Large hierarchies from attractor vacua

We discuss a mechanism through which the multivacua theories, such as string theory, could solve the hierarchy problem, without any UV-regulating physics at low energies. Because of symmetry the number density of vacua with a certain hierarchically small Higgs mass diverges, and is an attractor on the vacuum landscape. The hierarchy problem is solved in two steps. It is first promoted to a problem of the superselection rule among the infinite number of vacua (analogous to theta vacua in QCD) that are finely scanned by the Higgs mass. This rule is lifted by heavy branes, which effectively convert the Higgs mass into a dynamical variable. The key point is that a discrete "brane-charge-conjugation" symmetry guarantees that the fineness of the vacuum scanning is set by the Higgs mass itself. On a resulting landscape, in all but a measure-zero set of vacua, the Higgs mass has a common hierarchically small value. In minimal models this value is controlled by the QCD scale and is of the right magnitude. Although in each particular vacuum there is no visible UV-regulating low energy physics, the realistic models are predictive. For example, we show that in the minimal case the "charge-conjugation" symmetry is automatically a family symmetry, and imposes severe restrictions on quark Yukawa matrices.