The anthropic principle and the mass scale of the Standard Model

TL;DR

The paper applies the anthropic principle to the Higgs mass parameter $μ^2$, treating different regions of the universe as having different $μ^2$ values while holding other SM parameters fixed. It derives how the Higgs vev $v$ and ensuing particle masses vary with $μ^2$ in both negative and positive regimes, then analyzes the implications for nuclear stability, nucleosynthesis, and stellar evolution. For $μ^2<0$, life-friendly chemistry requires $v$ to stay within a modest range around $v_0$; for $μ^2>0$, life imposes an extreme suppression of $|μ^2|$ compared to the Planck scale, making anthropically allowed regions very narrow. The study argues that the observed $μ^2$ is reasonably typical within the anthropically allowed set, offering a plausible explanation for the proximity of the QCD and weak scales and suggesting environmental selection as a potential resolution to the hierarchy problem when multiple domains exist.

Abstract

In theories in which different regions of the universe can have different values of the the physical parameters, we would naturally find ourselves in a region which has parameters favorable for life. We explore the range of anthropically allowed values of the mass parameter in the Higgs potential, $μ^2$. For $μ^2<0$, the requirement that complex elements be formed suggests that the Higgs vacuum expectation value $v$ must have a magnitude less than 5 times its observed value. For $μ^2>0$, baryon stability requires that $|μ|<<M_P$, the Planck Mass. Smaller values of $|μ^2|$ may or may not be allowed depending on issues of element synthesis and stellar evolution. We conclude that the observed value of $μ^2$ is reasonably typical of the anthropically allowed range, and that anthropic arguments provide a plausible explanation for the closeness of the QCD scale and the weak scale.