Integral Transformations for Conformally Invariant Celestial Amplitudes

We propose an integral transformation for celestial gluon amplitudes that maps the celestial coordinates $(z_i,{\bar{z}}_i)$ to a new set of complex variables $(s_i,{\bar{s}}_i)$, inspired by the structure of closed string scattering amplitudes. A consistent inverse transformation is constructed by regulating a divergence associated with translational redundancy and absorbing it into an overall normalization. Applying this transformation to celestial MHV amplitudes, we derive constraints on $(s_i,{\bar{s}}_i)$ for three-, four-, and general $n$-point amplitudes, and show that these conditions are necessary for invariance under global conformal transformations.