The hierarchy problem and fine-tuning in a decoupling approach to multi-scale effective potentials

TL;DR

This paper investigates the decoupling method for handling multiple energy scales in quantum field theories, finding it acceptable but insufficient to resolve the hierarchy and fine-tuning problems, and proposes a new parameter fixing prescription.

Contribution

It analyzes the effectiveness of the decoupling approach in multi-scale effective potentials and introduces a new prescription to mitigate fine-tuning issues.

Findings

Decoupling method yields a convergent effective potential.

Decoupling does not fully solve the hierarchy fine-tuning problem.

Alternative decoupling implementations lead to similar fine-tuning conclusions.

Abstract

In many realizations of beyond the Standard Model theories, new massive particles are introduced, leading to a multi-scale system with widely separated energy scales. In this setting the Coleman-Weinberg effective potential, which describes the vacuum of the theory at the quantum level, has to be supplemented with a prescription to handle the hierarchy in mass scales. In any quantum field theory involving scalar fields and multiple, highly differing mass scales, it is in general not possible to choose a single renormalization scale that will remove all the large logarithms in the effective potential. In this paper, we focus on the so-called decoupling method, which freezes the effects of heavy particles on the renormalization group running of the light degrees of freedom at low energies. We study this for a simple two-scalar theory and find that, while the decoupling method leads to an acceptable and convergent effective potential, the method does not solve the fine-tuning problem that is inherent to the hierarchy problem of multi-scale theories. We also consider an alternative implementation of the decoupling approach, which gives different results for the shape of the potential, but still leads to similar conclusions on the amount of fine-tuning in the model. We suggest a way to avoid running into this fine-tuning problem by adopting a prescription on how to fix parameters in such decoupling approaches.