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We consider 2-2 scattering in four spacetime dimensions in Celestialvariables. Using the crossing symmetric dispersion relation (CSDR), we recastthe Celestial amplitudes in terms of crossing symmetric partial waves. Thesepartial waves have spurious singularities in the complex Celestial variable,which need to be removed in local theories. The locality constraints (nullconstraints) admit closed form expressions, which lead to novel bounds onpartial wave moments. These bounds allow us to quantify the degree of low spindominance(LSD) for scalar theories. We study a new kind of positivity thatseems to be present in a wide class of theories. We prove that this positivityarises only in theories with a spin-0 dominance. The crossing symmetric partialwaves with spurious singularities removed, dubbed as Feynman blocks, haveremarkable properties in the Celestial variable, namely typically realness, inthe sense of Geometric Function Theory (GFT). Using GFT techniques we derivenon-projective bounds on Wilson coefficients in terms of partial wave moments.

Celestial insights into the S-matrix bootstrap