Bananas are Unparticles: Differential Equations and Cosmological Bootstrap

We establish an exact correspondence between tree-level cosmological correlators with unparticle exchange (at integer scaling dimensions) and banana diagrams of conformally coupled scalars. This duality enables us to systematically solve the governing differential equations through the application of shift relations and de Sitter bootstrap techniques. Furthermore, we adapt a dimensional regularization scheme to cosmological correlators, demonstrating how renormalization conditions uniquely fix the regularization prescription. Our results provide new insights into the analytic structure of higher-order loop corrections to inflationary correlation functions.