Celestial amplitude for 2d theory

We explore celestial amplitude corresponding to $2d$ bulk $\mathcal{S}$-matrix. We consider scalar particles with identical mass and show that the celestial amplitude becomes the fourier transform of the $2d$ $\mathcal{S}$-matrix written in the rapidity variable. We translate the crossing and unitarity conditions into the conditions on the celestial amplitude. For the $2d$ Sinh-Gordon model, we calculate the celestial amplitude perturbatively in coupling constant and check that the crossing and unitarity conditions are satisfied for the celestial amplitude. Imposing the crossing and unitarity conditions to the celestial amplitude, we want to find amplitudes to the higher order in perturbation theory from the lower order i.e., to provide a "\textit{proof of principle}" to show we can apply the bootstrap idea to the celestial amplitude. We find that imposing the crossing and unitarity conditions is not enough for bootstrapping celestial amplitude, there is an extra term which can't be fixed by the crossing and unitarity conditions. We also study the gravitational dressing condition in $2d$ QFT for massless particles in celestial space and see that for the gravitationally dressed celestial amplitude, the poles on the right half-plane get erased for several ansatzes.