Relations between basis sets of fields in the renormalization procedure

TL;DR

The paper discusses the conflicting approaches to renormalizing mixing angles in field theories, advocating for a basis-independent method where bare and renormalized angles are identified to ensure consistency and gauge-independence.

Contribution

It proposes a basis-independent renormalization scheme by equating bare and renormalized mixing angles, resolving issues of gauge dependence and singularities.

Findings

Basis-independent approach ensures gauge invariance.

Identifying bare and renormalized angles simplifies renormalization.

Avoids basis-dependent inconsistencies and singularities.

Abstract

It seems that the literature suggests to go in two opposing directions simultaneously. On the one hand, many papers construct basis-independent quantities, since exactly these quantities appear in the expressions for observables. This means that the mixing angles such as $\tan \beta$ in the Two Higgs Doublet Model must drop out when calculating anything physical. On the other hand, there are many attempts to renormalize such mixing angles -- this is in the opposite direction to basis-independence. This basis-dependent approach seems to bring gauge-dependence and singular behaviour, both of which are required to be absent in mixing renormalization. Most importantly, mixing angle counterterms single out a preferred basis and further basis rotations lead to inconsistencies. In contrast, we argue that the bare mixing angles should be identified with the renormalized ones -- this is the basis-independent approach -- such that all the mixing renormalization requirements are fulfilled in a trivial and consistent manner.