Attracting the Electroweak Scale to a Tachyonic Trap

We propose a new mechanism to dynamically select the electroweak scale during inflation. An axion-like field $\phi$ that couples quadratically to the Higgs with a large initial velocity towards a critical point $\phi_c$ where the Higgs becomes massless. When $\phi$ crosses this point, it enters a region where the Higgs mass is tachyonic and this results into an explosive production of Higgs particles. Consequently, a back-reaction potential is generated and the field $\phi$ is attracted back to $\phi_c$. After a series of oscillations around this point it is eventually trapped in its vicinity due to the periodic term of the potential. The model avoids transplanckian field excursions, requires very few e-folds of inflation and it is compatible with inflation scales up to $10^5~\rm GeV$. The mass of $\phi$ lies in the range of hundreds of GeV to a few TeV and it can be potentially probed in future colliders.