Viable range of the mass scale of the standard model

Abstract

In theories in which different regions of the universe can have different values of certain physical parameters, we would naturally find ourselves in a region where they take values favorable for life. We explore the range of such viable values of the mass parameter in the Higgs potential, ${\ensuremath{\mu}}^{2}$. For ${\ensuremath{\mu}}^{2}<0$, 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 ${\ensuremath{\mu}}^{2}>0$, baryon stability requires that $|\ensuremath{\mu}|\ensuremath{\ll}{M}_{P}$, the Planck mass. Smaller values of $|{\ensuremath{\mu}}^{2}|$ may or may not be allowed depending on issues of element synthesis and stellar evolution. We conclude that the observed value of ${\ensuremath{\mu}}^{2}$ appears reasonably typical of the viable range, and a multiple-domain scenario may provide a plausible explanation for the closeness of the QCD scale and the weak scale.