Spectral Flow in Instanton Computations and the \boldmath{$\b$} functions

TL;DR

This paper compares instanton effects on beta function calculations across different quantum field theories, highlighting spectral differences between topologically trivial and nontrivial backgrounds and their impact on renormalization.

Contribution

It reveals how spectral structures differ in topologically trivial versus nontrivial instanton backgrounds, affecting beta function computations in various theories.

Findings

Spectral structures can be deformed into flat backgrounds in trivial cases.

Topologically nontrivial backgrounds cause spectrum restructuring.

Mismatch occurs between spectra around instantons and trivial vacua in certain theories.

Abstract

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.