Renormalization and running in the 2D $CP(1)$ model

TL;DR

This paper computes the scattering amplitude in the 2D $CP(1)$ model in a scheme-independent manner, revealing how the physical running coupling and beta function emerge and discussing implications for asymptotic freedom.

Contribution

It provides a regularization scheme independent calculation of the scattering amplitude and clarifies the scheme dependence of the beta function in the 2D $CP(1)$ model.

Findings

Reproduces asymptotic freedom in the $CP(1)$ model.

Shows the pathway to the beta function varies with regularization scheme.

Comments on evasion of Landau's argument against asymptotic freedom.

Abstract

We calculate the scattering amplitude in the two dimensional $CP(1)$ model in a regularization scheme independent way. When using cutoff regularization, a new Feynman rule from the path integral measure is required if one is to preserve the symmetry. The physical running of the coupling with renormalization scale arises from a UV finite Feynman integral in all schemes. We reproduce the usual result with asymptotic freedom, but the pathway to obtaining the beta function can be different in different schemes. We also comment on the way that this model evades the classic argument by Landau against asymptotic freedom in non-gauge theories.