Effect of Scheme Transformations on a Beta Function with Vanishing One-Loop Term

TL;DR

This paper proves that in theories where the one-loop term of the beta function vanishes, it is generally impossible to remove higher-loop scheme-dependent terms through scheme transformations.

Contribution

It demonstrates that the common assumption about removing higher-loop terms via scheme transformations does not hold when the one-loop term of the beta function is zero.

Findings

Scheme transformations cannot generally eliminate higher-loop terms when the one-loop beta function vanishes.

The paper clarifies limitations on scheme dependence in quantum field theories.

It provides a rigorous proof of the impossibility in this specific scenario.

Abstract

It is commonly stated that because terms in the beta function of a theory at the level of $\ell \ge 3$ loops and higher are scheme-dependent, it is possible to define scheme transformations that can be used to remove these terms, at least in the vicinity of zero coupling. We prove that this is not, in general, possible in the situation where a beta function is not identically zero but has a vanishing one-loop term.