Naturalness and Fisher Information

TL;DR

This paper introduces a new information-theoretic measure of fine-tuning in physical theories, based on the Fisher information metric, providing a geometric interpretation and generalizing existing criteria.

Contribution

It proposes a Fisher information-based fine-tuning matrix that extends the Barbieri--Giudice criterion to multiple correlated parameters with a geometric perspective.

Findings

The measure aligns with physical intuition in various models.

It generalizes traditional fine-tuning criteria to complex parameter spaces.

The approach offers a geometric interpretation of sensitivity in theories.

Abstract

Fine-tuning and naturalness, the sensitivity of low-energy observables to small changes in the fundamental parameters of a theory, are cornerstones of physics beyond the Standard Model. We propose a new measure of fine-tuning based on information theory. To each point in parameter space we associate a probability distribution over observables. Divergence measures encode the sensitivity of observables to model parameters and determine a Riemannian metric on parameter space. By Chentsov's theorem, the physically motivated metric is the Fisher information metric, up to scaling. We propose a rescaled fine-tuning matrix $\mathcal{F}_{ij}$ derived from the Fisher information matrix, whose non-zero eigenvalues serve as our measure of fine-tuning. When the number of observables exceeds the number of parameters, $\mathcal{F}_{ij}$ admits a natural geometric interpretation as the pullback of the Euclidean metric from observable space to the submanifold of admissible predictions, with large eigenvalues corresponding to highly stretched directions and indicative of fine-tuning. Our measure reproduces the familiar Barbieri--Giudice criterion as a special case, while generalising it to multiple correlated parameters. We illustrate its behaviour on dimensional transmutation, the Wilson--Fisher fixed point, a simple model of the hierarchy problem, and the electron Yukawa coupling, finding agreement with physical intuition in each case.