcal{P}_2)W_{d-1,d-2,d-1} \nonumber\\
& -\big(1\leftrightarrow 2\big)+ \big(2\rightarrow1, 1\rightarrow3\big)\Big)f^{123}\,,\label{eq: YM 3-pt}
\end{align}
where $W_{\Delta_i}$ refers to the scalar contact Witten diagram with no derivatives, and $i\leftrightarrow j$ also permute the corresponding legs for contact Witten diagram. $\mathcal{T}_{\mathcal{O}}$ means the operator ordering, which always place $\mathcal{E}$ on the left hand side of $\mathcal{P}$ for the same point. For simplicity in this letter, we define the amplitudes by pure differential forms with contact seeds slipped off $\hat{\mathcal{M}}$. It is easy to recover the contact seeds and go from $\hat{\mathcal{M}}$ to $\mathcal{M}$ by power counting $\mathcal{P}$. By dividing appropriate normalization, \eqref{eq: YM 3-pt} is a trivial uplift from flat-space by taking $\epsilon\rightarrow \mathcal{E}$, $p\rightarrow \mathcal{P}$ followed by operator reordering.

\subsubsection{Graviton}

For graviton three-point amplitude, we find (see also \cite{Raju:2012zs})
\begin{equation}
\mathcal{M}_{3,{\rm grav}}= 4\sqrt{8\pi G}\int D^{d+1}Y V_{h,12}^{\mu\nu}(Y)\delta_3 h_{\mu\nu}\,,
\end{equation}
where the vertex function $V_{h,12}^{\mu\nu}$ is lengthy and we leave its explicit expression to appendix \ref{app: vertex}. Similarly, $\delta_i h_{\mu\nu}$ is the bulk-to-boundary propagator for graviton, given by (we dot it into bulk embedding polarizations to keep it light)
\begin{equation}
\delta_i h_{\mu\nu}W^{\mu}W^\nu = \mathcal{C}_{d,2}\fft{4\left(W\cdot X_i\, Y\cdot Z_i-W\cdot Z_i\, X_i\cdot Y\right)^2}{(-2X_i\cdot Y)^{d+2}}\,.
\end{equation}
As we promise, we find
\begin{align}
&\delta_i h_{\mu\nu} = \mathcal{E}_{i,\mu}\mathcal{E}_{i,\nu} \delta_i \phi_{d}\,,\nonumber\\
&  \nabla_{\mu}\delta_i h_{\nu\rho} =  \mathcal{E}_{i,\nu}\mathcal{E}_{i,\rho} \mathcal{P}_{i\mu}\delta_i \phi_{d-1}+\mathcal{O}(Y)\,,\nonumber\\
& \nabla_{\mu}\nabla_{\nu}\delta_i h_{\rho\sigma}=\mathcal{E}_{i,\rho}\mathcal{E}_{i,\sigma} \mathcal{P}_{i\mu}\mathcal{P}_{i\nu} \delta_i \phi_{d-2} + \mathcal{O}(Y,g)\,,\label{eq: ddh rep}
\end{align}
where we drop out the lengthy terms depending on bulk coordinates and metric $Y_\mu$ and $g_{\mu\nu}$. We record the complete expressions in appendix \ref{app: ddh rep and id}. These terms $\mathcal{O}(Y,g)$, as contracted, can either be annihilated or give rise to less derivatives contact terms.  For three-point function, they are completely canceled and we arrive at precisely flat-space uplift
\begin{equation}
\hat{\mathcal{M}}_{3,{\rm grav}} = \mathcal{T}_{\mathcal{O}}\Big(\mathcal{M}_{3,{\rm grav}}^{\rm flat}\Big|_{\epsilon\rightarrow\mathcal{E},p\rightarrow\mathcal{P}}\Big)\,.
\end{equation} 

\subsubsection{The differential double copy}

Using differential operators \eqref{eq: ops}, three-point amplitudes have the same structure as in flat-space up to universal reordering. Therefore, the double copy structure at three-point level should be straightforward. To show this, we find although $\mathcal{P}$ itself is not conserved as the momentum, an effective conservation emerges at the level of three-point amplitudes for both YM and graviton, e.g., three-point amplitudes are invariant under following replacement
\begin{equation}
\mathcal{E}_3\cdot \mathcal{P}_2 \rightarrow - \mathcal{E}_3\cdot \mathcal{P}_1\,,\quad \mathcal{P}_1\cdot \mathcal{P}_2\rightarrow 0\,.
\end{equation}
Keeping these identities in mind, three-point amplitudes in AdS then make no much difference from flat-space and the differential double copy is valid
\begin{equation}
\hat{\mathcal{M}}_{3,\rm grav}=\fft{4}{(d-1)^2}\fft{8\sqrt{8\pi G}}{g_{\rm YM}^2}\mathcal{T}_{\mathcal{O}}\Big[(\hat{\mathcal{M}}_{3,\rm YM})^2\Big]\,.
\end{equation}

\subsection{Four-point amplitudes}

\subsubsection{Yang-Mills}

Let us start with considering only $s$-channel exchange diagram in YM theory
\begin{align}
\mathcal{M}_{4{\rm ex},{\rm YM}}^{(s)}=& g_{\rm YM}^2\int D^{d+1}Y_1D^{d+1}Y_2 \,f^{abc}f^{deg}\times\nonumber\\
 &  V_{g,12}^{\mu,ab}(Y_1)\left\langle A_{\mu}^{c}(Y_1)A_{\nu}^{g}(Y_2)\right\rangle_{\rm bulk}V_{g,34}^{\nu,de}(Y_2)\,,
\end{align}
where the expectation value of the remaining bulk fields gives rise to the bulk-to-bulk propagator
\begin{equation}
\left\langle A_\mu(Y_1) A_\nu(Y_2)\right\rangle:= \Pi_{g,\mu\nu}(Y_1,Y_2)\,,
\end{equation}
which satisfies the equation
\begin{equation}
\nabla^2_{Y_1} \Pi_{g,\mu\nu}(Y_1,Y_2)= -\delta_{\mu\nu}\,\delta(Y_1-Y_2)\,.\label{eq: YM bulk-to-bulk eq}
\end{equation}

The trick to find the differential representation is to eliminate the bulk-to-bulk propagator by the conformal Casimir operator minus its eigenvalue for propagating field. This procedure produces an effective contact diagrams that include only the bulk-to-boundary propagators. Indeed, we find
\begin{equation}
\mathcal{D}_{12}^{d-1,1} V_{12}^{\mu,ab}= \nabla_{Y_1}^2 V_{12}^{\mu,ab}\,,
\end{equation}
where 
\begin{equation}
\mathcal{D}_{12}^{\Delta,J}=\mathcal{C}_{12}-\big(\Delta(\Delta-d)+J(J+d-2)\big)\,,
\end{equation}
The conformal Casimir $\mathcal{C}_{12}$ is
\begin{equation}
\mathcal{C}_{12} = - \fft{1}{2}(L_1+L_2)^2\,,\quad L_{i}^{\mu\nu}= X_{i}^{[\mu}  \partial_{X_i}^{\nu]}+Z_{i}^{[\mu}  \partial_{Z_i}^{\nu]}\,.
\end{equation}
By integration-by-parts, we can move this bulk Laplacian to act on the bulk-to-bulk propagator and \eqref{eq: YM bulk-to-bulk eq} then reduces it to effective contact interactions
\begin{equation}
\mathcal{C}_{12}\mathcal{M}_{4{\rm ex},{\rm YM}}^{(s)} = -g_{\rm YM}^2 \int D^{d+1}Y f^{abe}f^{cde}V_{12}^{ab}(Y)\cdot V_{34}^{cd}(Y)\,.
\end{equation}
We can then represent this effective contact terms by using \eqref{eq: dA rep}. In addition to the exchanged diagram, the YM theory also provides four-point contact diagram $A^4$. This contact diagram can now be trivially uplifted from flat-space by using \eqref{eq: dA rep}. Combining together with permutations to include all channels, we obtain
\begin{equation}
\hat{\mathcal{M}}_{4,{\rm YM}}=\fft{(d-1)^2}{4}\mathcal{T}_{\mathcal{O}}\Big(\hat{\mathcal{M}}_{4,{\rm YM}}\Big|_{\epsilon\rightarrow\mathcal{E},p\rightarrow\mathcal{P},1/s_{ij}\rightarrow 1/(2\mathcal{D}_{ij}^{d-1,1})}\Big)\,,
\end{equation}
where we follow \cite{Herderschee:2022ntr,Cheung:2022pdk} to define the operator $1/\mathcal{D}_{ij}$ satisfying $1/\mathcal{D}_{ij} \, \mathcal{D}_{ij}\equiv 1$, and now the operator ordering $\mathcal{T}_{\mathcal{O}}$ always keeps $1/\mathcal{D}_{ij}$ on the most left.

\subsubsection{Graviton}

We follow the same logic for evaluating graviton amplitudes
\begin{align}
\mathcal{M}_{4{\rm ex},{\rm grav}}^{(s)}=& 16\times 8\pi G \int D^{d+1}Y_1 D^{d+1}Y_2\times \nonumber\\
& V_{h,12}^{\mu\nu}(Y_1) \big\langle h_{\mu\nu}(Y_1)h_{\rho\sigma}(Y_2)\big\rangle_{\rm bulk} V_{h,34}^{\rho\sigma}(Y_2)\,.
\end{align}
To correctly deal with graviton, we have to be careful about the trace part of graviton and the vertex. We adopt the de-Donder gauge for propagating graviton and careful analysis shows \cite{Herderschee:2022ntr, YueZhou:future}
\begin{align}
\mathcal{D}_{12}^{d,2}\mathcal{M}_{4{\rm ex},{\rm grav}}^{(s)}=&16\times 8\pi G \int D^{d+1}Y \times \nonumber\\
& V_{h,12}^{\mu\nu}(Y) P_{\mu\nu,\rho\sigma} V_{h,34}^{\rho\sigma}(Y)\,,
\end{align}
where $P_{\mu\nu,\rho\sigma}$ is the projector precisely the same as the flat-space propagator
\begin{equation}
P_{\mu\nu,\rho\sigma}=-\fft{1}{2}\Big(g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\rho\sigma}-\fft{2}{d-1}g_{\mu\nu}g_{\rho\sigma}\Big)\,.
\end{equation}
The resulting contact terms can then be represented by using \eqref{eq: ddh rep} (precisely one is \eqref{eq: compete ddh rep}). To have well-defined amplitudes, we should also include four graviton contact contributions given by Einstein-Hilbert action (\eqref{eq: 4-pt ct} in appendix \ref{app: vertex}), which can again be rewritten by using \eqref{eq: ddh rep}. We can use a lot of identities to eliminate all $Y$ dependences (see appendix \ref{app: ddh rep and id}), and in the end we find it can be uplifted from the flat-space graviton amplitude up to an extra contact contribution from AdS (with zero derivatives)
\begin{equation}
\hat{\mathcal{M}}_{4,{\rm grav}}=& \mathcal{T}_{\mathcal{O}}\Big(\hat{\mathcal{M}}_{4,{\rm grav}}\Big|_{\epsilon\rightarrow\mathcal{E},p\rightarrow\mathcal{P},1/s_{ij}\rightarrow 1/(2\mathcal{D}_{ij}^{d,2})}\Big) \nonumber\\
& + \fft{1}{R_{\rm AdS}^2} \hat{\mathcal{M}}_{4,{\rm grav}}^{\rm AdS}\,.\label{eq: grav 4-pt}
\end{equation}
We explicitly write down $1/R_{\rm AdS}^2$ to emphasize the corresponding term is purely contributed from AdS and shall shut down at the flat-space limit. This extra term is
\begin{equation}
& \hat{\mathcal{M}}_{4,{\rm grav}}^{\rm AdS}=4\pi G d \Big(\big((\mathcal{E}_1\cdot\mathcal{E}_2)^2(\mathcal{E}_3\cdot\mathcal{E}_4)^2\nonumber\\
& -4 \mathcal{E}_1\cdot\mathcal{E}_2\,\mathcal{E}_2\cdot\mathcal{E}_3\,\mathcal{E}_3\cdot\mathcal{E}_4\,\mathcal{E}_4\cdot\mathcal{E}_1 \big)+{\rm perm} \Big)\,.
\end{equation}
The existence of this term arises from the gauge invariance in the bulk, which is served as the conservation law for stress-tensor on the CFT side. This provides an idea to ``bootstrap'' stress-tensor correlators from flat-space amplitudes: we speculate that we can directly uplift the flat-space amplitudes and then enumerate crossing symmetric structures with less $\mathcal{P}$ multiplied with undetermined coefficients; those coefficients should be fixed by requiring the conservation of stress-tensors. 

\subsection{Comment on gauge invariance and conservation}

Before we end this section, we would like to discuss and comment on the relation between bulk gauge invariance and boundary conservation law by using our uplift operators \eqref{eq: ops}.

From bulk perspective, the action is invariant under the gauge transformation
\begin{equation}
A_\mu\rightarrow A_\mu +\nabla_\mu\chi\,,\quad h_{\mu\nu}\rightarrow h_{\mu\nu}+\nabla_{(\mu} \xi_{\nu)}\,,
\end{equation}
where $\chi$ is arbitrary scalar and $\xi$ vector. According to our statement that bulk derivatives can be replaced by $\mathcal{P}$ modulo bulk coordinates, we expect this gauge invariance, as in flat-space, is represented by the invariance of boundary correlator under 
\begin{equation}
\mathcal{E}_\mu\rightarrow \mathcal{E}_\mu + \# \mathcal{P}_\mu\,.\label{eq: gauge inv}
\end{equation}
In other words, if we replace $\mathcal{E}$ by $\mathcal{P}$ (without changing the scaling dimension of the building scalar contact), the correlator should be completely vanishing
\begin{equation}
\mathcal{M}\Big|_{\mathcal{E}_i\rightarrow \mathcal{P}_i}=0\,.\label{eq: conservation}
\end{equation}
This statement reminds us the conservation of conserved current and stress-tensor, as promised by AdS/CFT correspondence \cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}. To show this, we should first recover the boundary tensor indices by using 
\begin{equation}
\mathcal{D}_Z^\mu=\fft{d-2}{2} \partial_Z^\mu + Z\cdot\partial_Z \partial_Z^\mu -\fft{1}{2}Z^\mu \partial_Z^2\,,
\end{equation}
then we contract one index with boundary derivative. For the same point, we can show
\begin{equation}
& \partial_X\cdot \mathcal{D}_Z\, \mathcal{E}_{\mu} = -\fft{2}{d}\mathcal{P}_\mu\,,\nonumber\\
& \partial_X\cdot \mathcal{D}_Z \,  \mathcal{E}_{\mu}  \mathcal{E}_{\nu} =-\fft{2d}{(d+2)(d-1)(d-2)}
\mathcal{P}_{\mu}\mathcal{E}_\nu\,,
\end{equation}
where first line is the identity for conserved current and the second line is for stress-tensor \footnote{These identities can trivially pass the strange operator $1/\mathcal{D}_{ij}$, because we can always change $(ij)$ legs to other legs that $\partial_X\cdot \mathcal{D}_Z$ does not act by using the Ward identity.}. These identities confirm that the conservation is equivalent to   \eqref{eq: conservation} and is consistent with expectation from bulk gauge invariance. We then claim, we can even uplift the flat-space gauge invariance condition!

However, it is hard to impose or verify the conservation condition \eqref{eq: conservation} for differential representation. Because there are several difficulties arise after we do the replacement $\mathcal{E}\rightarrow \mathcal{P}$. Firstly, the ordering of differential operators is no longer in the operator ordering, which makes the organization a bit messy. Secondly, it is not trivial to move differential operators to remove the poles $1/\mathcal{D}_{ij}$. In the end, we do not find conservation of $\mathcal{P}$ analogous to flat-space \footnote{Since the replacement $\mathcal{E}\rightarrow \mathcal{P}$ makes even YM amplitude not linear in $\mathcal{P}$, the simple representation using conformal generator proposed in sec \ref{sec: BCJ} does not work.}. We leave these problems to be resolved in the future works.

\section{Differential BCJ relation for YM amplitudes}
\label{sec: BCJ}

\subsection{Differential representation using conformal generators}

We have already found a differential representation for YM amplitudes in terms of the weight-shifting operators. However, for four-point case we do not have simple analogy of the momentum conservation of flat-space in terms of $\mathcal{P}$. On the other hand, the conformal generators enjoy the analogy of ``momentum conservation'' because of conformal symmetry
\begin{equation}
\sum_{i=1}^n L_{i}^{\mu\nu} f(X_i) \equiv 0\,,
\end{equation}
where $f(X_i)$ is any conformal invariant function. To make hidden structures of YM amplitude manifest, we propose that we can replace $\mathcal{P}$ by the conformal generators. Although it is not obvious, we indeed find such replacement
\begin{align}
 \mathcal{E}_i\cdot P_j \mathcal{E}_k\cdot P_l \rightarrow 2\mathcal{E}_i^\mu
\mathcal{E}_k^{\nu} L_{j\mu}\,^\rho L_{l\rho\nu}\,,\quad P_i\cdot P_j \rightarrow -\fft{1}{2} L_i\cdot L_j\,,\label{eq: conserve replace}
\end{align}
simultaneously the scalar contact seeds are now uniform $W_{d-1,d-1,d-1,d-1}$. Note this is a replacement rather than identities. The real identity is more complicated but all unwanted terms are canceled out. We now arrive at an differential representation that enjoys the ``momentum conservation'' (that we can replace $L_4$ by $-L_1-L_2-L_3$) and the transversity
\begin{align}
 \mathcal{E}_i^\mu
\mathcal{E}_k^{\nu} L_{i\mu}\,^\rho L_{l\rho\nu}= 0\,,\quad \mathcal{E}_i^\mu
\mathcal{E}_k^{\nu} L_{j\mu}\,^\rho L_{k\rho\nu}= 0\,.\label{eq: trans}
\end{align}

\subsection{Establishing the differential BCJ}

Similar to the flat-space,  it is instructive to study the colour-ordered amplitudes. We can easily extract the colour-ordered amplitudes by recalling
\begin{align}
 f^{12e}f^{34e} &={\rm Tr}\big(T^{1}T^2 T^3 T^4\big)-{\rm Tr}\big(T^{1}T^2 T^4 T^3\big)\nonumber\\
& -{\rm Tr}\big(T^{1}T^3 T^4 T^2\big)+{\rm Tr}\big(T^{1}T^4 T^3 T^2\big)\,.
\end{align}
The colour-ordered amplitude $\mathcal{M}[i_1,i_2,i_3,i_4]$ is the coefficient of ${\rm Tr}\big(T^{i_1}T^{i_2}T^{i_3}T^{i_4}\big)$.

Let us take $\mathcal{M}[1234]$ as an example. We find, similar to the flat-space, the colour-slipped amplitude can be written as
 \begin{equation}
 \hat{\mathcal{M}}[1234]= \fft{1}{\mathcal{D}_{12}^{d-1,1}}\mathcal{N}_s - \fft{1}{\mathcal{D}_{23}^{d-1,1}}\mathcal{N}_t\,.
 \end{equation}
As in flat-space, the differential numerator $\mathcal{N}$ is also ambiguous. For example, shifting $\mathcal{N}_s$ by $\mathcal{N}_s\rightarrow \mathcal{N}_s+{\rm const} \times \mathcal{D}_{12}^{d-1,1}$ and similarly for $\mathcal{N}_t$ doesn't change the amplitudes. It is not hard to find such numerators satisfying following permutation properties
\begin{equation}
\mathcal{N}_s\big|_{2\leftrightarrow4}=-\mathcal{N}_t\,,\quad \mathcal{N}_{s}\big|_{2\rightarrow 4,3\rightarrow2,4\rightarrow3}= \mathcal{N}_t\,,
\end{equation}
which could be obtained by uplifting flat-space numerators
\begin{equation}
\mathcal{N}_s = \mathcal{T}_{\mathcal{O}}\Big(n_s^{\rm flat}\Big|_{\epsilon\rightarrow\mathcal{E},p\rightarrow\mathcal{P}}\Big)\,,
\end{equation}
followed by replacement \eqref{eq: conserve replace}. Nevertheless, even though we uplift the flat BCJ numerators that satisfy the kinematic Jacobi relation \cite{Bern:2008qj}
\begin{equation}
n_s+n_t+n_u=0\,,
\end{equation}
it still does not guarantee the differential numerators satisfy these kinematic Jacobi relations. The culprit is the ordering of differential operators, more specifically terms like $f(\mathcal{E})L_i\cdot L_j$ does not manifestly cancel the denominator $\mathcal{D}^{d-1,1}$. To resolve this problem, we should move $L_i\cdot L_j$ to the most left. We can easily do this by noting
\begin{equation}
&\mathcal{E}_{i_1}\cdot \mathcal{E}_{i_2}\,\mathcal{E}_{i_3}\cdot \mathcal{E}_{i_4}\, L_{i_1}\cdot L_{i_3}=\mathcal{D}_{i_1i_3}^{d-1,1} \mathcal{E}_{i_1}\cdot \mathcal{E}_{i_2}\,\mathcal{E}_{i_3}\cdot \mathcal{E}_{i_4}\nonumber\\
& - 2\big(\mathcal{E}_{i_2}\cdot \mathcal{E}_{i_4}\,\mathcal{E}_{i_1}\cdot \mathcal{E}_{i_3}-\mathcal{E}_{i_2}\cdot \mathcal{E}_{i_3}\,\mathcal{E}_{i_1}\cdot \mathcal{E}_{i_4}\big)\,.
\end{equation}
It turns out that extra terms such as the second line above would generally cancel out in the final expression and we then trivially move $L_i\cdot L_j$ to the most left as $\mathcal{D}_{ij}^{d-1,1}$. After this operation, we can then use the Ward identity to rewrite $L_4=-L_1-L_2-L_3$ followed by using the transversity \eqref{eq: trans} to enimilate unwanted terms. We also have to show
\begin{equation}
\big(\mathcal{D}^{d-1,1}_{12}+\mathcal{D}^{d-1,1}_{13}+\mathcal{D}^{d-1,1}_{23}\big)f_{1,d-1}(X_i)=0\,,
\end{equation}
where $f_{1,d-1}$ is any conformal invariant function with spin weights $J=1$ and scaling weights $\Delta=d-1$. This statement is equivalent to $L_i^2 f_{1,d-1}=0$, which can be easily proved by acting $L_i^2$ on shadow representation \cite{SimmonsDuffin:2012uy,Karateev:2018oml} of any such function $f_{1,d-1}$
\begin{equation}
f_{1,d-1}=& \sum_J \int d\Delta I(\Delta,J)\times\nonumber\\
& \int D^dX_5 \langle V_1 V_2 \mathcal{O}_{\Delta,J}(X_5)\rangle \langle  \tilde{\mathcal{O}}_{d-\Delta,J}(X_5)V_3 V_4\rangle\,.
\end{equation}
Taking all of these into account, we can then prove the differential kinematic Jacobi identity
\begin{equation}
\mathcal{N}_s+\mathcal{N}_t+\mathcal{N}_u=0\,.\label{eq: diff BCJ num}
\end{equation}

Following the exact same operations described above, we can readily prove the differential BCJ relation
\begin{equation}
\mathcal{D}_{12}^{d-1,1}\hat{\mathcal{M}}[1234]-\mathcal{D}_{13}^{d-1,1}\hat{\mathcal{M}}[1324]
=0\,.
\end{equation}



\subsection{Comment on four-point double copy}

Since we can uplift the BCJ numerators \eqref{eq: diff BCJ num}, it is highly speculative that the differential double copy can be proved and the graviton amplitude should be given by
\begin{equation}
\hat{\mathcal{M}}_{4,{\rm grav}} \stackrel{?}{\propto} \mathcal{T}_{\mathcal{O}}\Big(\fft{1}{\mathcal{D}_{12}^{d,2}}\mathcal{N}_s^2+\fft{1}{\mathcal{D}_{23}^{d,2}}\mathcal{N}_t^2+\fft{1}{\mathcal{D}_{24}^{d,2}}\mathcal{N}_u^2\Big)\,.\label{eq: guess double copy}
\end{equation}
However, this statement is hard to prove.  The most important reason is that the momentum conservation is built into the double copy relation, but we do not manage to find clean way to replace $\mathcal{P}$ in \eqref{eq: grav 4-pt} by $L$. In fact, there are large redundancies to rewrite \eqref{eq: grav 4-pt} in terms of conformal generators, and it is difficult to locate a nice minimum basis that allows us to prove or disprove \eqref{eq: guess double copy}. Another way to explore \eqref{eq: guess double copy} might be generalizing the algorithm in \cite{YueZhou:future} for translating the differential representation to final amplitudes in the Mellin space, the resulting Mellin amplitudes  \cite{Mack:2009mi,Penedones:2010ue} may help up explicitly verify the relation. Fortunately, either the power counting rule or the flat-space limit ensures the double copy up to contact terms suppressed by AdS radius, namely
\begin{equation}
\hat{\mathcal{M}}_{4,{\rm grav}} \propto \mathcal{T}_{\mathcal{O}}\Big(\fft{1}{\mathcal{D}_{12}^{d,2}}\mathcal{N}_s^2+\fft{1}{\mathcal{D}_{23}^{d,2}}\mathcal{N}_t^2+\fft{1}{\mathcal{D}_{24}^{d,2}}\mathcal{N}_u^2\Big) + \text{contacts}\,.\label{eq: double copy contact}
\end{equation}
By enumerating all possible contact structures followed by requiring the conservation law for stress-tensors, the contact terms may be completely fixed. We leave this interesting question to future studies.

\section{Summary}
\label{sec: summary}

We proposed the differential representation for tree-level gluon and graviton scattering from YM and Einstein gravity in AdS. The essential differential operators are proportional to dimension-up and spin-up weight-shifting operators, which shall provide power counting for (A)dS amplitudes and large-$N$ conformal correlators. Using these differential operators, three-point and four-point amplitudes in AdS are straightforwardly uplifted from flat-space cousins. For three-point amplitudes, such uplift makes double copy relation straightforward. At four-point level, we find a rather different differential representation for YM amplitudes by using spin-up weight-shifting operators and the conformal generators, for which differential BCJ numerators can be uplifted from flat-space ones, building differential BCJ relations. The differential BCJ numerators essentially follow the kinematic Jacobi identity, for which we could argue that double copy structure for four-point function should be valid up to remaining contact terms. It would be interesting to make connection between our findings and the similar structures in momentum space \cite{Armstrong:2020woi,Albayrak:2020fyp,Lee:2022fgr} or Mellin space (for supersymmetric theories) \cite{Alday:2021odx,Zhou:2021gnu,Bissi:2022wuh}.

This paper is the first step toward revealing hidden structures of spinning correlators. Most importantly, our surprising findings rely on more or less guessing work. It is thus crucial to develop a more systematic way for uplifting by using \eqref{eq: ops} and relevant operators, similar to the scalar case \cite{Cheung:2022pdk}. Besides, ordering of differential operators and non-conservation of operators $\mathcal{P}$ prevent one from proving or imposing the conservation for current and stress-tensor operators. One possible way to resolve this problem is to carefully think about algebra that \eqref{eq: ops} may form together with other differential operators (such as emergent ${\rm SO}(5,5)$ algebra for bispinor representation of AdS$_4$/CFT$_3$ \cite{Binder:2020raz}). Besides, the uplift from flat-space convinces us there should also exist Parke-Taylor formula, and we believe the bispinor formalism \cite{Binder:2020raz} would be the correct tool. Presumably, as these challenges are overcame, we believe four-point double copy in AdS can be precisely established. 

It would be also interesting to understand why this differential representation makes the flat-space limit manifest by explicit investigation on inönü-wigner contraction of conformal group and its representations \cite{Goncalves:2014rfa}. This exploration can be useful for understanding many aspects of S-matrix as flat-space limit of conformal correlators, following the lines of \cite{Komatsu:2020sag,vanRees:2022itk,Caron-Huot:2021enk,Duary:2022pyv}.

In the end, we want to emphasize that the differential representation might be useful for bootstrapping holographic CFTs beyond the scope of Lorentzian inversion formula \cite{Caron-Huot:2017vep,Simmons-Duffin:2017nub,Kravchuk:2018htv}. The Lorentzian inversion formula does not work well for spin-zero trajectories, while the differential representation precisely captures the contact terms with complete OPE data built into on the numerators.

\begin{acknowledgments}
We thank Simon Caron-Huot, Julio Parra-Martinez, Jiajie Mei, Joao Penedones and David Simmons-Duffin for discussions. We thank Yi Wang for the hospitality during my visit to HKUST Jockey Club Institute for Advanced Study. YZL is supported by the Simons Foundation through the Simons Collaboration on the Nonperturbative Bootstrap.
\end{acknowledgments}



\bibliographystyle{apsrev4-1}
