Diagram dependance of the $\mathbb{C}\mathbb{P}(1)$ beta function

We revisit the computation of the beta function in the two-dimensional $\mathbb{C}\mathbb{P}(1)$ sigma model. We show that in different schemes, different diagrams are responsible for the running, such as momentum-independent tadpoles or even UV-finite bubble diagrams. We also comment on the relation between the beta functions and the energy dependence of scattering amplitudes.