The properties of the Higgs potential are determined by three parameters: the mass term, the quartic self-coupling, and a constant term. Remarkably, all three of these parameters seem subject to a significant amount of fine-tuning, relating to the hierarchy problem, the metastability of the electroweak vacuum, and the cosmological constant problem, respectively. A curious feature of these tunings is that they can be understood as their corresponding parameters being close to critical values marking quantum phase transitions. While such behavior would be surprising from a conventional particle physics perspective, it is a common feature of dynamical systems. This has motivated the conjecture that the values of the Higgs' parameters are the result of some dynamical mechanism.

This idea naturally motivates the construction of explicit mechanisms dynamically choosing sets of Higgs parameters. In these notes, I discuss a complementary approach. Taking seriously the possibility that such a mechanism could exist, it is plausible to assume that it also influences Beyond-Standard-Model physics. This suggests considering near-critical combinations of parameters in any model of interest and investigating their physics more generally, in particular independent of a concrete mechanism responsible for their near-criticality.

In these notes, I first explain what it means for the parameters of the Higgs potential to be near-critical. This includes the discussion of the recently discovered "metastability bound" on the Higgs mass, which can be understood through a critical point.

I then review two concrete examples of mechanisms in which the parameters of the Higgs potential are dynamically driven towards critical values. While interesting on their own, these mechanisms also serve as a proof on concept for the feasibility of the assumptions at the foundation of these notes. Using a simple example for concreteness, the final part of these notes then explicitly demonstrates how to approach a given model in the light of the near-criticality conjecture.