Flat limit of massless scalar scattering in $\mathrm{AdS}_2$

We explore the flat limit of massless scalar scattering in $\mathrm{AdS}_2$. We derive the $1 \to 1$ $\mathcal{S}$-matrix from the CFT $2$-point function. We show a key property of the $2 \to 2$ $\mathcal{S}$-matrix in $2d$, where the contact interaction in the flat limit gives momentum conserving delta function. We show the factorization of the $n \to n$ $\mathcal{S}$-matrix for integrable models in the flat limit, focusing on contact interactions. We calculate the $\mathcal{S}$-matrix by linking the CFT operator on the AdS boundary to the scattering state in flat-space. We use bulk operator reconstruction to study massless scalar scattering in the flat limit and solve the Klein-Gordon equation in global $\mathrm{AdS}_2$ for the massless scalar field. The solution is simple, involving a pure phase in global time and a sinusoidal function in the radial coordinate. This simplicity also extends to the smearing function, allowing us to map the scattering state to the CFT operator while taking AdS corrections into account.