  We discuss various differences in the instanton-based calculations of the
$\beta$ functions in theories such as Yang-Mills and $\mathbb{CP}(N\!-\!1)$ on
one hand, and $\lambda\phi^4$ theory with Symanzik's sign-reversed prescription
for the coupling constant $\lambda$ 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 $\mathbb{CP}(N\!-\!1)$ theories, the spectrum undergoes restructuring. In
these cases, a mismatch between the spectra around the instanton and the
trivial vacuum occurs.
