Spectral flow in instanton computations and the β functions

We discuss various differences in the instanton-based calculations of the $\ensuremath{\beta}$ functions in theories such as Yang-Mills and $\mathbb{CP}(N\ensuremath{-}1)$ on one hand, and $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ theory with Symanzik's sign-reversed prescription for the coupling constant $\ensuremath{\lambda}$ on the other hand. Although the aforementioned theories are asymptotically free, in the first two theories, instantons are topological, whereas the Fubini-Lipatov instanton in the third theory is topologically trivial. The spectral structure in the background of the Fubini-Lipatov instanton can be continuously deformed into that in the flat background, establishing a one-to-one correspondence between the two spectra. However, when considering topologically nontrivial backgrounds for Yang-Mills and $\mathbb{CP}(N\ensuremath{-}1)$ theories, the spectrum undergoes restructuring. In these cases, a mismatch between the spectra around the instanton and the trivial vacuum occurs.