Realizing the relaxion from multiple axions and its UV completion with high scale supersymmetry

We discuss a scheme to implement the relaxion solution to the hierarchy problem with multiple axions, and present a UV-completed model realizing the scheme. All of the N axions in our model are periodic with a similar decay constant integral well below the Planck scale. In the limit N >> 1, the relaxion phi corresponds to an exponentially long multi-helical flat direction which is shaped by a series of mass mixing between nearby axions in the compact field space of N axions. With the length of flat direction given by Delta phi = 2 pi f(eff) similar to e(xi N) f for xi = O(1), both the scalar potential driving the evolution of f during the inflationary epoch and the phi-dependent Higgs boson mass vary with an exponentially large periodicity of O(f(eff)), while the back reaction potential stabilizing the relaxion has a periodicity of O(f). A natural UV completion of our scheme can be found in high scale or (mini) split supersymmetry (SUSY) scenario with the axion scales generated by SUSY breaking as f similar to root m(SUSY)M(*), where the soft SUSY breaking scalar mass m(SUSY) can be well above the weak scale, and the fundamental scale M-* can be identified as the Planck scale or the GUT scale.