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 $f$ well below the Planck scale. In the limit $N\gg 1$, the relaxion $ϕ$ 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 $Δϕ=2πf_{\rm eff} \sim e^{ξN} f$ for $ξ={\cal O}(1)$, both the scalar potential driving the evolution of $ϕ$ during the inflationary epoch and the $ϕ$-dependent Higgs boson mass vary with an exponentially large periodicity of ${\cal O}(f_{\rm eff})$, while the back reaction potential stabilizing the relaxion has a periodicity of ${\cal O}( f)$. 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 $f\sim \sqrt{m_{\rm SUSY}M_*}$, where the soft SUSY breaking scalar mass $m_{\rm SUSY}$ can be well above the weak scale, and the fundamental scale $M_*$ can be identified as the Planck scale or the GUT scale.

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