# Shadow transform For an operator $\mathcal{O}(x)$ with dimension $\Delta$ in a $d$-dimensional CFT, one can define a nonlocal shadow operator $\widetilde{\mathcal{O}}(x)$ with dimension $\widetilde{\Delta}=d-\Delta$. Then $\int d^{d} x \mathcal{O}(x)|0\rangle\langle 0| \widetilde{\mathcal{O}}(x)$ is dimensionless and conformally invariant. It *almost* projects a 4-pt function to a [[0031 Conformal block|conformal block]]: $\int d^{d} x\left\langle\phi_{1}\left(x_{1}\right) \phi_{2}\left(x_{2}\right) \mathcal{O}(x)\right\rangle\langle\widetilde{\mathcal{O}}(x) \phi_{3}\left(x_{3}\right) \phi_{4}\left(x_{4}\right)\rangle=g_{\mathcal{O}}\left(x_{i}\right)+\text {shadow block}.$The shadow block has different behaviour near $x_{12} \rightarrow 0$. In a 2D CFT, it not not possible to have both an operator and its shadow being local. And this gives rise to issues in CCFT. ## Celestial shadow transforms Following notations in [[2021#Pasterski, Puhm, Trevisani (May, a)]], a shadow transform does the following$\widetilde{\Phi}_{\Delta, J}^s=\widetilde{\Phi_{2-\Delta,-J}^s},$where$\tilde{\Phi}_{\Delta,+s}^s=(-1)^s\left(-X^2\right)^{\Delta-1} m_{\mu_1} \ldots m_{\mu_s} \varphi^{\Delta}.$ An important object in CCFT is the shadow of the $\Delta=0$ operator associated to the graviton: this is the stress tensor of CCFT with weight $\Delta=d-0=2$. ## Refs - CFT - Ferrara, Gatto, Grillo and Parisi - reviewed in [[2012#Simmons-Duffin]] - CCFT - studied in [[2017#Pasterski, Shao]] - scattering amplitudes in shadow basis can have advantages - [[ChangCuiMaShuZou2022]][](https://arxiv.org/abs/2210.04725) - modified prescription - [[2022#Banerjee, Pasterski]] - massive particles and the massless limit - [[2023#Furugori, Ogawa, Sugishita, Waki]]