On the Renormalization Group flow of distributions

TL;DR

This paper introduces an evolution equation for distributions on the space of couplings in Renormalization Group flows, enabling analysis of error propagation and emergent structures in coupling sets, with applications to the Standard Model.

Contribution

It develops a novel distribution-level Renormalization Group equation and demonstrates its utility in error propagation and understanding coupling structure emergence.

Findings

Error distribution properties cannot be propagated from individual couplings.

Most probable metastability scale in the Higgs sector is calculated.

Emergence of structure in couplings, with observed values near predicted maxima.

Abstract

Renormalization Group flows relate the values of couplings at different scales. Here, we go beyond the Renormalization Group flow of individual trajectories and derive an evolution equation for a distribution on the space of couplings. This shift in perspective can provide new insights, even in theories for which the Renormalization Group flow of individual couplings is well understood. As a first application, we propagate errors under the Renormalization Group flow. Characteristic properties of an error distribution, such as its maximum or highest density region, cannot be propagated at the level of individual couplings, but require our evolution equation for the distribution on the space of couplings. We demonstrate this by calculating the most probable value for the metastability scale in the Higgs sector of the Standard Model. Our second application is the emergence of structure in sets of couplings. We discover that infrared-attractive fixed points do not necessarily attract the maximum of the distribution when the Renormalization Group is evolved over a finite range of scales. Instead, sharply peaked maxima can build up at coupling values that cannot be inferred from individual trajectories. We demonstrate this emergence of structure for the Standard Model, starting from a broad distribution at the Planck scale. The Renormalization Group flow favors the phenomenological ordering of third-generation Yukawa couplings and, strikingly, we find that the most probable values for the Abelian hypercharge and Higgs quartic coupling lie close to their observed values.