Challenges to Understanding Celestial Holography from the Bottom Up

In the bottom-up approach to celestial holography, it is tempting to define celestial amplitudes by transforming momentum-space amplitudes order by order in perturbation theory. We test this prescription in the exactly solvable two-dimensional Sinh-Gordon model. Because the full non-perturbative S-matrix is known, we can compare two operations directly: first transform and then expand, or first expand and then transform. They do not agree, already at leading nontrivial order in the coupling. More broadly, this suggests that naive term-by-term celestial transforms should not be assumed valid in generic quantum field theories with asymptotic weak-coupling expansions. This has an immediate consequence for proposed CCFT duals: if one tries to test them by taking celestial transforms of perturbative bulk amplitudes term-by-term, a mismatch need not falsify the proposal. This makes bottom-up tests of celestial dualities far more subtle than one might have expected.